key: cord-0973794-kypccuip authors: dos Reis, E. V. M.; Savi, M. A. title: A Dynamical Map to Describe Covid-19 Epidemics date: 2021-03-12 journal: nan DOI: 10.1101/2021.03.10.21253322 sha: 5a618f0af3ec47e8dd16074307353d88874bbb6f doc_id: 973794 cord_uid: kypccuip This paper proposes a dynamical map to describe COVID-19 epidemics based on the classical susceptible-exposed-infected-recovered (SEIR) model. The novel map represents Covid-19 discrete-time dynamics standing for the infected, cumulative infected and vaccinated populations. The simplicity of the discrete description allows the analytical calculation of useful information to evaluate the epidemic stage and to support decision making. In this regard, it should be pointed out the estimation of the number death cases and the herd immunization point. Numerical simulations show the model capacity to describe Covid-19 dynamics properly representing real data and describing different scenario patterns. Real data of Germany, Italy and Brazil are of concern to verify the model ability to describe Covid-19 dynamics. The model showed to be useful to describe the epidemic evolution and the effect of vaccination, being able to predict different pandemic scenarios. The novel coronavirus disease (Covid-19) has been a major subject of research since it was characterized as pandemic by the World Health Organization in March 11th 2020, receiving a lot of attention of scientists in different areas of human knowledge. Covid-19 dynamics is one of these subjects that has an increasing interest due the possibility to understand its evolution and to establish proper health strategy plans. Dynamical perspective is an interesting approach to deal with biomedical systems [1] . The literature presents several approaches regarding dynamical epidemic models [2] , generally employed An alternative to continuous models can be established by maps, that are discrete-time and governed by a system of algebraic equations. Dynamical maps have advantages due to their simplicity. Alonso-Quesada et al. [18] pointed out that the use of discrete-time instead of continuous-time models is preferred since the amount of necessary computation effort can be considerably reduced. Since epidemic statistics take place on fixed time intervals, it makes easier to parameterize a discretetime than a continuous-time epidemic model. Enatsu et al. [19] stated that there are situations that constructing discrete epidemic models are more appropriate to understand disease transmission dynamics and to evaluate eradication policies since they permit arbitrary time-step units, preserving the basic features of corresponding continuous-time models. Kwon and Jung [20] employed a discrete version of SEIR model to characterize the spread of coronavirus MERS in Korea, showing that an effective quarentine plan would reduce the maximum number of infected population by about 69% and MERS fade-out period may be shortened by about 30%. Din [21] analyzed the global stability analysis of the equilibrium points in the discrete-time form of SIR model. Enatsu et al. [19] used a backward differential scheme to discretize a class of SIR differential models showing that the effect of discretization is harmless to the global stability of the epidemic equilibrium. Alonso-Quesada et al. [18] proposed another method to discretize the SEIR method also considering natural births, deaths and reinfection. Cui et al. [22] built a discretized version of the SIR model using a nonstandard finite difference scheme. The model was applied to childhood diseases carrying out the vaccination effect of new borns, showing that the discrete and continuous models have the same equilibrium points. All these models are discrete versions of the continuous models. The effect of vaccination is often considered in epidemic models in order describe the disease spread control. Different vaccination strategies can be imagined for each disease and their models consider that vaccination rate can be a function of either time or the susceptible population. Pulse vaccination is one kind of strategy characterized to be periodic in time [23, 24] . On the other hand, continuous vaccination strategy is the alternative that is descbribed by Gumel et al. [25] that showed the potential impact of SARS vaccine over the pandemic that spread to over 32 countries in 2003. Alexander et al. [26] built a model to study the transmission of influenza virus, computing the threshold vaccination rate necessary for community wide control. Kabir and Tanimoto [27] considered the effects of information buzz and information costs on the vaccination effect. Other references also discussed the effects of this kind of vaccination [28, 29] . 3 This work deals with a novel Covid-19 dynamical map that represents the epidemics from infected, cumulative infected and vaccinated populations. The discrete-time model is developed based on the SEIRV model that considers susceptible-exposed-infected-removed-vaccinated populations. The novel map reduces the six coupled ordinary differential equation into three algebraic equations, being capable to capture the main features of the Covid-19 epidemics. A model verification is established using real data from Covid-19 pandemic in Germany, Italy and Brazil, showing good agreements. Different scenarios are investigated showing a multi-wave pattern. Due to the simplicity of the model, it is possible to obtain analytic expressions regarding the infectious ratio and the herd immunization point. Stability analysis is performed showing the conditions to control the epidemic spread. The effect of vaccination is carried out showing its importance to reduce the number of deaths. This work has the main goal to develop a dynamical map to describe Covid-19 epidemics based on the classical SEIR framework. An extra population is incorporated considering the effect of vaccination, defining an SEIRV model, which considers the following populations: susceptible, S; exposed, E; active infected, I; removed, R; and vaccinated, V . In addition, the cumulative infected population C is incorporated as a useful model information. On this basis, the governing equations are defined as follows.