key: cord-0885192-rqkj65wc authors: Huang, Sen-Zhong title: A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R 0 date: 2008-09-30 journal: Mathematical Biosciences DOI: 10.1016/j.mbs.2008.06.005 sha: 650cae3c285240bc19d3592b477e5138e27e74dd doc_id: 885192 cord_uid: rqkj65wc Abstract We present a novel SEIR (susceptible–exposure–infective–recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R 0 in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R 0 for most known diseases, that are calculated by our formulas, coincide very well with the values of R 0 estimated by the usual method of fitting the models to observed data. R 0 , the so-called basic reproductive (reproduction) number of a communicable disease, defined as the expected number of secondary infectious cases generated by one typical primary case in an entirely susceptible and sufficiently large population (Anderson and May [1] ), is a key for controlling the epidemic spread. For instance, the general guideline for mass vaccination is that the herd immunity of a population against the disease must be above some critical immunization threshold q c that is a function of R 0 and some other factors like vaccine efficacy. A very simple but illustrative choice of such a critical immunization threshold is q c ¼ 1 À 1=R 0 , used in the traditional theory of epidemiology [1] under the unrealistic assumption (Benenson [3] , CDC [6] ) that the vaccine is perfectly effective. Although epidemiologists and theoreticians ( [3, 6] , Halloran et al. [24, 25] , Longini et al. [41] ) have been aware of this for a long time, no confident relation between the critical immunization threshold q c and the vaccine efficacy was drawn. It gives rise to the need of establishing more realistic epidemic models for the mass vaccination theory. A further motivation for establishing new epidemic models is the estimation of the basic reproductive number R 0 . As shown by the very recent case of the 2002-2003 epidemic SARS (Donnelly et al. [15] , Lipsitch et al. [36] , Riley et al. [53] , Bauch et al. [2] ), an accurate estimation of R 0 is essential for a successful control of newly emerging diseases by the public heath control measures. A large number of compartmental (both deterministic and stochastic) epidemic models (Anderson and May [1] , Hethcote [28] , Diekmann and Heesterbeek [14] ) have been proposed for this purpose (see Heffernan et al. [27] for a comprehensive review). But most of these models rely on the traditionally used assumption that both transmission processes of latency and recovery states are exponentially distributed. This assumption is biologically unrealistic, because it corresponds to assuming that such a transmission process is memoryless in the sense that the chance of change from a state 'A' to another state 'B' in a given time interval is independent of the time since the entrance into the previous state 'A'. More realistic distributions are, for example, the gamma or the log-normal distributions (Sartwell [54] ). However, the inclusion of gamma distributions, as shown by Lloyd [38, 39] and Wearing et al. [59] , could change drastically the patterns and dynamics of the epidemic models, as well as the estimate of the basic reproductive number R 0 . In this work, we will present (in Section 2) a new SEIR (susceptible-exposure-infective-recovered) epidemic model, which covers the classical SEIR model (Hethcote [28] ) as a special case and has the following advantage than most of the known mathematical epidemic models: our new SEIR model is suitable for the aim of incorporating epidemiological factors like the vaccine efficacy and the varying of transmission patterns into the model settings. Hence, it is reasonable to expect that our new SEIR model fits better to reality. Below we highlight the following three results obtained by our new SEIR model: a novel theory for the eradication of diseases by mass vaccination, a refined theory for the control of newly emerging diseases by the public heath control measures of isolating infectious individuals combined with contact tracing and quarantining suspected individuals, and four formulas for calculating the basic reproductive number R 0 . For simplicity, we assume as usual that the population under discussion is homogeneously mixed. Let VE S be the vaccine efficacy against susceptibility, and VE I the vaccine efficacy against infectiousness (Halloran et al. [24, 25] , Longini et al. [41] ). Moreover, we assume that the vaccination is so effective that the following inequality is satisfied: Let f c and 0 < f c < 1, be the solution of the following quadratic equation Then f c is a critical immunization threshold; that is, if the coverage f of the herd immunity in a homogeneously mixed population exceeds f c , then it can be expected that the disease will be eradicated by the mass vaccination, no matter how complicated the transmission patterns of the disease could be. In the absence of effective vaccine or treatment, the symptombased public control measures follow the following baselines (Fracer et al. [21] ): (i) effective isolation of symptomatic individuals and (ii) tracing and quarantining the contacts of symptomatic cases. Let h be the proportion of asymptomatic infectives whose transmission occurs prior to symptoms. Let e I be the efficacy of isolation/quarantine and e T be the efficacy of contact tracing. (We assume that the duration of isolation/quarantine is long enough so that the infection is cleared and no longer infectious.) The value e I could equally be thought of as the probability that an infected individual will be isolated immediately after he/she becomes symptomatic, and the value e T could equally be thought of as the probability that an asymptomatic infective will be first detected by contact tracing and then isolated. Let r D ¼ dr (with d 2 ð0; 1) be the mean time of delay between isolation and the starting point of infection for those asymptomatic infectives who are detected by contact tracing. Here r is the mean infectious period for those infectives who are neither isolated nor detected by contact tracing. If the control measures are such that the following inequality ð1 À hÞe I þ hð1 À dÞe T > 1 À 1 R 0 = ð1:2Þ is satisfied, then our theory ensures that the outbreak will be brought under control, no matter how complicated the transmission patterns of the disease could be. (We remark that formulas similar to (1.2) have been obtained previously by Müller et al. [45] and Fracer et al. [21] using different methods.) This result indicates in particular that a newly emerging disease could be contained by isolation combined with contact tracing and quarantine alone, provided that these control measures are sufficiently effective when compared to the basic reproductive number R 0 of the disease. The newest example for this is the control of the 2002-2003 epidemic SARS (see e.g., Donnelly et al. [15] , Lipsitch et al. [36] ). The above theory also confirms the usually employed strategy for controlling an epidemic: containing the infection sources (infectives), controlling the epidemic. As indicated above, a core for a successful eradication or control of diseases is the estimation of the basic reproductive number R 0 . In our present work, we follow the usual guideline [27, 28] used for both the classical SIR model of Kermack and McKendrick [33] and the classical SEIR model [28] by defining R 0 as the product of the intrinsic mean infection rate (denoted by b) and the mean infectious period (denoted by r), i.e., R 0 :¼ b  r. The usual way to estimate R 0 is to estimate both b and r by fitting observed data from known outbreaks into corresponding models; see, e.g., [1, 7, 9, 11, 13, 16, 18, 36, 43] . As the disease spreads, the mean infectious period r of a disease can be directly estimated, by using a few data, with or without models. However, estimation of the intrinsic mean infection rate b depends strongly on the model and the fruitfulness of data used and is in general very subtle; cf. [1, 7, 9, 11, 13, 16, 18, 36, 43] . Our present method for estimating b is purely theoretical and relies on our new SEIR model. Based on some sophisticated probabilistic arguments (cf. Section 4), our method yields that the intrinsic mean infection rate b can be determined in terms of the three main factors of a disease: the mean latent period (denoted by s), the mean infectious period, and the transmission patterns of the latency and recovery processes. The typical transmission patterns of latency and recovery will be divided into four types: light (Type I), moderate (Type II), severe (Type III), extremely severe (Type IV). Our method yields the following four formulas (1.4)-(1.7) for calculating the basic reproductive numbers R I-IV 0 ðz) associated with the four types of transmission patterns, in terms of the ratio z :¼ mean infectious period : mean latent period: ð1:3Þ R I 0 ðz), the minimal reproductive number associated with transmission patterns of Type I (with the slowest latency process and the fastest recovery process). R IV 0 ðz), the largest reproductive number associated with transmission patterns of Type IV (with the fastest latency process and the extremely slow recovery process). All four functions R I-IV 0 ðz) are strictly increasing functions of z ¼ r=s, the ratio of the mean infectious period r to the mean latent period s, and they satisfy the following inequalities: These four formulas will be used as follows: according to the type of its transmission patterns, we assign one of the main categories 'mild' and 'severe' to the disease and then estimate its basic reproductive number. We assume that diseases of the category 'mild' have transmission patterns which are between the light type (Type I) and the moderate type (Type II), and the superspreading events (SSEs) generated by a disease of the category 'mild' correspond to the case that the transmission patterns are of the severe type (Type III). Assuming that a disease of the category 'mild' has a mean latent period ranging from k to l units of time (e.g., days), and a mean infectious period ranging from m to n units of time, then the basic reproductive numbers R 0 and the reproductive number R SSEs 0 generated by SSEs for this 'mild' disease are estimated by We assume that diseases of the category 'severe' have transmission patterns which are between the severe type (Type III) and the extremely severe type (Type IV), and the superspreading events (SSEs) generated by a disease of the category 'severe' correspond to the case that the transmission patterns are of the extremely severe type (Type IV). Assuming that a disease of the category 'severe' has a mean latent period ranging from k to l units of time (e.g., days), and a mean infectious period ranging from m to n units of time, then the basic reproductive numbers R 0 and the reproductive number R SSEs 0 generated by SSEs for this 'severe' disease are estimated by In applying the above method to most known diseases, our classification is as follows: The category 'mild' includes the mild diseases like Hepatitis B, Polio, Scarlet fever and HIV (Aids) which have a long infectious period (when compared to their mean latent periods) but have an observed small basic reproductive number R 0 . The newly (re)-emerging diseases SARS, Ebola, AHC, FMD, influenza, influenza pandemic and avian influenza are put also into the category 'mild', since they have an observed relatively small basic reproductive number R 0 . The category 'severe' includes the rest severe and extremely severe diseases like Chickenpox, Mumps, Rubella, Measles, etc. which have an observed large basic reproductive number R 0 . The newly emerging