key: cord-0997715-xw6rfuo9 authors: Rebhan, E. title: Modified SIR-model applied to covid-19, similarity solutions and projections to further development date: 2020-08-03 journal: nan DOI: 10.1101/2020.07.30.20165035 sha: a4b4ea62bf5dc61bff5237ab88cb99fe1398327a doc_id: 997715 cord_uid: xw6rfuo9 The SIR-model is adapted to the covid-19 pandemic through a modification that consists in making the basic reproduction number variable. Independent of it, another reproduction number is introduced, which is defined similarly to the usual net reproduction number. Due to its simple analytic form, it enables a clear interpretation for all values. A further parameter, provisionally called acceleration parameter, is introduced and applied, which enables a more differentiated characterization of the infection number dynamics. By a variable transformation the 3 equations of the modified SIR-model can be reduced to 2. The latter are solved up to ordinary integrations. The solutions are evaluated for current situations, yielding a pretty good match with the data reported. Encouraged by this, a variety of possible future developments is examined, including linear and exponential growth of the infection numbers as well as sub- and super-exponential growth. In particular, the behavior of the two reproduction numbers and the acceleration parameter is studied, which in some cases leads to surprising results. With regard to the number of unreported infections it is shown, that from the solution for a special one solutions for others can be derived by similarity transformations. In this paper it is shown that the SIR-model [1] can pretty well be adjusted to the data reported for the covid-19 pandemic by means of a modification, which consists in allowing a dynamic variability of the usually constant basis reproduction number R 0 . An additional reproduction number is introduced which is similarly defined as the sensitive R-number of the Robert Koch Institute (RKI) or the net reproduction rate. Another number turns out to be particularly useful, the values of which allow even more differentiated informations about the progression of the infection numbers. For it, the term acceleration parameter is proposed. First, by a suitable transformation of variables the system of 3 coupled nonlinear differential equations of the SIR-model is converted into a system with only 2 equations. The determination of the general solution of the latter is then carried on so far that only ordinary integrations remain. This makes it unnecessary to develop a numerical program for the solutions. Rather, the simplicity and clarity of the solutions obtained make it possible to quickly achieve results for concrete problems, especially when programs like MATHEMATICA are employed. With regard to the number of unreported infections it turns out, that despite the nonlinearity of the equations, from the solution for a special one the solutions for other numbers can be derived by similarity transformations under rather unobtrusive assumptions. In applying the solutions to the covid-19 pandemic, the reported data for one variable are replaced by a best least-squares fit function and thus integrated in the expanded SIR-model. Thereupon, from this the associated solution for a further variable is determined and compared with the corresponding reported data. It turns out that at least for more advanced states there is fairly good agreement. This encourages solutions to be explored, that continue preceding solutions to some extent into the future. Various assumptions about the further course of the infection or reproduction numbers are thus examined and compared. Also considered are the conditions under which the pandemic comes to an end. In the concluding section, some more qualitative aspects of the pandemic are discussed. In order to keep the health damage and death cases caused by the pandemic as low as possible, significant restrictions and unfamiliar actions are expected of the population. So that people understand why these are necessary, it is important that the reasons for them are communicated as precise and intelligible as possible. This in turn presupposes that the knowledge about the various aspects of the pandemic is as extensive as possible, while being precise and thorough. This article was written with the intention and hope of making a small contribution to this. It is mainly based on the system of SIR-equations which were taken from Wikipedia [2] . Therefore, only a few references are given. (2) where the following notations are used: S=susceptible individuals, I=infectious individuals, R=removed individuals (recovered with acquired immunity to the disease or deceased), N=(invariable) total number of individuals, t=time with the unit day, andṠ(t)=dS/dt etc. Furthermore, γ is the daily recovery rate of infected individuals, and β is the daily rate of new infections caused by an infected individual. The SIR-model is based on the assumptions that individuals can be infected only once, become contagious immediately after infection and remain so until they gain immunity or die. The time-dependent variables of the SIR-equations satisfy the relation We also use the