infections Acute HIV-1 and Acute SIV belong also to this category. Although our method for estimating R 0 is purely theoretical, the resulting estimates of R 0 (given in Table 2 of Section 3 and Table 4 of Section 4) fits very well to the published estimates of R 0 (see, e.g., [1, 7, 9, 11, 13, 16, 18, 36, 43] ) using data from known outbreaks and using other models. This indicates that the four functions R I-IV 0 ðz) given above can be used as empirical formulas for estimating R 0 for unknown diseases -a point that is especially valuable for the control of a newly emerging disease, because the three main factors (the mean latent period, the mean infectious period and the types of the transmission patterns), which are needed for estimating the values of R 0 for that disease, are observable as the disease spreads. The present study can be extended in several directions. These include explicit incorporation of the heterogeneity of the population (cf. Anderson and May [1] , Hethcote [28] ), the spatial and temporal varying of the transmission patterns, the control measures as well as the population growth. Some examples are presented in Appendix D. We point out that this framework can also be applied to the eradication and control of network viruses. Further content of this article is organized as follows: in Section 2 we present the new SEIR model. In Section 3 we establish the aforementioned novel theory for the eradication and control of diseases. In Section 4 we describe in details our method for estimating R 0 and present the numerical results. In the appendix section we give the detailed mathematical proofs of the results mentioned in the previous sections. We consider the situation that an epidemic with a mean latent period s and a mean infectious period r emerges and spreads in a homogeneously mixed population for which time-dependent control measures (including vaccination, isolation and contact tracing, etc.) have been implemented during the epidemic. The new SEIR model and its derivation: The dynamics of such an epidemic can be described by the following novel susceptibleexposure-infective-recovered (SEIR) model S ! E ! I ! R: is called the initial (seed) epidemic size. In the sequel, we assume that each of the four functions (SðtÞ; FðtÞ; gðtÞ; RðtÞ) is an absolutely continuous function with an almost everywhere existing and bounded derivative. We note that the derivatives F 0 ðt) and g 0 ðt) can be interpreted as follows: The value of F 0 ðt) (resp., g 0 ðt)) can be thought of as the number of new exposures (resp., new infectives) at time t, i.e. the number of new individuals that are infected (resp., have become being infectious) at time t. Similarly, the value of R 0 ðt) can be thought of as the number of newly recovered infectives at time t. In the practical use, the value of R 0 ðt) can be thought of as the number of newly detected and reported cases at time t. Before further explaining things involved in the model Eq. (2.1), we point out that the new SEIR model Eq. (2.1) in terms of the four functions (SðtÞ; FðtÞ; gðtÞ; RðtÞ) can be reformulated (though with relatively complicated representations) in terms of the four functions (SðtÞ; EðtÞ; IðtÞ; RðtÞ) that come from the usual formalism of the classical SEIR model [1, 14, 28] . To see this, we recall that in the usual formalism of the classical SEIR model one considers four possible subsequent states for individuals in the population with a constant size N: susceptibility (S), latency (E) (i.e., infected but not yet infectious), infectiousness (I) (i.e., infectious but not yet recovered), and recovery (R) (recovered or removed by isolation, death, etc.). Thus, if SðtÞ; EðtÞ; Iðt) and Rðt) are the number of individuals at time t in the corresponding (S)/(E)/(I)/(R) state, then we have SðtÞ þ EðtÞ þ IðtÞ þ RðtÞ ¼ N: ð2:2aÞ We note that in the formalism of model Eq. (2.1), both functions Sðt) and Rðt) keep their classical meaning. Moreover, the number Eðt) (resp., Iðt)) of exposures (resp., infectives) at time t can be calculated by The function rðt) is assumed to be bounded and piecewise continuous. At a time point t P 0, the value rðt) is the mean infection rate per unit time per active infective. The function W is an absolutely continuous (a.c., for short) cumulative distribution function (CDF, for short) on R þ ½0; 1) with expectation (mean) value s and zero initial derivative, i.e., The absolute continuity of W means that W has an almost everywhere existing derivative W 0 (called the probability density function of W) which is Lebesgue-integrable on R þ . The requirement W 0 ð0Þ ¼ 0 corresponds to the fact that no exposed individual will immediately become infectious after infection. For the function A we assume that for each fixed t P 0 the function Aðt; s) in s P 0 is an a.c. CDF on R þ ½0; 1) with a finite expectation The value Rðt) is called effective mean infectious period. In general, Rðt) depends on the time-varying control measures and is not greater than the intrinsic mean infectious period r : RðtÞ 6 r ðt P 0Þ: The functions Wðt) and Aðt; s) have the following interpretation. The value 1 À Wðt) is the probability that an exposed individual will remain in the exposure state after its emergence of t units of time. For each fixed t P 0, the value Aðt; s) is the probability that an infective individual that emerges at t will be recovered at the later time point t þ s, i.e., after an emergence period of s units of time. Hence, we call A also the (time-dependent) recovery cumulative distribution function. In this article, by a CDF (cumulative distribution function) P on R þ we mean a function that is monotone increasing, right-continuous for all t > 0 and that satisfies Pð0Þ P 0 and PðtÞ ! 1 as t ! 1. Moreover, the integral with respect to CDFs is interpreted as the Lebesgue-Stieltjes integral, as commonly done in the probability theory. To finish the model setting, we explain below how our integrodifferential model Eq. (2.1) is derived. First we do this for both Eqs. (2.1a) and (2.1b). Using (2.2b) we see that Eq. (2.1b) is purely a restatement of the classical balance Eq. (2.2a). We point out that there is a direct way to derive Eq. (2.1b). In fact, at t P 0, the difference N À Sðt) is equal to the number of individuals that have been first exposed to the disease and then infected within the time interval ½0; t. This yields the balance Eq. (2.1b) stating N À SðtÞ ¼ Fðt), since Fðt) is the cumulative number of exposures that emerge newly within the time interval [0, t]. Our derivation of Eq. (2.1a) just follows the usual formalism of the classical SEIR model [1, 28] by assuming that the decrease of susceptibles follows the law of mass action in the sense that S 0 ðtÞ ¼ ÀkðtÞSðtÞ; ð2:4aÞ where kðt) is the so-called 'force of infection' (Anderson and May [1, p. 63] ). That is, at time t the product kðtÞDt is the probability that a given susceptible host will become infected in the small time interval Dt. Since only those infectives that are active (i.e., infectious but yet not removed) will be able to transmit their infection, the force of infection kðt) can be calculated as follows: N  kðt) is equal to the product of rðt), the time-dependent mean infection rate per unit time per active infective, with the number Iðt) of infectives that are still active at time t, i.e., Both latency and recovery processes follow linear laws. We explain them in a more detailed way. (i) For Eq. (2.1c) we assume that the latent process E ! I is linear and determined by the transmission CDF W as follows: g 0 ðtÞ ¼ Fð0ÞW 0 ðtÞ þ Z t 0 F 0 ðt À sÞW 0 ðsÞ ds ðt P 0Þ: ð2:5Þ The term Fð0ÞW 0 ðt) in (2.5) represents the transmission of the initial exposures (amounting to Fð0)) to new infectives at time t, and the integral term in (2.5) represents the transmission of the exposures that have emerged within the time interval ½0; t to new infectives at t. Eq. (2.1c) is the result of integrating (2.5) and using the initial condition gð0Þ ¼ 0. The first equality in condition (2.3a) corresponds to the requirement that the mean latent period be s. (ii) Eq. (2.1d) is derived under the assumption that the transmission process I ! R for the infectives is linear and determined by the time-dependent CDFs Aðt; Á). A more illustrative explanation is as follows: at a fixed time point t > 0, the density of infectives that emerge s units of time before t and that will be removed at t is the product g 0 ðt À sÞ Â Aðt À s; s). Hence, the cumulative number Rðt) of removed infectives at time t is given by RðtÞ ¼ Z t 0 g 0 ðt À sÞAðt À s; sÞ ds; which is just Eq. (2.1d). We give some comments to the new SEIR model Eq. (2.1). As seen above, both Eqs. (2.1a and 2.1b) are derived under the usual formalism for the classical SEIR model [28, 1] . What are really new in our model are Eqs. (2.1c) and (2.1d) for describing the latency and recovery processes in terms of the general CDFs W and A. We point out that many possible choices for the CDFs W and A (e.g., multi-staged exponentially distributed CDFs) that are met in practical applications, will lead our original model Eq. (2.1) to systems of ordinary differential equations (odes). Some such examples are given in Appendix D. In particular, we find that such systems of odes contain the classical SEIR model as a special case (see Appendix E). With general choices of the CDFs W and A, the new SEIR model Eq. (2.1) is an integro-differential system. However, we can show, with some more manipulations (details given in Appendix A), that the original system Eq. (2.1) can be simplified to a single equation for the function F of the cumulative number of exposures combined with the relations for calculating the cumulative numbers g and R of infectives and removed infectives. Because the equation for F is a non-linear integral equation of the Volterra type, the usual fixed-point method (see Appendix A) is used to establish the wellposedness of the integral equation for F, and thus the well-posedness of Eq. (2.1). Particularly, it yields that for each initial epidemic size g 0 (0 < g 0 < N), the system Eq. (2.1) admits a unique, global and increasing solution F such that Fð0Þ ¼ g 0 and FðtÞ < N for all t P 0. R eff ðt), the effective reproductive number, and R 0 , the basic reproductive number. The number R eff ðt) given by R eff ðtÞ :¼ Z 1 0 rðt þ sÞð1 À Aðt; sÞÞ ds ðt P 0Þ ð 2:6Þ will be called the effective reproductive number at time t. The reason why we use this terminology is based on the following observation. We consider the case where prior immunity and control measures are absent. For this case we have that rðtÞ b; Aðt; sÞ ¼ A 0 ðsÞ ðs; t P 0Þ; ð2:7aÞ where b is the intrinsic mean infection rate and A 0 is the intrinsic recovery CDF. Correspondingly, the mean value is just the mean infectious period of the disease. Under the choices (2.7a) for r and A, the function R eff ðt) given by (2.6) is identical to a constant R 0 given by R 0 :¼ br: ð2:7cÞ R 0 is called the basic reproductive number. Our definition of R 0 is consistent with the general baseline (cf. Anderson and May [1] , Hethcote [28] ) where one defines the basic reproductive number as the expected number of secondary infectious cases generated by one typical primary case in an entirely susceptible and sufficiently large population. The effective reproductive numbers R eff ðt) and control of epidemic: the threshold theorem. The asymptotic behavior of solutions to Eq. (2.1) is as follows: a direct consequence of the well-posedness of Eq. (2.1) is that the epidemic course described by Eq. (2.1) will develop in such a way that RðtÞ 6 gðtÞ 6 FðtÞ < N ð2:9aÞ for all t P 0, and will be stopped eventually with a finite final size of g 1 infectives in such a way that F 0 ðtÞ þ FðtÞ À g 1 j jþgðtÞ À g 1 j jþRðtÞ À g 1 j j!0 as t ! 1; Note that the value of F 0 ðt) is the number of new exposures at time t. Hence, the statement F 0 ðtÞ ! 0 as t ! 1 in (2.9b) says that the epidemic course will be stopped eventually. (Clearly, one reason for the stopping of the epidemic course is the boundedness of the total population size.) The other statement in (2.9b) stating that the three functions Fðt), gðt) and Rðt) have the same limit as t ! 1 restates exactly the fact that all exposures will become infectious and will eventually be removed from the disease either by death or discharge. In the literature, Eq. (2.9c) is called the final size equation and is also satisfied by many other models [28, 27] . Historically, the final size equation was derived for the first time by Kermack and McKendrick [33] in 1927 for the classical SIR model. In a realistic situation, the control measures will be implemented at a certain time point t 0 P 0 after the disease has spread. The baseline for the implementation of control measures is to reduce the effective reproductive number R eff ðt) (given by (2.6)). It is expected that the epidemic will be brought under control if the effective reproductive number will be maintained below some level d < 1 after t 0 , a certain time point after the intervention. Our following result confirms the theoretical correctness of this expectation. The proof of Theorem 2.1 will be given in Appendix B. We remark that the so-called 'threshold property' of the basic reproductive number R 0 given by (2.7c) is interpreted as follows: In the absence of prior immunity and control measures we have that R eff ðtÞ R 0 . Hence, if R 0 < 1, then Theorem 2.1 yields that the final epidemic size g 1 is independent of the population size N, and is bounded by the product of the initial epidemic size Fð0Þ ¼ g 0 with the number 1=ð1 À R 0 ). Conversely, if R 0 P 1, then Theorem 2.1 yields that the final epidemic size g 1 will increase with the population size N. In general, the population size must be set to be very large (e.g., 10 million) so that the condition for applying the law of mass action will be justified. Therefore, in the second case (R 0 P 1), the final epidemic size will be very large and the epidemic is very severe. The result in Theorem 2.1 (i) gives the baseline for bringing an epidemic course under control: (a) reducing the mean infection rate rðt). (b) Shortening the effective mean infectious period Rðt). The goal is to reduce the controlled reproductive number R c (given by (2.11a)) in such a way that R c < 1. In case a control strategy achieves this, then the ratio g 1 =Fðt 0 ) of the final epidemic size to the epidemic size at t 0 (the intervention time point) is bounded by the constant 1=ð1 À R c ), which is independent of the population size. In Table 1 we give some computed values of the solution p of Eq. (2.12c). It is certainly very surprising to see that a disease with a relatively small basic reproductive number R 0 ¼ 3:0 (P R 1 ) will be able to infect the great part (> 94%) of the population if no control measures have been applied. Which level of herd immunity will be safe? How effective do the control measures have to be, so that an epidemic will be brought under control? These two problems are crucial for the maker of public heath policy. We tackle these two problems by virtue of our new SEIR model Eq. (2.1). Our theory (Theorem 2.1) reveals that the condition for the eradication or control of diseases is that there exists some time point t 0 P 0 such that the controlled reproductive number R c given by is maintained below unity, i.e., R c < 1: ð3:1bÞ More exactly, we have the following conclusion as a consequence of Theorem 2.1(i): If (3.1b) is satisfied, then an epidemic course will be brought under control after the time point t 0 in such a way that the ratio of the final epidemic size to the epidemic size at t 0 (of Fðt 0 ) individuals) is bounded by the constant 1=ð1 À R c ) (which is independent of the population size), no matter how complicated the transmission patterns of the disease are. In the traditional epidemic theory, it is expected that a disease will be eradicated or an epidemic course will be brought under control if the controlled reproductive number R c is maintained below unity. However, one important point, namely, the varying patterns of the infection transmission, was ignored by this classical way of thinking. By contrast, in deriving condition Eq. (3.1) we have taken into account all transmission patterns that are most likely to exist and thus condition Eq. (3.1) must be the right one for eradicating the diseases or controlling their outbreaks. For practical uses, our control condition Eq. (3.1) re-establishes the following baseline of implementing control measures for eradicating or containing a disease: (i) controlling the contact to disease of the public networks. These measures include vaccination, travel restrictions and barrier precautions (e.g., masks, gloves), which lead to reduce the mean infection rate rðt). (ii) Controlling the sources of infection. These measures include vaccination, quarantine, case isolation and contact tracing of infectives, which lead to reduce the effective mean infectious period Rðt). We take a closer look at the impact of control measures on the latency and recovery processes. The latent process, i.e., the process that an exposure becomes being infectious, is mainly an intrinsic thing. More exactly, the latent process depends mostly on the disease itself and on the corresponding circumstances like seasons. Certainly, control measures like vaccination can also yield impact on the latent process. However, this impact can be negligible in the practice. On the contrast, the removed (recovery) process will be strongly affected by the extrinsic control measures. For the purpose of practical applications of our theory, we have to know how the control measures impact the functions rðt) and Aðt; s). Although it is complicated, the assessment of the impact of control measures on infection sources can be measured by the socalled instant efficacy and delayed efficacy, which are defined as follows: The instant efficacy qðt). At time t, we define qðt) to be the proportion of infectives that will be blocked by the control measures in such a way that their transmission of infection will be prevented immediately after their emergence and permanently. The efficacies of vaccine, quarantine and case isolation can be put into this category. The delayed efficacy jðt) and the delay r D . At time t, we define jðt) to be the proportion of infectives that will be blocked by the control measures after its emergence of a mean lifespan of r D units of time. The efficacy of contact tracing is a typical example of these delay types. Alternatively, we can define the parameters (qðtÞ; jðtÞ; r D ) by decomposing the infectives into three groups: Group I (blocked): At time t, this group of proportion qðt) consists of all infectives whose infection is blocked by the control measures immediately and permanently. Group II (blocked after 'diagnosis'): At time t, this group of proportion jðt) consists of all infectives whose infection will be blocked immediately and permanently after 'diagnosis'. Here we understand the term 'diagnosis' as the ensemble of all control measures like contact tracing or clinical diagnosis that lead to the detection and isolation of infectives. In this sense, the mean delay r D can be equally thought of as the waiting time for an infective to be 'diagnosed'. Let A D ðt) be the recovery CDF for this group. Then we have Group III (unblocked and 'undiagnosed'): At time t, this group of proportion 1 À qðtÞ À jðt) consists of the remaining infectives whose recovery is intrinsic; that is, this group consists of all infectives for which the control measures do not impact their recovery and thus their recovery process is an intrinsic thing that depends only on the disease itself and on the corresponding circumstances like seasons. The recovery CDF for this group is just the intrinsic CDF A 0 . The mean value r :¼ is the mean infectious period of the disease. It follows that r D 6 r: Under these assumptions, the effective recovery CDF Aðt; s) is given by Table 1 The least proportion p of infectives in terms of the reproductive number R1. To continue, we assume that there is a time point t 0 P 0 after intervention such that rðtÞ 6 r c ; qðtÞ P q c ; jðtÞ P j c ð3:3aÞ for all t P t 0 , where r c ; q c and j c are positive constants such that q c þ j c 6 1. Using condition (3.3a), the effective reproductive number R eff ðt) given by (2.10) can be estimated as follows: is called the controlled mean infectious period. It follows that the controlled reproductive number R c can be estimated by ð3:3dÞ Below we will treat two more concrete cases: eradication of diseases by mass vaccination, and control of outbreaks by isolation combined with contact tracing and quarantine. We understand the vaccination effect as the ensemble of all impacts of vaccination on the control of diseases that are made before and during an epidemic course. We consider the situation that in a homogeneously mixed population a proportion f of the population has been vaccinated after a certain time point t 0 . In order to measure the vaccination effect, we follow Halloran et al. [25] (cf. also [24, 41] ) by defining two quantities, the vaccine efficacy against infectiousness (VE I ) and the vaccine efficacy against susceptibility (VE S ), as follows: 1 À VE I :¼ infectiousness of vaccinated infectives infectiousness of unvaccinated infectives ; and 1 À VE S :¼ susceptibility of vaccinated susceptibles susceptibility of unvaccinated susceptibles : For simplicity, we assume that the vaccination effect is immediate and permanent. Hence, we can set r D :¼ 0 and j c :¼ 0 in (3.3) and thus the effective mean infectious period r c given by (3.3c) is where q c is the proportion of infectives whose transmission of infection will be blocked by the vaccine immediately and permanently. The quantity 1 À q c is equal to the fraction of infectives whose transmission of infection has not been blocked by prior immunity and thus can be identified with the effective infectiousness. By setting the infectiousness of unvaccinated infectives to be unity, the value 1 À q c is calculated by Similarly, the effective susceptibility is equal to 1 À f  VE S and thus the controlled mean infection rate r c is given by where b is the mean infection rate in the absence of prior immunity. Remember that the basic reproductive number R 0 is given by where r is the mean infectious period. By (3.4a)-(3.4c), the controlled reproductive number R c 6 r c r c can be estimated by We say that the level f of herd immunity is safe for the vaccine efficacy of the values (VE S ; VE I ) if the controllability condition R c < 1 is satisfied. By (3.5) , this is the case if It follows that the level f of herd immunity is safe for the vaccine efficacy ( where f c , called the critical immunization threshold, is the unique positive solution of the equation The term 'critical immunization threshold' means that the disease can be expected to be eliminated if the herd immunity f exceeds f c , no matter how complicated the transmission patterns of the disease vary. Considering the solution f c of Eq. (3.7b) as a function of the basic reproductive number R 0 and the vaccine efficacy (VE S ; VE I ), we see that f c increases with R 0 , decreases with VE S and VE I . A necessary and sufficient condition for the solution f c of Eq. (3.7b) being not greater than 1 is which determines a