basic reproduction number thereby omitting the subscript 0 of the usual notation R 0 in view of a modification specified further on. Clearly, the numerical solution of the above equations is not a problem. However, considerable simplifications are possible that make it easier to answer specific questions and enable a better understanding of solution properties. In a first step, the system of 3 equations (1)-(3) can be reduced to a system of 2 equations by introducing the variable which is also used in the infection diagrams of institutions determining the infection numbers like the Robert Koch Institute (RKI) or the Johns Hopkins University (JHU). Eliminating S and R by use of S=N−Z and R=Z−I (which follows from Eqs. (4) and (6)), and using β =γR (which follows from Eq. (5)), Eqs. (2)-(3) becomeİ With the definitions we switch from the extensive variables I and Z to the intensive variables a and z, thus finally ending up with the equationṡ With the help of Eqs. (4), (6) and (9), the solutions a(t) and z(t) of these equations immediately lead to S(t)=N(1−z(t)) and R(t)=N(z(t)−a(t)). Dividing Eq. (10) by Eq. (11), with a (z)=da/dz=ȧ(t)/ż(t) we obtain For fixed R this is a differential equation for a with the solution to the initial condition a(z 0 )=a 0 . We also want to consider situations in which a(t) and/or z(t) are predetermined by empirical data on the progress of the pandemic. In this case Eq. (13) cannot be expected to be a suitable solution. However, it turns out that the SIR-model can be adapted to the covid-19 pandemic by allowing the basic reproduction number to be variable. In practice, such changes are brought about by ordering measures to protect the population, such as quarantine, social distancing, hygienic washing of the hands or mandatory wearing of respirators, but also through sensible and responsible behavior of the population. Only experience can teach how this quantitatively affects R or the reproduction number R z introduced further down. Specifically, we introduce the replacement R → R(z), where R(z) is obtained by resolving Eq. (12) with respect to R, . Inserting this in Eq. (11) yieldṡ 4 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. . where t(z) is the inverse function of z(t).ż(t) is the temporal increase of z(t), which can be looked up in the daily records of the RKI [3] or JHU [4]. Integration of Eq. (15b) 1 yields with t(z)=t 0 for z=z 0 . (Eqs. (15)-(16) also apply to the case of fixed R.) Eq. (16) is the solution for given a(z). We also consider the case where t(z) is given and a solution for a(z) is sought. The differential equation from which a(z) must be determined is Eq. (15a), which withż(t)=1/t (z) can be converted into The solution to this equation is readily obtained by using the method of variation of constants and is where a * and z * are arbitrary fixed values in the range of possible aand z-values. Annotation: From a mathematical point of view, the modification R= const to R=R(z) in the SIR-model means that the system (10)-(11) of 2 equations is under-determined because it contains 3 unknown functions, z(t), a(t) and R(z). However, this is not a problem because it would not even be desirable for R(z) to be determined by the equations. In practical applications, there is the possibility of specifying any of the 3 functions, either by adaption to empirical data or by making assumptions about the future development The variable reduction number introduced in Eq. (14) does not allow a clear interpretation of its values, even though we will see later on that it has values and time variations similar to usual ones. For better comparison with the latter and to enable better interpretability, we introduce a second reproduction number that is similarly defined as the sensitive R-number of the RKI or the net reproduction rate. For this purpose we assume that a list of successive daily values Z n for the total number of infected individuals is given. First, a best least-squares fit function z(t) [5] to the corresponding data {t n , z n }, n=0, 1, 2, . . . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. . https://doi.org/10.1101/2020.07.30.20165035 doi: medRxiv preprint in many cases can be done using a power series z(t)= ∑ N 0 a n t n . (In this way, the data is smoothed, as is also done in a similar way in the usual calculation of R values by averaging). The current increase in z(t) isż(t) ∆t, and furthermorė holds. With this, we define our new reproduction factor as We still relate it to the time unit day by putting ∆t=1, and after a slight reshaping we finally get (v z is the growth velocity of z(t).) On cursory inspection, it looks like R z (t) is an instantaneous value related only to the moment t. Contrary to this, in the calculation of the sensitive R-value of the RKI, due to a time delay in the reported infection data past values are included, and the data are smoothed by averaging over several days to compensate for statistical fluctuations. On closer inspection, however, this also applies to our R z (t) because to determine the function z(t) via a best least-squares fit to a data set {t n , z n }, past data are included as well as smoothing takes place. For the