valid domain for the vaccine efficacy (VE S ; VE I ). We consider the generic but less unrealistic case that VE I ¼ VE S =2, i.e., the vaccine efficacy for an infected individual is only half the vaccine efficacy for an non-infected. Then the inequality (3.8a) is satisfied if and only if This implies that the possibility for eradicating the disease by mass vaccination exists if VE S > VE c and VE I > VE c =2. Because of this, we will call VE c the generic critical level of vaccine efficacy. As in the traditional theory of epidemiology (Anderson and May [1] ), we consider the somehow idealized case that vaccination works as follows: (i) VE S ¼ 1, i.e., a vaccinated individual will have a perfect protection against the disease. (ii) VE I ¼ 0, i.e., the vaccination effect falls completely on an infected individual. Under (i) and (ii), we solve Eq. (3.7b) and obtain that Remember that in the traditional theory of epidemiology [1] the value q c ¼ 1 À 1=R 0 is used as the critical immunization threshold for the case that the vaccine efficacy against susceptibility is 100% effective and the vaccine efficacy against infectiousness as well as the varying of transmission patterns can be ignored. Our above theory establishes in a more detailed way the correctness of this traditional theory of epidemiology. In the following Table 2 , we present our estimation for several known diseases. Comparison: The following short list gives the basic reproductive number R 0 for the known diseases estimated by other different models and methods (in the spirit of fitting the models to concrete data): Chickenpox: 7-12 [ Compared to our results presented in Table 2 , we see that the ranges of R 0 obtained by our purely theoretical method (given in the next section) coincide very well with the known ranges obtained by others using observed data and different models. A comprehensive review of these known estimates and the mathematical methods used for the estimation can be found in the recent work of Heffernan et al. [27] (cf. also Bauch et al. [2] ). Discussion: The results in Table 2 reveal that the commonly required coverage of vaccination for eradicating diseases by mass vaccination (taking into account the worst case of SSEs) must be more than 87%. Among all of the selected diseases, there is an almost perfect (% 100%) observed vaccine efficacy against Smallpox. This probably explains why Smallpox can be eradicated globally by vaccination. The observed high efficacy (90-100%) of vaccination against Polio should also be sufficient for the eradication of Polio. However, there is a great difference between Smallpox and Polio: Our result in Table 2 implies that the reproductive number generated by SSEs with Polio might be as large as 22.0 and thus the critical immunization threshold q c might be needed to be as large as 95%. This indicates that the global eradication of Polio must be more difficult than the global eradication of Smallpox. The situation is similar for the remaining diseases. Hepatitis B, HIV and the newly (re-)emerging diseases SARS, Ebola and AHC have a basic reproductive number R 0 between 1.6 and 5.3 (without SSEs; 6 16:2 with SSEs), which is distinctly smal- Haydon et al. [26] , Chowell et al. [11] . As in [18, 32, 31, 26, 11] , our model unit is the farm. This is because the spread of FMD between animals within a farm is so rapid that a total farm will be infected immediately if one individual is infected. Therefore, our above estimation for R 0 is calculated as in [18, 32, 31, 26, 11] at the farm level. g According to Stegeman et al. [56] , the avian influenza (H7N7) has a mean infectious period of 10-14 days and a mean latent period of about 2 days. Here we assumed that the mean latent period for avian influenza is comparable to the mean latent period for human influenza, ranging from 1 to 3 days. h Our estimation of the mean latent period s and the infectious period r for HIV-1 cells is based on the data given in [49] , [50] and [37] , by which the latent period s is equal to the reciprocal of the clearance rate c (i.e., s ¼ 1=c) and the infectious period r is equal to the reciprocal of the infected cell loss rate d (i.e., r ¼ 1=d). It was estimated (see Table 5 .1 in [50] ) that the values of c have a range of 3:1 AE 0:6 which yields an estimate for s ¼ 1=c of 1=3:7 % 0:3 to 1=2:5 ¼ 0:4 days. However, as pointed out in [49] , these values of s are in general underestimated. It is very likely (cf. [37] ) that the values of s range from 0.5 to 1.0 days. On the other hand, different models and methods ([49,50,37] ) yield different estimation of the values of d. In [49] (see also [50] ), it was estimated that the values of d vary from 0.3 to 0.7 (with a mean value of 0.5), while in [37] (Table 4 there) the values of d vary from 0.2 to 0.5 (with a mean of value 0.3). Based on these considerations, we assume, as given above, that the values of s range from 0.5 to 1.0 days, and the values of r ¼ 1=d range from 1=0:7 % 1:4 to 1=0:3 % 3:3 days.Our result suggests that each HIV-1 infected human cell will infect on average (7:6 þ 27:7Þ=2 $ 18 cells over the course of its life span when CD4 target cells are not limiting. In [37] , a similar result (of $ 19 cells) was obtained. i SIV stands for simian immunodeficiency virus. Our estimation of the mean infectious period r for SIV cells is based on [47] from SIV infected macaques, by which the infectious period r is equal to the reciprocal of the infected cell loss rate a (i.e., r ¼ 1=a). It was estimated (see Table I of [47] ) that the values of a range from 0.51 to 0.96. Moreover, it was known (cf. [47, 37] ) that the latent period s for SIV is between 0.5 and 1.0 days. Therefore, we assume, as given above, that s ranges from 0.5 and 1.0 days, and r ranges from 1.0 to 2.0 days.Our result suggests that each SIV infected macaque cell will infect in average (6:8 þ 22:1Þ=2 $ 15 cells over the course of its life span when CD4 target cells are not limiting. In [47] , the estimated R 0 values vary drastically under different models: R 0 ranges from 4.2 to 17 in the exponentially distributed model with a mean latent period of 0.5 days, and R 0 ranges from 5.4 to 68 in the fixed time delay model with a mean latent period of 1.0 days. * Sources: Benenson [3] and 'The Pink Book' of CDC, 9th Ed., 2006 [6] . ** The subjects under study are cells. Therefore, the basic reproductive number is defined for the micro-parasites (cells) here. s is the mean latent period and r the mean infectious period. Unit of time: day (except HIV). The sources for the estimation of s and r of the known diseases (except SARS, Ebola, AHC, FMD, Avian influenza, Acute HIV-1 and Acute SIV) are Anderson and May [1] and 'The Pink Book' of CDC, 9th Ed., 2006 [6] . à We have previously used our results on the estimation of the basic reproductive number R 0 (with or without SSEs (superspreading events)), which are obtained by a method presented in the next section. In Table 2 , the types I-IV are defined by the categories to which the diseases belong: Type I-II corresponds to the category 'mild' and Type III-IV corresponds to the category 'severe'. ler than the basic reproductive numbers generated by the most dangerous diseases like Measles and Pertussis. This might indicate that these diseases do not cause epidemic courses as often as Measles and Pertussis. The results in Table 2 show that the observed relatively high vaccine efficacy for the diseases Diphtheria, Measles, Mumps, Pertussis, Polio and Rubella should be sufficient for protecting the global spreading of these diseases. Our estimation of the critical level of vaccine efficacy given in Table 2 reveals that a high vaccine efficacy (P 90%) should be sufficient for eradicating the selected known diseases. This point might be valuable for the medical industry. Control of outbreaks by isolation combined with contact tracing and quarantine The 2002-2003 epidemic SARS provided an example that a newly emerging disease can be contained under the implementation of isolation combined with contact tracing and quarantine; see [53, 36, 15, 58, 21, 35, 2] . Using our models, we can give an explanation for this. We decompose the infectives into two groups: (a) Asymptomatic infectives whose transmission occurs prior to symptoms; let h be the proportion of this group. (b) Symptomatic infectives whose transmission occurs just after symptoms. This fraction is equal to 1 À h. The symptom-based public control measures start with a certain time point t 0 (after the outbreak) and work as follows: (a) Isolation: A proportion of symptomatic infectives are isolated for a long enough duration. (b) Contact tracing and quarantine: Symptomatic individuals who have been isolated have their contacts traced, and a proportion of their contacts will be quarantined for a long enough duration. In order to describe the efficacy of these control measures, we follow the ideas of Fracer et al. [21] and define e I (efficacy of isolation) to be the probability that an infected will be isolated immediately after he/she becomes symptomatic, and e T (efficacy of contact tracing) to be the probability that an asymptomatic infective will be detected first by contact tracing and subsequently isolated. For an asymptomatic infective who is detected by contact tracing, generally, there is a time delay between isolation and the starting point of infection. We let r D be the mean time of delays of those infectives. Set d :¼ r D =r 2 ð0; 1: It is reasonable to consider d to be an increasing function of h, the proportion of asymptomatic infectives. Under above assumptions, the proportion q c of blocked symptomatic infectives and the proportion j c of asymptomatic infectives that are blocked by contact tracing are calculated as follows: Correspondingly, the controlled mean infectious period r c given by (3.3c) is calculated by r c ¼ ½1 À ð1 À hÞe I À hð1 À dÞe T Á r: To complete the model settings, we assume, for simplicity, that the scale of the public controls is relatively small and thus the value rðt), the mean infection rate per unit time per active (unblocked) infective, will be reduced very few, i.e., we assume that rðtÞ r c ¼ b, where b is the mean infection rate in the absence of control measures. It follows that the controlled reproductive number R c 6 r c r c can be estimated by The controllability condition R c < 1 will be satisfied, if This implies that an outbreak can be brought under control by isolation alone (i.e., e I ¼ 1 and e T ¼ 0) if the fraction h of asymptomatic infectives is not greater than 1=R 0 . Particularly, we find that it is possible to control SARS and relatively mild Smallpox and Influenza by isolation alone, while HIV (h > 80%) can be controlled only under very effective isolation combined with very effective contact tracing [42] ; cf. Table 3 . Similar controllability conditions are obtained by Müller et al. [45] and Fracer et al. [21] using different methods. For example, one result of [21] says that an outbreak can be brought under control by isolation alone, if the proportion e I of isolated symptomatic individuals is greater than (1 À 1=R 0 Þ=ð1 À h), which is the same result as given by our inequality (3.11b). However, this conclusion of [21] is based on the unrealistic but mathematically simplifying assumption that both latency and recovery processes are exponentially distributed. By contrast, in our derivation of the controllability condition (3.11b) we have taken into account all possible transmission patterns. Thus, our condition (3.11b) must be the right one for controlling an outbreak by isolation combined with contact tracing and quarantine. We consider the special (but idealized) case that the isolation of symptomatic individuals is perfectly effective and every asymptomatic individual will be detected by contact tracing, i.e., e I ¼ 100%; e T ¼ 100%: ð3:12aÞ Under assumption (3.12a), we find that the controllability condition We call d c the maximally allowed ratio of delay in contact tracing. For controlling a disease, the smaller the value of d c is, the faster the contacts should be traced. As seen in the previous sections, calculating the basic reproductive number R 0 is equivalent to breaking the codes of communicable diseases: One key for a successful eradication program or a control strategy is to know the value R 0 of the disease. In this section, we will present a method which will give a fairly correct estimate of R 0 for most known diseases, and probably more important, for coming unknown diseases. We prepare some tools. We will use the convolution to simplify the representations. Recall that the convolution u à v of two Lebesgue-integrable functions u and v on R þ is defined by the Lebesgue integral u à vðtÞ :¼ Z t 0 uðt À sÞvðsÞ ds ðt P 0Þ: In particular, the convolution 1 à u is the integral of u : In particular, if the CDF P has a density function v P , then the Laplace transform of v P coincides with the Laplace-Stieltjes transform of P. Below we will use this fact without further explanation. We now turn to our main topic in this section. We note that two ingredients that determine the epidemic spreading are the transmission patterns and the intrinsic growth rate with which the epidemic has started. It was shown by Wearing et al. [59] that different choices of the transmission patterns and different estimates of the intrinsic growth rates can yield sometimes drastically different estimates of the basic reproductive numbers. Because of this, we postulate that the so-called basic reproductive number, R 0 , must be an overall mean, i.e., it is the result of averaging the reproductive numbers generated by all most likely transmission patterns and all most likely intrinsic growth rates. The formula for calculating the basic reproductive number R 0 as the product of the mean infection rate b and the mean infectious period r means that the value of R 0 calculated in this way is a long-time average. Another baseline for defining R 0 (Anderson and May [1] ) is to define R 0 as the expected number of secondary infectious cases generated by one typical primary case in an entirely susceptible and sufficiently large population. Since only the primary case is under consideration, the value of R 0 calculated in this way is completely determined by the dynamic behavior of the epidemic course in the early stages. It is generally known (cf. [1] ) that in the early stages the epidemic course grows exponentially with a positive growth rate, K, the so-called intrinsic growth rate of the epidemic. The estimate of the intrinsic growth rate of an epidemic depends on the models used. In our approach, we will do this by means of our model Eq. (2.1). More exactly, we use Eq. (2.1) to model the course of an epidemic spreading in a population that is homogeneously mixed and without prior immunity and control measures. This means that the recovery CDF A in Eq. (2.1d) is just equal to the intrinsic recovery CDF. In order to save the notation, below we also use A to denote the intrinsic recovery CDF. First we need to give a concrete definition of the so-called intrinsic growth rate K of an epidemic whose course is described by our model Eq. (2.1). For this purpose, we note that in our present case the mean infection rate rðt) in Eq. (2.1a) is equal to the (constant) intrinsic mean infection rate b, i.e., rðtÞ b. Using rðtÞ b and integrating Eq. (2.1a), we obtain Substituting SðtÞ ¼ ðN À g 0 Þe ÀHðtÞ=N into Eq. (2.1b) and using H ¼ bð1 À AÞ Ã W 0 à F, we obtain ð4:1aÞ where the kernel G is given by By setting F 1 :¼ F=N < 1 and a :¼ g 0 =N < 1, we rewrite Eq. (4.1) as logð1 À F 1 Þ ¼ logð1 À aÞ À G à F 1 : In the early stages of the epidemic course, the approximation holds. Therefore, in the early stages of the epidemic course, the solution of the non-linear Eq. (4.2) can be approximated by the solution of the following linear Volterra equation u ¼ À logð1 À aÞ þ G à u: The solution u of (4.3) can be expressed as uðtÞ ¼ ðÀ logð1 À aÞÞÁ wðtÞ ðt P 0), where w is the solution of the linear Volterra equation Positivity of the kernel G ¼ bð1 À AÞ Ã W 0 implies that the solution w of (4.4a) satisfies wðtÞ P 1 for all t P 0. Moreover, the exponential growth rate K of w defined by K :¼ inffx 2 R : 9M > 0 such that wðtÞ 6 Me xt 8t P 0g ð4:4bÞ is finite and non-negative. (To see the finiteness of K, we note that 0 6 G 6 b. Therefore, by choosing a sufficiently large constant x > 0 we have c :¼ R 1 0 e Àxt GðtÞ dt < 1. Consider the functions vðtÞ :¼ wðtÞe Àxt (t P 0) and aðtÞ :¼ sup 06s6t vðs). Then a calculation yields that v P 0 satisfies the equation vðtÞ ¼ e Àxt þ R t 0 ðe Àxs GðsÞÞ vðt À sÞ ds (t P 0). It follows that vðtÞ 6 1 þ aðtÞ Z t 0 e Àxs GðsÞ ds 6 1 þ caðtÞ for all t P 0. Consequently, we have aðtÞ ¼ sup 06s6t vðsÞ 6 1 þ caðt) for all t P 0 and thus aðtÞ 6 1=ð1 À cÞ ¼: M. This implies that wðtÞ 6 Me xt for all t P 0, showing the finiteness of K.) We define the value K given by (4.4b) as the intrinsic growth rate of the epidemic. Next we derive a relation between K and the basic reproduction number R 0 . We use the basic theory of linear Volterra equations (cf. [ [52] Chapt. I]) to conclude that (4.4a) has a solution with the exponential growth rate K > 0 if and only if the Laplace transform e Gðk) of G P 0 is equal to 1 at k ¼ K. As consequence of this, we have e GðKÞ ¼ 1: ð4:5aÞ By calculating the Laplace transform of G we obtain e GðkÞ ¼ b g ð1 À AÞðkÞ f W 0 ðkÞ ð 4:5bÞ for all k P 0. Thus, by replacing the Laplace transform f W 0 by the Laplace-Stieltjes transform g dW , we find from (4.5b) that e GðkÞ ¼ b g ð1 À AÞðkÞ g dW ðkÞ ðk P 0Þ: ð4:5cÞ As we will see later, the representation (4.5c) in terms of the Laplace-Stieltjes transform is sometimes more appropriate for our purposes. Finally, by setting k ¼ K in (4.5c), using condition (4.5a) and replacing b by R 0 =r, we obtain that R 0 ¼ R 0 ðK), as a function of the intrinsic growth rate K, is given by R 0 ðkÞ :¼ r g ð1 À AÞðkÞ g dW ðkÞ ðk P 0Þ: ð4:6aÞ It can be seen that the function R 0 ðk) given by (4.6a) is continuously differentiable for all k P 0 and satisfies the estimate: R 0 ðkÞ P rk 8k P 0: Moreover, the function R 0 ðk) is strictly increasing for k P 0, since the functions g ð1 À AÞðkÞ ¼ for all k P 0. In particular, for the case g ¼ 1 (i.e., where A is exponentially distributed) we have R 0 ðkÞ ¼ ð1 þ askÞ 1 a ð1 þ rk) which is a strictly convex function of k P 0 (for every fixed a 2 ð0; 1). The strict convexity of such functions R 0 ðÁ) has important implications, see Proposition 4.3 below. For the special case a ¼ 1 ¼ g (i.e., where both W and A are exponentially distributed) we obtain R 0 ðkÞ ¼ ð1 þ skÞð1 þ rkÞ ¼ 1 þ ðs þ rÞk þ srk 2 : This simple formula was used by Lipsitch et al. [36] to estimate the basic reproductive number for SARS. Average over the intrinsic growth rates: For the moment, we fix the CDFs W and A that determine the transmission pattern E ! W I ! A R. Theoretically, an epidemic is allowed to start with an arbitrary intrinsic growth rate K P 0. However, as observed in the real world, the most likely epidemic courses will start with some typical values of the intrinsic growth rates and thus the basic reproductive number R 0 will be given by some typical intrinsic growth rate K in the sense that R 0 ¼ R 0 ðK). A crucial problem now is: Which values of the intrinsic growth rates are typical? Or, more concretely, which values of intrinsic growth rates are such that the epidemic will start with them most likely? Mathematically, such typical values of the intrinsic growth rates must obey some kind of abundance. Below we give a method to determine such typical values of the intrinsic growth rates that will be suitable for calculating the basic reproductive number R 0 . We consider an epidemic course with a fixed transmission pattern E ! W I ! A R. An important property, that characterizes also the epidemic course, is the distribution of the secondary cases. Stimulated by the ideas used by Lloyd where F : R þ ! ½0; 1 is some CDF such that Fð0Þ ¼ 0 and FðxÞ ! 1 as x ! 1. In the sequel, for convenience reasons, we will identify an epidemic course with its distribution law F of secondary cases. (Please do not confuse the notation 'F' used here with the ones used early in model Eq. (2.1). From now on to the end of this section, the early meaning of 'F' as the cumulative number of exposures will be dropped. We re-employ the notation 'F' only for the reason of saving notations.) In general, the form of F can be arbitrary. We take the mean value as the averaged reproductive number generated by an epidemic course that obeys the distribution law F. A relation between the distribution law F and the distribution of the averaged reproductive numbers R 0 ðK) given by the intrinsic growth rates K P 0 can be established as follows. We decompose the group of infected individuals into two subgroups. Group A consists of those infected individuals, called typical infected individuals, which have generated at least one secondary case. Group B consists of those infected individuals, called atypical infected individuals, which have generated less than one secondary case. In this way, the distribution of the number m in the range ½0; 1) is determined by the averaged numbers of secondary cases generated by the untypical infected individuals, and the distribution of the number m in the range ½1; 1) is determined by the averaged numbers of secondary cases generated by the typical infected individuals. We consider a particular typical infected individual which generates on average k secondary cases (k P 1). Then there exists a unique n k P 0 such that R 0 ðn k Þ ¼ k. Using our previous formulation by means of model (2.1), it is plausible to think that this typical infected individual generates an epidemic course with the intrinsic growth rate n k . Correspondingly, we call n k an individual intrinsic growth rate. In this way, the distribution of the individual intrinsic growth rates n P 0 is determined as follows: Probðn P 0jR 0 ðnÞ 6 xÞ ¼ FðxÞ À Fð1Þ ð 4:8aÞ for all x P 1. Since the function R 0 ðn) is strictly increasing for all n P 0, we find from (4.7a) that the probability distribution of the individual intrinsic growth rates is given by Probðn P 0jn 6 kÞ ¼ FðR 0 ðkÞÞ À Fð1Þ ð 4:8bÞ for all k P 0. We call the mean value kðFÞ :¼ the mean intrinsic growth rate of the epidemic course that obeys the distribution law F. The mean intrinsic growth rate kðF) yields the number R 0 ðkðFÞ) that represents the expected number of secondary cases generated by a particular infected individual which has started its infection chain with the individual intrinsic growth rate n ¼ kðF). This local reproductive number R 0 ðkðFÞ) will coincide with the global reproductive number F if, say, the pre-assigned distribution law F is a point distribution. By a point distribution concentrated at a given point z > 0 we mean the CDF F z satisfying where dðÁ) is the Dirac delta function concentrated at the origin. Equivalently, the function F z is given by F z ðxÞ :¼ 0 for all 0 6 x < z and F z ðxÞ :¼ 1 for all x P z. We have for all K P 0 that kðF R 0 ðKÞ Þ ¼ K; F R 0 ðKÞ ¼ R 0 ðKÞ: ð4:9bÞ Before going to the next step, we make an observation. Let K P 0 be a fixed intrinsic growth rate which yields the averaged reproductive number R 0 ðK). By (4.9b) we see that the epidemic course obeying the point