R-number of RKI and the usual net R-number the value 1 is easy to understand, because on average every infected individual infects another one during its contagious stage, whence the number of newly infected individuals remains constant. For values = 1 a similarly simple explanation is not possible, except that a deviation downwards or upwards is more favorable or unfavorable the larger it is. For our R z -number the situation is slightly better, because Eq. (18) can be used to quantify the conditions under which a deviation increases or decreases: the greater the relative accelerationv z /v z , the greater the deviation from 1 where R z (t)≡1 for constantż(t) as above. It would even be possible to use the number as another number for characterizing the state of the pandemic. However, in the next section it is shown that a modification of Q(t) is a much better choice. We would like to apply our reproduction number also where not z(t) is specified but its inverse t(z). For this purpose, we differentiate the identity t (z)ż(t) = 6 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. . For the case of given a(z) addressed in Section 3.1.1, we still determine how R(z) can be expressed in terms of a(z). From Eq. (11) with R → R(z) we geẗ Dividing byż(t) and using which follows from Eq. (14), we get (21) γ must be determined so that R z (z 0 ) coincides with the R z (t 0 ) of Eq. (18). The terms linear and exponential growth are often used to characterize the increase in the number of infections, the net reproduction number being used to distinguish them. As we will see, this is a rather vague and possibly misleading characterization. In the following, another parameter is proposed that enables a more differentiated assessment. Combining Eq. (18) and Eq. (19) yields from which follows that R z and Q offer the same information content. The acceleration parameter proposed by us consists in a modification of Q=z/ż and is The relationship between A(t) and Q(t) shows that A(t) contains more information than Q(t). 7 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. . Now, we first calculate A(t) for the case of exponential growth, z(t)=z 0 e α t . . For other growth patterns we expand z(t) around an arbitrary z 0 =z(t 0 ) according to Let us now consider different values of A 0 for the same values of z 0 and v 0 . In the case of exponential growth we have v 0 =α z 0 , A 0 =α 2 z 0 and from this is curved stronger upward than an exponential curve of the same slope through the same point z 0 and therefore (locally) shows a super-exponential growth. For A 0 0 we have δ >0, R z (0)=1/(1−1/δ )>1 and R z (0)>R z (t)>1. Asymptotically we get R z (t) In spite of an accelerated increase of z(t) the reproduction number R z decreases. This illustrates that the specification of a reproduction number, R z in our case, is not sufficient to characterize the pandemic. For α<0 or δ <0 resp. we can write whence R z (t)1, the growth becomes super-exponential, and almost linear around day 60. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. . https://doi.org/10.1101/2020.07.30.20165035 doi: medRxiv preprint We have seen that the modified SIR-model is well applicable to the pandemic, although in some cases rather only to its more advanced stage. We therefore assume that it can also be applied to future developments to a certain extent. Of particular interest is how a(t) and z(t) develop when certain assumptions about the further course of R(t), R z (t) or A(t) are made. Predefined R(z): In this case, from Eq. (10) results the equation which is solved by The associated solution z(t) is given by its inverse function t(z) to be calculated from Eq. (16). For γ, the value of the preceding solution can be used, to which the forecast solution is linked. Predefined R z (t): For given R z (t), from Eq. (18) results the equatioṅ and integration ofż(t)=v z (t) yields with v z (τ) given by Eq. (49). For determining the associated function a(t), we draw on Eq. (15a) in the form a (z) + γa/ż(t) = a (z) + γa/v z (t) = 1 . which is solved by 17 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. R z (t) oscillations around R z =1: In practice, after the number of newly infected individuals has dropped to a lower level, it is variously observed that the net reproduction number is close to 1 and alternately exceeds this value and falls below it. We simulate this situation by making the ansatz R z (t) = 1 + ε sin(ω t) . (52) According to this and Eq. (49) the relative rate of new infections iṡ The integral in the exponent can indeed be expressed through analytic functions, It is, however, rather tedious to get the various branches of the trigonometric functions aligned correctly adjacent to each other. We have therefore evaluated Eq. (53) numerically. The result is shown in Fig. 5 . The surprising outcome is that v z (t) gradually decreases. The upward exceedings of limit 1 by v z are therefore harmless if they are compensated for by undershoots of the same duration and strength. 18 . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. Case of constant R: In this case Eq. (13) applies, and for a=0 we obtain With the definitions and by transition from the logarithm to the exponential function this results in The resolution of this equation with respect to W is called Lambert-W-function, omega function or product logarithm. It has several branches, of which the one we need is called the principal solution W 0 (V ). Thus, in terms of W and V the solution of our problem is W =W 0 (V ). Returning to our original variables by use of Eqs. (54) and (55b), for the value z=z f , at which the pandemic ends, we get Case of constant R z : According to Eq. (18) for constant R z we geṫ and from this v z (t) = v 0 e α t with v 0 ≥ 0, because z(t) cannot decrease. This means that the constancy of R z implies exponential growth of v z (t). According to Eq. (11) the condition a=0 for ending the pandemic calls forż(t)=v z =0 which is not possible with α ≥ 0, and for α<0 this is only achieved after an infinitely long time. With v z =ż(t) from Eq. (58) results the differential equationż(t)=v 0 e α t which is solved by According to the termination condition v z =0, from this and Eq. (58) we obtain The final value z f that z(t) assumes at the end of the pandemic remains below its maximum 1 only for |α|>v 0 /(1 − z 0 ). . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted August 3, 2020. . https://doi.org/10.1101/2020.07.30.20165035 doi: medRxiv preprint By many, the exponential growth in infection numbers is considered as the worst of all possibilities, and it is assumed to occur immediately when the net reproduction number exceeds 1. The first assumption overlooks the fact that the term exponential growth does not represent a complete and adequate characterization, because there is slow and fast exponential growth (e α t with a small or large α) as there can be slow and fast linear growth (ż(t)=α with a small or large α). Accordingly, rapid linear growth can be worse than slow exponential growth over a long period of time. So the growth of Z(t) in the world was almost linear from the beginning of April to the middle of May, but unfortunately at very high speed. That the second assumption is wrong can be seen from Eqs. (25)-(26). According to them A(t)=0 for R z (t)=1. If R z (t) were to become a little larger from this value, then A(t) would have to jump to 1 immediately for exponential growth, which is impossible according to Eq. (26). 4 As we have seen, the modified SIR-model allows for all growth possibilities, from sub-linear up to super-exponential. How dangerous the current growth is depends, however, not only on its type as characterized by A(t), but also on its current growth rate v z , and on how long this kind of growth will last. A question, that has been asked many times and the answer to which is important for the acceptance of restrictions that must be endured to contain the pandemic, is: Why do the latter still have to be maintained when the rate of new infections (i.e. v z ) has become sufficiently low. The answer to this is not easy, but there is an analogy familiar to all of us which maybe contributes to some understanding. Imagine driving a car and pressing the accelerator halfway to maintain a certain speed. If you want to drive faster, you have to push it further, e.g. three quarters. If you would release the pedal now, you would fall back to lower speed. In a similar way, you either have to maintain the measures undertaken for keeping low the number of new infections, or to replace them with equally effective, but more targeted and less restrictive measures. Note that this analogy is of a more symbolic kind, because neither frictional forces nor inertia and energy supply can be attributed to the pandemic. At most, in the case R z <1 or A(t)<0 a constant negative accelerationv z (t)=α<0 (treated at the end of Section 2.5) can to a limited extent be compared with the acceleration of a car. However, according to Eq. (5), independently of α the final velocity v z is always zero due to the lack of friction. The fact that it is so difficult to bring and keep R z well below 1 or A(t) far below zero can possibly be explained as follows: Either the effort to maintain a negative acceleration increases with decreasing v z (t), or else, there are processes that work like negative friction. A third mechanism is discussed in the next paragraph. When looking at the infection rates which the JHU releases for the different countries of the world, one notices that for those who have weathered the crisis rather well (e.g. Germany, Austria, Japan, South Korea or New Zealand),ż(t) does not decrease completely down to zero, but only to a low and almost constant level. One reason for this could also be, that the different countries do not form closed systems, so that due to the pandemic nature of the covid-19 crisis infections are always brought in from outside. Since the SIR-model only applies to closed systems, this factor cannot be treated with it. That would be possible for the world as a whole, but the latter is still far from low infection rates. A contribution to the mathematical theory of epidemics Mathematica: A System for Doing Mathematics by Computer Mathematica: A System for Doing Mathematics by Computer Acknowledgements: For calculations and plots MATHEMATICA 10 was used.