distribution law F R 0 ðKÞ has a mean intrinsic growth rate of value K and an averaged reproductive number of value R 0 ðK). We consider a general distribution law G that generates the pair (kðGÞ; G) of mean intrinsic growth rate kðG) and averaged reproductive number G. If it occurs that the closedness of the pair (kðGÞ; G) to the given pair (K; R 0 ðKÞ) implies the closedness of the distribution law G to the special point distribution law F R 0 ðKÞ , then we can intuitively imagine that there is only one epidemic course, namely the one obeying the special point distribution law F R 0 ðKÞ , that will start with the mean intrinsic growth rate K and hit the target R 0 ðK). In other words, the event that an epidemic will start with such a mean intrinsic growth rate K is rare. This observation leads to the following definition. If an intrinsic growth rate K is not rare in the above sense, then we briefly say that K is typical. In order to estimate the basic reproductive number of a disease by the averaged reproductive numbers R 0 ðK), only typical intrinsic growth rates K should be taken into account, because the so-called basic reproductive number of a disease is the result of averaging many epidemic courses that might be drastically different. There is another way to describe rareness. Given an intrinsic growth rate K P 0, we consider the solutions to the following inequalities: where F is a CDF on R þ . Clearly, the point distribution F R 0 ðKÞ is a solution of (4.12). Below (in Proposition 4.3) we will see that the rareness of K is equivalent to F ¼ F R 0 ðKÞ being the uniqueness solution of (4.12). This has a very interesting implication. Since for each fixed K P 0 the set S K of CDFs that solve (4.12) is closed under convex combinations, we conclude that if K is typical (i.e., not rare), then the solution set S K is convex and contains infinitely many elements. A concrete interval containing only typical intrinsic growth rates can be determined as follows. We consider the continuous function (ii) Every K < l is typical. (iii) l is rare. (iv) Assume that the function R 0 ðk) is strictly convex on R þ . Then all intrinsic growth rates K with K P l are rare. Moreover, k=R 0 ðk) attains its absolute maximum at (and only at) k ¼ l. From the above we see that the special intrinsic growth rate l obeys the following minimax property: l is the smallest number such that every intrinsic growth rate K < l is typical, and the largest number among all intrinsic growth rates K such that the ratio k=R 0 ðk) (of the intrinsic growth rate to the averaged reproductive number) attains its absolute maximum at k ¼ K. The proof of Proposition 4.3 will be given in Appendix C. We remark that in most instances the estimator R 0 ðk) given by (4.6a) can be chosen to be strictly convex for all k P 0. One example was already given in Example 4.1. Based on the above detailed discussion, we come finally to the following assertion: The basic reproductive number R 0 is given by some typical intrinsic growth rate K < l in the sense that ð4:14Þ In general, the scaling factor s is not specified. Average over transmission patterns. We remember that the function R 0 ðk) given by (4. We recall also that the pair (W; A) has the following meanings: (i) Wðt) measures how an exposure is far from an infective after his infection of t units of time. (ii) Aðt) represents the level of recovery, i.e., Aðt) is the probability that an infective will be recovered after having being infective for t units of time. We try to find some special pairs (W; A) that are 'typical', i.e., 'representatives' for most of the possible cases. We remark that the 'representatives' to be chosen below may be not absolutely continuous (a.c., for short), for two reasons as: (i) These 'representatives' have simple forms and with immediate probabilistic interpretations. (ii) These 'representatives' describe the cluster of typical a.c. CDFs and thus are limits of such a.c. CDFs. We characterize a typical transmission pattern E ! W I ! A R based on the following three properties (P1)-(P3): that is, We give some comments on these baselines. First, it was demonstrated by P.E. Sartwell [54] that the latent periods of many known diseases are typically 'right-skewed' or 'log-normal' distributed with a long right-hand tail. There are many known 'right-skewed' distributions. However, the gamma distributions are more commonly used and more important, they are more appropriate for our present purpose because their Laplace transforms can be calculated explicitly. Note that in (P1) we have dropped the restriction (4.15b) (saying W 0 ð0Þ ¼ 0) so that the exponential distribution can be adopted. Second, in epidemiology, very often an infected person will show a rash after the latent period. As commonly accepted in the natural sciences, such a rapid change can be modeled by the exponential growth law. Our assumption (P2) follows this basic principle. The assumption (P3) emphasizes the observable fact that the recovery process must, as successor of the latent process, change for large times more quickly than its former. In order to make the empirical study of the property (P3) more transparent, we rewrite (4.16c) as z að1 À bÞ ¼ 1 þ z að1 À gÞ with some g 2 ½0; 1Þ: Correspondingly, we define the class fðW a ; A a;g Þ : 0 < a 6 1; 0 6 g < The Laplace-Stieltjes transforms of W a and A a;g are g dW a ðkÞ ¼ ð1 þ askÞ À 1 a ; g ð1 À A a;g ÞðkÞ ¼ r 1 þ skða þ z=ð1 À gÞÞ ð4:17cÞ for all k P 0. Remember that the variable z :¼ r s ð4:17dÞ used above is the ratio of the mean infectious period to the mean latent period. We have that lim a!0 g dW a ðkÞ ¼ e Àsk ðk P 0Þ ð 4:18aÞ which is just equal to the Laplace transform g dW 0 ðk) of the point distribution W 0 given by W 0 ðtÞ :¼ 0 ð0 6 t < sÞ; W 0 ðtÞ :¼ 1 ðt P sÞ: The point distribution W 0 corresponds to the case of a complete delay of the latent process in the time interval ½0; s). Such a delay effect has been observed in practice. Hence, the point distribution W 0 can be accepted as the representative of cases with delay. In short, the 'representatives' for the transmission patterns of latency are fW a : 0 6 a 6 1g. Summing up, we have the following assertion: For each pair (a; gÞ 2 ½0; 1  ½0; 1) of parameters the transmission pattern E ! W I ! A R with the choice (W; AÞ ¼ ðW a ; A a;g ) satisfies the required properties (P1)-(P3) and thus is a typical transmission pattern. To continue, we fix a pair (a; gÞ 2 ½0; 1  ½0; 1), and denote by R 0 ða; g; s; r; k) the function R 0 ðkÞ ¼ r=ð g ð1 À AÞðkÞ g dW ðkÞ) given by (4.6a) with W ¼ W a and A ¼ A a;g . Substituting the Laplace transforms g dW a ðk) and g ð1 À A g Þðk) given by (4.17c), we obtain that R 0 ða; g; s; r; kÞ ¼ ð1 þ askÞ 1 a 1 þ skða þ z=ð1 À gÞÞ ½ ð k P 0Þ: It is easy to see that the function R 0 ða; g; s; r; k) is a strictly convex function for all k P 0. Therefore, the value l > 0, which yields the estimate of the basic reproductive number R 0 by R 0 ¼ R 0 ða; g; s; r; slÞ with some 0 6 s < 1; We rewrite the function given by (4.19a) as R 0 ða; g; s; r; kÞ R 0 ða; g; z; x) with R 0 ða; g; z; xÞ :¼ ð1 þ axÞ 1 a 1 þ xða þ z=ð1 À gÞÞ ½ ð x P 0Þ: ð4:20bÞ Now a calculation yields that the Eq. (4.19c) becomes a quadratic equation in the new variable x ¼ sl of the form x þ x 2 ða þ z=ð1 À gÞÞ ¼ 1 þ ax: ð4:20cÞ Solving (4.20c), we obtain the unique positive solution x ¼ 2=ð1 À a þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ aÞ 2 þ 4z=ð1 À gÞ q Þ: ð4:20dÞ It follows that the basic reproductive number R 0 ¼ R 0 ða; g; s; r; slÞ ¼ R 0 ða; g; z; sx) is given by R 0 ¼ ð1 þ asxÞ 1 a 1 þ sxða þ z=ð1 À gÞÞ ½ ð 4:21aÞ with x given by (4.20d) and some s, 0 6 s < 1. We use the parameters (a; g) and the ratio z ¼ r=s to parameterize the scaling factor s, i.e., we take the form s ¼ sða; g; z) as a continuous function of its variables. On the one hand, it is reasonable to assume that sða; g; zÞ ! 1 as z ! 1. On the other hand, we need that sða; g; 0Þ ¼ 0. To see this, we note that the solution x given by (4.20d) has the property that x ! 1 as z ! 0. It follows that the function R 0 given by (4.21a) has the property that R 0 ! ð1 þ asða; g; 0ÞÞ 1þ 1 a as z ! 0. However, one has to require that R 0 ! 1 as z ! 0. Hence, we must have sða; g; 0Þ ¼ 0. A parameterization s ¼ sða; g; z) that satisfies the above two requirements (sða; g; 0Þ ¼ 0 and sða; g; zÞ ! 1 as z ! 1) is, e.g., We choose such a relatively complicated parameterization form for s because an s of the above form, which is closely related to the solution x given by (4.20d), will yield a very simple representation for R 0 (see below). We set The function R 0 ða; g; z) is strictly increasing with z P 0. Moreover, a numerical study reveals that R 0 ða; g; z) is also a strictly increasing function of the parameters a 2 ½0; 1 and g 2 ½0; 1). We remember that R 0 ða; g; z) is the basic reproductive number associated with the typical transmission pattern E ! Wa I ! Aa;g R. Our final step is to give a method for comparing the different transmission patterns. Our goal for doing this is to derive suitable formulas for estimating the basic reproductive numbers generated by different epidemic courses. We consider a transmission pattern E ! Wa I ! Aa;g R with the parameters a 2 ½0; 1 and g 2 ½0; 1). Recall that the mode of the gamma distribution W a is equal to (1 À aÞs. Clearly, the smaller the mode (1 À aÞs, the faster the latent process E ! Wa I. On the other hand, we note that the growth rate of the recovery probability function A a;g ðt) is equal to z aþz=ð1ÀgÞ Á 1 r . It follows that the smaller the growth rate z aþz=ð1ÀgÞ Á 1 r , the slower the recovery process I ! Aa;g R. In short, the transmission pattern E ! Wa I ! Aa;g R is characterized by the vector Dða; gÞ :¼ ð1 À a; z=ða þ z=ð1 À gÞÞÞ in the sense that the smaller the vector Dða; g), the faster the latent process E ! Wa I and the slower the recovery process I ! Aa;g R. In particular, the larger the parameter a, the faster the latent process E ! Wa I; and the larger the parameter g, the slower the recovery process I ! Aa;g R. It is an observable fact that a disease with a faster latent process and a slower recovery process will generate more secondary cases than that with a slower latent process and a faster recovery process. Our function R 0 ða; g; z) for estimating the basic reproductive numbers reflects this observed fact very well, since R 0 ða; g; z) is a strictly increasing function of the parameters a 2 ½0; 1 and g 2 ½0; 1). Therefore, the combinations of the three typical states slow, mean, fast of the latent process E ! Wa I and the four typical states fast, mean, slow, extremely slow of the recovery process I ! Aa;g R yield four typical transmission patterns: The pair (a; gÞ ¼ ð0; 0) yields the slowest transmission pattern of latency and the fastest transmission pattern of recovery and thus generates the minimal reproductive number R 0 ð0; 0; z). We call the transmission patterns E ! W 0 I ! A 0;0 R to be of Type I (the light type). The pair (a; gÞ ¼ ð0:5; 0:5) yields the mean transmission patterns of latency and recovery and thus generates a mean reproductive number Rð0:5; 0:5; z). We call the transmission patterns E ! W 0:5 I ! A 0:5;0:5 R to be of Type II (the moderate type). The pair (a; gÞ ¼ ð1; 0:95) yields the fastest transmission pattern of latency and a slow transmission pattern of recovery and thus generates the maximal reproductive number R 0 ð1; 0:95; z). We call the transmission patterns E ! W 1 I ! A 1;0:95 R to be of Type III (the severe type). The pair (a; gÞ ¼ ð1; 0:99) yields the fastest transmission pattern of latency and the extremely slow transmission pattern of recovery and thus generates the largest reproductive number R 0 ð1; 0:99; z). We call the transmission patterns E ! W 1 I ! A 1;0:99 R to be of Type IV (the extremely severe type). In the above, we have used the 'fifty-fifty rule' to represent the mean processes of latency and recovery. Moreover, we propose to use the '95-99% rule' to determine the slow recovery process: The slow recovery processes are characterized by the CDFs A a;g with g in the range from 0.95 (¼ 95%) to 0.99 (¼ 99%), i.e., the general slow recovery process is corresponding to the choice g ¼ 0:95 and the extremely slow recovery process corresponds to the choice g ¼ 0:99. We remark that there are other possibilities to determine the slow recovery process by choosing other ranges for the parameter g. The range g 2 ½0:95; 0:99 chosen above can be considered as an empirical range, because it yields results which coincide very well with the known estimates, see Table 2 of Section 3 and Table 4 below. Summing up, we have the following concrete analytical formulas for calculating the basic reproductive numbers. coincides very well with the observed facts. For example, it reveals that acute diseases (i.e., diseases whose latent period is relatively short compared to the infectious period) are more dangerous than chronic diseases (i.e., diseases whose latent period is relatively long compared to the infectious period) in the sense that the former has a larger basic reproductive number than the latter. Estimation of R 0 for most known diseases. The method for this practice has been explained in the previous section Section 1, and the result has been given in Table 2 (Section 3), in connection with the study of vaccine efficacy and strategies of disease control. In short, we first put a disease, according to the types of its transmission patterns, into one of the two main categories 'mild' and 'severe' and then estimate its basic reproductive number in terms of the ratio z :¼ mean latent period : mean infectious period: Assuming that a disease of the category 'mild' has a mean latent period ranging from k to l units of time (e.g., days), and a mean infectious period ranging from m to n units of time, then the ratio z ranges from m=l to n=k and thus the basic reproductive numbers R 0 and the reproductive number R SSEs 0 generated by SSEs for this 'mild' disease are estimated by R I 0 ðm=lÞ 6 R 0 6 R II 0 ðn=kÞ; R SSEs 0 ¼ R III 0 ðn=kÞ: ð4:28Þ Assuming that a disease of the category 'severe' has a mean latent period ranging from k to l units of time (e.g., days), and a mean infectious period ranging from m to n units of time, then the ratio z ranges from m=l to n=k and thus the basic reproductive numbers R 0 and the reproductive number R SSEs 0 generated by SSEs for this 'severe' disease are estimated by Our classification for the known diseases is as follows. The category 'mild' includes the mild diseases like Hepatitis B, Polio, Scarlet fever and HIV (Aids) which have a long infectious period (when compared to their mean latent periods) but have an observed small basic reproductive number R 0 . The newly (re-)emerging diseases SARS, Ebola, AHC, FMD, influenza, influenza pandemic and avian influenza are put also into the category 'mild', since they have an observed relatively small basic reproductive number R 0 . The category 'severe' includes the rest severe and extremely severe diseases like Chickenpox, Mumps, Rubella, Measles etc. which have an observed large basic reproductive number R 0 . The newly emerging infections Acute HIV-1 and Acute SIV belong also to the category 'severe'. In the following Table 4 we give a more comprehensive overview of our estimates, with a comparison to results obtained by different models (e.g., age-structured models [1] ) and methods [27,2] (e.g., fitting the models to concrete data). We see that the ranges of R 0 obtained by our purely theoretical method (given in this section) coincide very well with the known ranges obtained by others using observed data. We call the reader's attention also to the fact that by our results the choice of time units is not crucial for estimating the basic reproductive number. In fact, what determines the estimation of the basic reproductive numbers is the ratio r=s of the mean infectious period to the mean latent period. The rðs þ xÞð1 À Aðs; xÞÞ dx ðs P 0; y P 0Þ: ð5:1cÞ By substituting (5.1b) into (5.1a) and using Eq. (2.5) (saying g 0 ¼ Fð0ÞW 0 þ F 0 à W 0 and recalling Fð0Þ ¼ g 0 ) we find finally that our original model Eq. (2.1) are simplified into the following single equation for the function F : F ¼ N À ðN À g 0 Þ expðÀH=NÞ with ð5:2aÞ Kðt À s; sÞðg 0 W 0 ðsÞ þ F 0 à W 0 ðsÞÞ ds: ð5:2bÞ We prove the well-posedness of the SEIR model Eq. (2.1) by showing that the integral (5.2) is well-posed. More exactly, we want to show the following assertion. Assertion: For any given g 0 , 0 < g 0 < N, Eq. (5.2) has a unique and global solution F such that Fð0Þ ¼ g 0 and Fðt) is a non-decreasing and absolutely continuous function of t P 0 with F 0 2 L 1 loc ðR þ ). To this end, we fix g 0 , 0 < g 0 < N. For each T P 0 we set X T :¼ fu 2 AC½0; T : uð0Þ ¼ g 0 ; u 6 N; 0 6 u 0 6 N Á krk L 1 ½0;T g: Here and below, we use AC½0; T to denote the Banach space of all absolutely continuous (real) functions over the interval ½0; T. By employing the distance dðu; vÞ :¼ ku 0 À v 0 k L 1 ½0;T ð8u; v 2 X T Þ; the set X T becomes a complete metric space. We consider the set X :¼ fT P 0 : 9F 2 X T satisfying Eq: ð5:2Þ for all 0 6 t 6 Tg: Clearly, 0 2 X and X is closed. We need to show that X is also open in R þ . Once the openness of X has been established, we conclude that X ¼ R þ by the connectedness of R þ . We now prove the openness of X. It suffices to show that for each T 2 X there exists some e > 0 such that ½T; T þ e & X. For this purpose, fix T 2 X and let q 2 X T be the unique solution of Eq. PuðtÞ ¼ N À ðN À g 0 Þ expðÀð1=NÞHqðtÞÞ ¼ qðtÞ ¼ uðtÞ: Therefore, P maps Y M into itself. We want to show that P is a contraction provided that the difference M À T is sufficiently small. To this end, we take u; v 2 X M . Then we have PuðtÞ ¼ qðtÞ ¼ Pvðt) for all t 2 ½0; T. For t 2 ½T; M we have QðtÞ :¼ PuðtÞ À PvðtÞ ¼ ðN À g 0 Þ e ÀHvðtÞ=N À e ÀHuðtÞ=N À Á : It follows that Q 0 ðtÞ ¼ ð1 À g 0 =NÞ½ðHuÞ 0 ðtÞe ÀHuðtÞ=N À ðHvÞ 0 ðtÞe ÀHvðtÞ=N and thus ð2=ð1 À g 0 =NÞÞQ 0 ðtÞ ¼ððHuÞ 0 ðtÞ À ðHvÞ 0 ðtÞÞðe ÀHuðtÞ=N þ e ÀHvðtÞ=N Þ þ ððHuÞ 0 ðtÞ þ ðHvÞ 0 ðtÞÞðe ÀHuðtÞ=N À e ÀHvðtÞ=N Þ: This implies by the elementary inequality je Ày À e Àz j 6 jy À zj (y; z P 0) that jQ 0 ðtÞj 6 jðHuÞ 0 ðtÞ À ðHvÞ 0 ðtÞj þ jðHuÞ 0 ðtÞ þ ðHvÞ 0 ðtÞj Á jHuðtÞ À HvðtÞj: ð5:3eÞ On the other hand, we have by (5.3c) that Before going to the proof of Theorem 2.1, we show first the conclusions in (2.9a)-(2.9c). By the well-posedness proved before, the non-decreasing a.c. function Fðt) is bounded by the population size N and such that F 0 2 L 1 loc ðR þ ). Using the monotonicity of F, we have gðtÞ ¼ Fðt À sÞW 0 ðsÞ ds 6 FðtÞWðtÞ 6 FðtÞ: On the other hand, by the equation g 0 ¼ Fð0ÞW 0 þ F 0 à W 0 (Fð0Þ ¼ g 0 ) again we find that g 0 ðtÞ P 0 for all t P 0. This implies by Eq. (2.1d) that RðtÞ ¼ Z t 0 g 0 ðt À sÞAðt; sÞ ds 6 Z t 0 g 0 ðt À sÞ ds ¼ gðtÞ; since 0 6 Aðt; sÞ 6 1 and gð0Þ ¼ 0. This proves Eq. (2.9a). The bounded and monotone function Fðt) has a finite limit g 1 as t ! 1. We have gðtÞ ¼ g 1 WðtÞ þ Z t 0 ðFðt À sÞ À g 1 ÞW 0 ðsÞ ds: It is routine to show that the above integral converges to 0 and thus gðtÞ ! g 1 as t ! 1. To prove the convergence RðtÞ ! g 1 as t ! 1, we note that RðtÞ ¼ gðtÞ À Z t 0 g 0 ðsÞð1 À Aðs; t À sÞÞ ds: It is routine to show, using the condition Aðs; t À sÞ ! 0 as t ! 1 (for each fixed s P 0, since Aðs; Á) is a CDF), that the above integral converges to 0 and thus RðtÞ ! g 1 as t ! 1. We now turn to the proof of Theorem 2.1. Case (i): R eff ðtÞ 6 R c < 1 for all t P t 0 . In this case we have (since gðtÞ ! g 1 as t ! 1) and thus, by Eqs. (5.4c and 5.4d), 1 À x P ð1 À aÞe Àgx P ð1 À aÞe ÀRcx : Using the elementary inequality e ÀRcx P 1 À R c x (x P 0) we find from the above that 1 À x P ð1 À aÞð1 À R c xÞ; which is equivalent to the inequality a P ½1 À ð1 À aÞR c x: In particular, we have which is the result (2.11b) of Theorem 2.1 (i). Case (ii): R eff ðtÞ P R 1 > 1 for all t P 0. In this case we choose t 0 ¼ 0 and find that g P This implies by (5.4c and 5.4d), by noting Hðt 0 Þ ¼ 0 with t 0 ¼ 0 and a > 0, that x > p, where p 2 ð0; 1) is the unique solution of the equation 1 À p ¼ e ÀpR 1 . This is the assertion in Theorem 2.1 (ii). Case (iii): R eff ðtÞ P 1 for all t P 0. In this case we choose t 0 ¼ 0 and find g P 1 g 1 Z 1 0 g 0 ðsÞ ds ¼ 1: It follows from (5.4c and 5.4d) (with Hðt 0 Þ ¼ 0 for the choice t 0 ¼ 0) that 1 À a ¼ ð1 À xÞe gx P ð1 À xÞe x P ð1 À xÞð1 þ xÞ; which yields that x P ffiffiffi a p . By the substitution of x ¼ g 1 =N and . This is the assertion in Theorem 2.1 (iii). h We first recall the following notations defined in Section 4: hðkÞ :¼ k=R 0 ðkÞ ðk P 0Þ; M :¼ maxfhðkÞ : k P 0g; ð5:5aÞ l :¼ max S with S :¼ fk P 0 : hðkÞ ¼ Mg: Moreover, we let uðx) (x P 1) be the inverse function of the strictly increasing function R 0 ðÁ). Proof of Proposition 4.3 (i): We prove the equivalent assertion stating that K P 0 is rare if and only if (4.12) has the unique solution F ¼ F R 0 ðKÞ . Clearly, if K P 0 is rare, then (4.12) has the unique solution F ¼ F R 0 ðKÞ . Assuming the uniqueness of solutions to (4.12) for a given K P 0, we want to show that K is rare in the sense of Definition 4.2. We consider a sequence (F n Þ nP1 of CDFs satisfying (4.10). Let (l Fn Þ nP1 be the sequence of the Borel measures induced by F n . By virtue of the theory of Prochorov [51] , it is easily shown that the uniform boundedness condition sup nP1 R 1 0 x dF n ðxÞ < 1 (see (4.10)) implies that the set fl Fn : n P 1g is relatively compact in the topology of weak convergence of measures, i.e., every subsequence of (F n Þ nP1 contains a weakly convergent subsequence. Hence, in order to show the weak convergence of (F n Þ nP1 to F, we may assume that the sequence (F n Þ nP1 itself is weakly convergent to some CDF, say G, in the sense that lim n!1 for all continuous and uniformly bounded functions f on R þ . We want to show that G ¼ F R 0 ðKÞ by proving that G is a solution of the inequalities (4.12). Consider the inverse function uðx) of the strictly increasing function R 0 ðk). Then we have uðxÞ=x ! 0 as x ! 1. It can be shown (details omitted) that the convergence uðxÞ=x ! 0 as x ! 1 combined with the boundedness sup nP1 R 1 0 x dF n ðxÞ < 1 implies that (5.6) holds also for the function f :¼ u. On the other hand, it is known (e.g., cf. Elstrodt [17] ) that (5.6) implies the inequality: Then a calculation yields H ¼ R 0 ðK) and kðH) is such that R 0 ðkðHÞ=aÞ ¼ m=a: By (5.7a) we have m=a ¼ R 0 ðK à ). Hence, R 0 ðkðHÞ=aÞ ¼ R 0 ðK à ) and thus kðHÞ ¼ aK à ¼ K; since the function R 0 ðÁ) is strictly increasing. The distribution H given by Eq. (5.7) is different from the point distribution F R 0 ðKÞ but satisfies the same inequalities (4.12). Therefore, K < l is typical by assertion (i). (iii): By (i), we need to show the uniqueness of the solution to (4.12) for K ¼ l. To this end, let G be a probability function satisfying (4.12) with K ¼ l, i.e., kðGÞ P l and G 6 R 0 ðl). Then we have, by using the equality l=R 0 ðlÞ ¼ M, that This implies that the composition G R 0 is a probability function that is concentrated at the unique point x ¼ l. Hence, we must have that G ¼ F R 0 ðlÞ . This is the desired result. (iv): Assuming R 0 ðk) to be strictly convex on R þ . We show that the function hðkÞ ¼ k=R 0 ðk) (k P 0) is strictly increasing for 0 6 k < l and strictly decreasing for k > l. Clearly, this implies that the set S ¼ fk P 0 : hðkÞ ¼ Mg is a singleton. Consider two points k 1 ; k 2 2 ½0; l) with k 1 < k 2 . Let a 2 ð0; 1) be such that k 2 ¼ ak 1 þ ð1 À aÞl: Then the strict convexity of R 0 ðÁ) implies that R 0 ðk 2 Þ < aR 0 ðk 1 Þ þ ð1 À aÞR 0 ðlÞ and thus, hðk 2 Þ > ak 1 þ ð1 À aÞl aR 0 ðk 1 Þ þ ð1 À aÞR 0 ðlÞ P min k 1 R 0 ðk 1 Þ ; . Thus, we have proved that hðk) is strictly increasing for k 2 ½0; l. The proof of that hðk) is strictly decreasing for k P l is similar and thus will be omitted. Next we show that every K > l is rare. To this end, let K > l be given and assume that G is a CDF that solves (4.12), i.e., kðGÞ ¼ x dGðxÞ 6 R 0 ðKÞ; Since R 0 ðÁ) is increasing, convex and R 1 0 dGðR 0 ðkÞÞ ¼ 1 À Gð1), we have by the Jensen inequality that ð1 À Gð1ÞÞR 0 ðK=ð1 À Gð1ÞÞ 6 x dGðxÞ: Equivalently, with y :¼ K=ð1 À Gð1Þ P K > l we have that hðyÞ P K=ðR 0 ðKÞ À Z 1 0 x dFðxÞÞ P K=R 0 ðKÞ ¼ hðKÞ: ð5:10Þ As shown before, the function hðk) is strictly increasing for all k > l. Hence, we must have y ¼ K, i.e., Gð1Þ ¼ 0. This yields which implies by the strict convexity of R 0 ðÁ) that G must be a point distribution and thus G ¼ F R 0 ðKÞ . For z > 0 we define the functions P z and e z by PðtÞ :¼ 1 À e Àt=z ; e z ðtÞ :¼ 1 z e Àt=z ðt P 0Þ: ð5:12aÞ The exponential function e z is the density function of the CDF P z . We consider an exponentially distributed transmission pattern u ! Pz v given by the CDF P z , i.e., vðtÞ ¼ Z t 0 uðsÞe z ðt À sÞ ds ðt P 0Þ: ð5:12bÞ By computing the derivative of v, we obtain that v 0 ðtÞ ¼ z À1 uðtÞ À z À1 Z t 0 uðsÞe z ðt À sÞ ds; which yields that zv 0 ðtÞ þ vðtÞ ¼ uðtÞ: ð5:12cÞ In short, the integral (convolution) equation (5.12b) is equivalent to the ordinary differential equation (ode) (5.12c). Let n be a positive integer. We call a CDF P on R þ to be n-staged exponentially distributed if it has the form P ¼ 1 à e j 1 à e j 2 à Á Á Á à e jn ð5:13aÞ with n positive constants j j > 0. Correspondingly, the transmission pattern u ! P v (i.e., v ¼ u à P 0 ) is called nÀstaged exponentially distributed if so is the CDF P. Let u ! P v be an n-staged exponentially distributed transmission pattern with a CDF P of the form (5.13a). Then we can decompose the pattern u ! P v into the composition of n exponentially distributed transmission pattern u 0 :¼ u ! Pj 1 u 1 ! Pj 2 u 2 Á Á Á ! Pj n u n :¼ v by setting u j :¼ u jÀ1 à e j j ðj ¼ 1; . . . ; nÞ: ð5:13bÞ As shown before, the system (5.13b) of integral equations can be translated into the following system of ordinary differential equations: j j u 0 j ðtÞ þ u j ðtÞ ¼ u jÀ1 ðtÞ ðj ¼ 1; . . . ; nÞ: ð5:13cÞ By defining D :¼ d dt to be differentiation, we see that the system (5.13c) can be simplified into one equation Y n j¼1 ð1 þ j j DÞuðtÞ ¼ vðtÞ; which is an n-th order ode for the function u. A consequence of the above observation is that our model Eqs. (2.1) in Section 2 can be translated equivalently to a system (or delayed system) of odes of the functions (F; g; R), provided that both transmission patterns E ! W I and I ! A R can be decomposed as the sums of multi-staged exponentially distributed patterns and delayed patterns. Below we compute some special cases. For this purpose, we rewrite Eq. (2.1) in the following more compact form: FðtÞ ¼ N À ðN À g 0 Þ expðÀð1=NÞHðtÞÞ with ð5:14aÞ H 0 ðtÞ ¼ rðtÞðgðtÞ À RðtÞÞ; Hð0Þ ¼ 0; ð5:14bÞ g ¼ W 0 à F and RðtÞ ¼ Z t 0 g 0 ðsÞAðs; t À sÞ ds: ð5:14cÞ We assume first that the latent pattern is 2Àstaged of the form: ð5:15aÞ where a; b are positives constants such that ð5:15bÞ is the mean latent period of the disease. Then the convolution equation g ¼ W 0 à F in (5.14c) is translated into the following ode: abg 00 ðtÞ þ sg 0 ðtÞ þ gðtÞ ¼ FðtÞ: We assume that the time-dependent recovery CDF Aðt; s) has the form (3.2d) which was used in Section 3 to study the eradication and control of outbreaks, i.e., we take Aðt; sÞ ¼ qðtÞ þ jðtÞA D ðsÞ þ ð1 À qðtÞ À jðtÞÞA 0 ðsÞ: ð5:16aÞ In the above, the function A 0 is the intrinsic recovery CDF, and qðt) is the time-varying proportion of blocked infectives. Moreover, the function jðt) is the proportion of infectives that are 'diagnosed' and subsequently blocked, and the function A D is the 'diagnosis' CDF. The mean value is the mean waiting time that an infective will be 'diagnosed'. We have that r D 6 r ¼ where r is the (intrinsic) mean infectious period of the disease. Under the choice (5.16a), the function Rðt) given in (5.14c) takes the form R ¼ R inst þ R D þ R uD with ð5:16bÞ R inst ¼ 1 à ðqg 0 Þ and ð5:16cÞ R D ¼ ðjg 0 Þ Ã A D ; R uD ¼ ðð1 À q À jÞg 0 Þ Ã A 0 : We assume finally that the intrinsic recovery CDF A 0 is 2-staged and the 'diagnosis' CDF A D is exponentially distributed: Summing up, we find that the original system (5.14) with 2-staged exponentially distributed transmission patterns is equivalent to the following second order non-autonomous ode system for the functions fg; R inst ; R D ; R uD ; Hg : abg 00 ðtÞ þ sg 0 ðtÞ þ gðtÞ ¼ N À ðN À g 0 Þe Àð1=NÞHðtÞ ; ð5:19aÞ H 0 ðtÞ ¼ rðtÞðgðtÞ À R inst ðtÞ À R D ðtÞ À R uD ðtÞÞ; The corresponding initial conditions are: Remember that all four numbers a; b; c; d are non-negative, and the sum s ¼ a þ b (resp., r ¼ c þ d) is the mean latent (resp., infectious) period of the infection. The number N is the (fixed) population size, g 0 < N is the initial epidemic size. Moreover, qðtÞ 2 ½0; 1 is the proportion of blocked infectives at t, jðtÞ 2 ½0; 1 is the proportion of 'diagnosed' infectives at t and r D is the mean 'diagnosis' time. Finally, RðtÞ ¼ R inst ðtÞ þ R D ðtÞ þ R uD ðt) is the cumulative number of removed infectives up to the time point t, and the value of R0ðt) can be equally thought of as the number of newly recovered (or newly detected/reported) infectives at time t. Under the choices (5.16a and 5.17a), the effective mean infectious period RðtÞ ¼ R 1 0 ð1 À Aðt; sÞÞ ds is RðtÞ ¼ jðtÞr D þ ð1 À qðtÞ À jðtÞÞr: ð5:20aÞ Assume that there exists a certain time t 0 P 0 such that the function rðt) is decreasing for all t P t 0 . Then the effective reproductive number R eff ðtÞ ¼ R 1 0 rðt þ sÞð1 À Aðt; sÞÞ ds can be estimated as follows: R eff ðtÞ 6 rðtÞ It follows from our theory (Theorem 2.1 (i), Section 2) that the epidemic course described by system (5.19) will be brought under control after time t 0 with a final epidemic size g 1 < Fðt 0 Þ=ð1 À R c ), where Fðt 0 ) is the cumulative number of exposures up to time t 0 . In the forthcoming works [29, 30] we will use model (5.19) or its variants to study numerically the 2002-2003 epidemic SARS and the vaccination strategy for emergency response to a biological terror attack. Infectious Diseases of Humans: Dynamics and Control Dynamically modeling SARS and other newly emerging respiratory illnesses: past, present, and future Control of Communicable Diseases Manual The epidemiology of Ebola hemorrhagic fever in Zaire Acute Hemorrhagic Conjunctivitis Prevention of Vaccine -Preventable Diseases, The Pink Book The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda Model parameters and outbreak control for SARS Characterization of an outbreak of acute hemorrhagic conjunctivitis in Mexico The role of spatial mixing in the spread of foot-and-mouth disease Comparative estimation of the reproduction number for pandemic influenza from daily case notification data Seasonal influenza in the United States, France and Australia: transmission and prospects for control Delaying the international spread of pandemic influenza Mathematical Epidemiology of Infectious Diseases: Model Building and 18 others, Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong Transmission potential of smallpox: estimates based on detailed data from an outbreak The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions Strategies for mitigating an influenza pandemic Modelling the 1985 influenza epidemic in France What makes an infectious disease outbreak controllable? Potential impact of antiviral drug use during influenza pandemic Transmission potential of smallpox in contemporary populations Estimability and interpretation of vaccine efficacy using frailty mixing models Study designs for evaluating different efficacy and effectiveness aspects of vaccine The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak Perspectives on the basic reproductive ratio The mathematics of infectious diseases Emergency response to a biological terror attack: mass vaccination (MV) or targeted vaccination (TV)? preprint A detailed study of the 2002-2003 SARS epidemic using a new SEIR model The impact of local heterogeneity on alternative control strategies for foot-and-mouth disease and nine others, Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape A contribution to the mathematical theory of epidemics The reemergence of Ebola hemorrhagic fever, Democratic Republic of the Congo The epidemiology of severe acute respiratory syndrome in the 2003 Hong Kong epidemic: an analysis of all 1755 patients and 11 others, Transmission dynamics and control of severe acute respiratory syndrome Viral dynamics of acute HIV-1 infection Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics, Theor Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods Superspreading and the effect of individual variation on disease emergence Containing the pandemic influenza with antiviral agents Infectious disease dynamics: what characterizes a successful invader? Transmissibility of 1918 pandemic influenza The 1918 influenza A epidemic in the city of Sao Paulo, Brazil Contact tracing in stochastic and deterministic epidemic models Time variations in the transmissibility of pandemic influenza in Prussia and 10 others, Viral dynamics of primary viremia and antiretroviral therapy in simian immunodeficiency virus infection Outbreak of Ebola hemorrhagic fever -Uganda HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time Mathematical analysis of HIV-1 dynamics in vivo Convergence of random processes and limit theorems in probability theory Evolutionary Integral Equations and Applications and 18 others, Transmission dynamics of the ethological agent of SARS in Hong Kong: impact of public heath interventions The distribution of incubation periods of infectious diseases Key transmission parameters of an institutional outbreak during the 1918 influenza pandemic estimated by mathematical modelling, Theor Avian influenza A virus (H7N7) in The Netherlands in 2003: course of the epidemic and effectiveness of control measures Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures Simulating the SARS outbreak in Beijing with limited data Appropriate models for the management of infectious diseases Predictions of the emergence of vaccineresistant hepatitis B in the Gambia using a mathematical model Appendix E. The classical SEIR model as a special case of the new model Eq. (2.1)We recall (Hethcote [28] , Anderson and May [1] ) that in the formulation of the classical SEIR model one takes SðtÞ; EðtÞ; IðtÞ; Rðt) to be the number of individuals that are at time t in the corresponding susceptibility, latency, infectiousness and recovery state. As observed in Section 2 (Eq. (2.2b)), both numbers Eðt) and Iðt) can be calculated by our cumulative numbers FðtÞ; gðt) and Rðt) as follows: We consider a special case of our model Eq. (2.1). We assume that both latency and recovery processes are exponentially distributed and the recovery process is time-independent, i.e., both CDFs W and A have the following forms On the other hand, by using (5.21) again, we have g 0 ðtÞ ¼ I 0 ðtÞ þ R 0 ðt) and FðtÞ ¼ gðtÞ þ Eðt). Substituting both relations into (5.24a) and using (5.25a), we obtain the following ode for the function IðtÞ :