Ṡ where dot represents time derivative; β is the transmission rate that is directly associated with social isolation; σ −1 is the mean latent period; γ −1 is the infectious period; and υ is the vaccination rate. It should be pointed out that dimensionless variables are considered, and therefore, (S, E, I, R, V ) ∈ [0, 1] and S + E + I + R + V = 1. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted March 12, 2021 The map is derived from these differential equations adopting some basic assumptions. By considering each one of the populations, represented by X, its time derivative is:Ẋ = lim ∆t→0 (X(t + ∆t) − X(t))/∆t, which means thatẊ ≈ X n+1 − X n if ∆t = 1 and n representing the n-th day. Under this assumption, the following steps are followed to build the dynamical map: 1. Substituting Eq. (1b) into Eq. (1c), one obtains:İ = βSI −Ė − γI. 2. It is assumed that the ratio E/I assumed in the beginning of the outbreak is kept constant throughout the whole epidemic period, which means that E = ΛI andĖ = Λİ. Substituting both Eq. (1b) and Eq. (1c) into it, and assuming S = 1, it yields Λ = γ − σ + (γ − σ) 2 + 4βσ /2σ. I i , together with step 2 assumption (E/I =Ė/İ = Λ) into S + E + I + R + V = 1, one can write the susceptible group S as a function of I: for the recovered group during the beginning of the outbreak. 4. Substituting Eq. (1f) into Eq. (1c), one obtainsĊ =İ + γI. Therefore, C n+1 = C n + I n+1 + (γ − 1)I n . 5. By considering a generic n, summing all the time steps, one can find that γ n i=0 I i = C n+1 − I n+1 , where it is assumed that I 0 ≈ C 0 , since the summation must consider all the active cases from the beginning of the outbreak. After this sequence, it is possible to isolate both I n+1 and C n+1 to present the Covid-19 map as follows where Λ = E/I is a constant estimated by a parametric condition (β, σ, γ). It is important to highlight that, despite this novel map is employed herein to describe Covid-19 dynamics, it can 5 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) also be employed to describe the dynamics of any other epidemics. The vaccination rate is assumed to be a function of the infected and vaccinated populations, υ = υ(V, C). Since 0 ≤ V ≤ 1, it is imposed the constraints 0 ≤ υ ≤ 1 and υ(C + V = 1) = 0. The population composed by the sum C + V constitutes the individuals that cannot become infected. Hence, neglecting the possibility of reinfection, the higher is the number of infectedvaccinated, the less is the number of susceptible individuals that can be infected. Furthermore, the absence of vaccination (V = 0) reduces the map to a two-population dynamics I-C. The total number of deaths, D, can be estimated based on the infected population. Therefore, the cumulative number of deaths is expressed by where µ is the death rate, usually around 2%. It should be pointed out that the current number of deaths can be determined by the difference D n − D n−1 . The transmission rate β is the critical parameter to characterize the Covid-19 dynamics, being related to social isolation and virus contagious capacity. Virus variants can be represented by changes on this parameters considering that a more contagious variant increases the value of the transmission rate for the same level of social isolation. Nevertheless, virus variants are not in the scope of this contribution and therefore, transmission rate is directly related to social isolation. Since the social isolation is clearly time dependent, it is convenient to define β = β(n). This time dependence can be established from an adjustment with real data, defining a proper fit. An interesting approach to match real data is the use step functions, defined as follows by Eq. (5) for m steps and presented in Fig. 1 . The other parameters, σ and γ, usually assumes typical values for Covid-19 dynamics [12, 14, 30] : σ = 1/3 and γ = 1/5. These parameters are employed in all simulations except when mentioned otherwise. 6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The stability of the map can be evaluated from the definition of the infectious rate r, defined as follows based on the ratio between two subsequent iterations, The infectious increases in a specific time if r n > 1, and decreases otherwise, if 0 < r n < 1. The case r n = 1 is the transition between both conditions. By taking the active infected given by Eq. (2a), one can obtain the ratio r n as a function of I n and (C n + V n ). Consider the system state space I-(C + V ) showing the curves standing for Eq. (7) with r = 1. This map is representative of the phase space being presented in Fig. 2a for various values of β and constant values of σ and γ. Since the curves are related to r = 1, the region below each curve is associated with values of r > 1 while above the curve is related to r < 1. Therefore, the region below the curve is associated with a growth of active cases. The peak of I occurs when r = 1 is reached. Fig. 2a allows one to obtain the number of total infected required to prevent the increase in the number of active cases regardless the number of infected or vaccinated. In other words, when the sum C + V reaches a critical value, the infected population I necessarily decreases. The herd immunization point, P h , is defined when this critical situation is achieved and I = 0 (Fig. 2b) . . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The value of the herd immunization point P h can be analytically defined as a function of the parameters (β, σ, γ). By considering r n = 1 and I n = 0 in Eq. (7) and after some algebraic manipulation, one obtains the following expression Note that this is a function of the transmission rate β and the infectious period γ −1 , being not dependent on the latent period σ −1 . Moreover, it is noticeable that an increase of the transmission rate β results in a higher value of P h . In other words, the higher is the transmission rate, the bigger is the infected-vaccinated population needed to achieve the herd immunization point. In the limit β → ∞, it yields P h → 1, as expected. This means that, in order to have r n < 1, it is necessary that 100% of the population becomes either infected or vaccinated. Finally, making P h = 0, one obtains β = γ, which means that for any β < γ the number of active cases necessarily decreases regardless the number of I or (C + V ). Therefore, the higher is the average infectious period, given by γ −1 , the lower is the transmission rate coefficient required. In order to present an interpretation of the Covid-19 map and the herd immunization point, consider the subspace (I n+1 -I n ), which is a function of (β, σ, γ, C), neglecting the vaccination effect. By assuming constant values of σ and γ, the map can be observed as a function of (β, C). Fig. 3 presents the influence of these parameters on the map curve showing a parabola that reduces its maximum value with the increase of either β or C. The dashed curve represents I n+1 = I n that is the 8 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint region where r = 1. Therefore, equilibrium points and their stability define the herd immunization point. Note that the fixed point (Ī,C) = (0, C) is stable if dI n+1 /dI n < 1. Thus, the threshold point P h is calculated for dI n+1 /dI n = 1, which results in that is the same result of Eq. (8), The stability can be observed from Fig. 4 that shows two different scenarios employing different parameters. The first scenario is related to an unstable case with β = 0.5, and the other one is a stable case with β = 0.05. Different initial conditions are of concern in order to illustrate the system evolution. In the sequence, the Covid-19 map is applied to describe the epidemic dynamics using real data from Germany, Italy and Brazil as references. The novel Covid-19 map is now employed to perform a dynamical analysis of the pandemic. Real data from the novel Covid-19 epidemics in Germany, Italy and Brazil are employed as reference 9 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Covid-19 map is able to represent the bell shape behavior by considering proper parameters. Based on this, consider a limit period of the real data given by the first 120 days. Fig. 6 presents results of the map simulations compared with real data, showing a good agreement between them. Transmission rate is described by step functions adjusted for each country: β ger for Germany, β ita for Italy and β bra for Brazil. These functions are displayed in Eq. (11), where n = 0 yields 6th 10 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. March 2020. In addition to real data, model verification establishes a comparison of the Covid-19 map with the SEIRV model that is integrated employing the fourth-order Runge-Kutta method with a 10 −2 time step. The vaccination effect is neglected in this stage, adopting V = υ = 0. Analytic considerations and an explicit discrete-continuous comparison are of concern. Due to the bell shape characteristics of active cases, three different aspects characterize an outbreak: the peak of the active cases curve, I max ; the time instant where the peak occurs, t max ; and the area below the curve, which is proportional to the total infected after the outbreak C(t → ∞). This latter characteristic can be confirmed by analyzing the SEIR model, substituting Eq. (1e) into 11 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Eq. (1c) and integrating from the beginning of the outbreak to its end, which gives and therefore, These three aspects can be used to build the model signature in 3D charts (I max -t max -C(t → ∞)). The variation of the three parameters (β, σ, γ) generates a solid object in the 3D chart that can be understood as the model signature. In order to facilitate the signature view, the 3D chart is split into two 2D maps. Additionally, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint σ and γ are assumed to be constant and β is free to vary. On this basis, a curve belonging to the model signature is obtained characterizing a scenario. Four different combinations for (σ, γ) are picked to compare the novel map and the SEIR model, as presented in Figs. 7 and 8 considering C 0 = I 0 = 10 −5 . In general, it is noticeable that the signature predicted by both models are in close agreement considering the total cases against t max . This convergence takes place regardless the parametric condition. On the other hand, Fig. 7 shows that the signature of both models diverge for bigger values of β, being the map less sensitive to β variation. Besides, the same effect is obtained with the increase of γ −1 . Nevertheless, there is a region where both models diverge from each other. This divergence does not imply that both models cannot be used to model the evolution of an outbreak, representing just a quantitative divergence. Moreover, within the convergence region, the same β value does not necessarily generate the same combination (I max , t max , C(t → ∞)) for both models. Therefore, there is a good qualitative agreement between the novel map and the classical SEIR model. β ranged from γ to 1.5. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The influence of the parameters on the map dynamics is now of concern. Fig. 9 presents the evolution of the outbreak simulation through the active cases I and the total cases C for different parameters. Fig. 9a presents the influence of transmission rate β showing that the peak of active cases is bigger and takes place sooner with the increase of the transmission rate. Additionally, the total number of infected increases as β increases. The influence of the mean latent period σ −1 is presented in Fig. 9b showing that the higher is the mean latent period, the later the peak of active cases takes place and the smaller is the peak reached by I. The total cases reached the same vale regardless of the value of σ. The variation of the latent period can change the course throughout time of I, but it does not influence the total number of infected. Finally, Fig. 9c presents the influence of the mean infectious period γ −1 on the outbreak evolution. The longer is the infectious period, the more time is available to someone infected to spread the virus among the others. Hence, the peak of active cases is higher and the total cases C is higher as well. 14 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint Based on epidemic real data, it is clear that Covid-19 dynamics has a multi-wave bell shape pattern. The previous section presented results showing the capability of the map to describe a single bell shape. This section is dedicated to evaluate the multi-wave pattern. Variations of the transmission rate is the most important parameter to capture this multi-wave behavior. In this regard, consider an epidemic scenario where β is described by the following step function, illustrated by Fig. 10 . This strategy is now employed to verify the capability of the Covid-19 map to represent real data. Once again, Covid-19 epidemic in Germany, Italy and Brazil are employed as reference. The comparison among numerical simulations and real data is now carried out for the whole period range. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint Transmission rate is defined by step functions presented in Eq. 15. Fig. 13 presents the comparison between numerical simulations and real data showing a good agreement for all countries. Based on that, it is possible to conclude that the Covid-19 map captures real data including the multi-wave scenario. The main difficulty is the proper determination of the transmission rate. 18 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The novel Covid-19 map is now employed to perform a dynamical analysis of the pandemic. As mentioned before, the transmission rate is the critical point for a proper description of the pandemic evolution. An approach to represent the transmission rate is to define a function of time β = β(n) that is adjusted from real data employing Eq. (2a). Based on real data time series, the Newton's method is employed to calculate β(n) at each time step for each country. Fig. 14 presents the estimated transmission rate, β(n), together with the infected active cases I showing that the real data is reproduced accordingly. It is observed that transmission rate are bigger during the first part of the outbreak. The reduction that follows is associated with social isolation policies. It should be pointed out that the growth of active cases are related to periods where β is bigger than ≈ 0.2, the 19 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. An extrapolation for future values are now of concern in order to evaluate the outbreak evolution on the state space I-C. The critical transmission rates is obtained for r = 1 (employing Eq. (8)) at the last day of the period, yielding β cr = 0.206 for Gemany, 0.210 for Italy and 0.209 for Brazil. Note that transmission rate values below β cr leads to the reduction of the number of active cases. Otherwise, if the transmission rate is bigger than β cr , the cases tend to increase. On this basis, transmission rate is the essential parameter to control the pandemic evolution, being possible to 20 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The analysis of deaths from Covid-19 infection is now of concern. The death ratio µ stands for the ratio between total cases and total deaths. Since the number of cases is directly associated with the transmission rate, it is expected that this ratio is a function of time, µ = µ(n). Fig. 16 presents µ adjusted from real data considering Germany, Italy and Brazil. Note that this ratio The relationship between transmission rate and herd immunization can be established by considering the number of total infected required to maintain the infectious rate less than one (r < 1). Eq. (7) can be used in order to build the herd immunization threshold point P h as a function of β (Fig. 17) . The average of transmission rate values during the first seven days of the period range is 0.91 for Germany, 0.62 for Italy and 1.26 for Brazil. Based on these values, the herd immunization point P h is achieved only with total infected cases C above 0.78, 0.68 and 0.84, respectively, as depicted in Fig. 17 . This scenario suggests the only way to avoid a high number of total infected without the need to maintain social isolation is employing population vaccination. Therefore the sum C + V can be high enough in order to allow a high β coefficient. The effect of vaccination is explored in the following section. This section investigates the effect of vaccination on the Covid-19 dynamics. It is adopted a situation where the vaccination starts on the last day of the real data period: 21st January 2021. Numerical simulations are carried out showing the outbreak evolution from this point on, assuming different vaccination conditions associated with three distinct rates: υ = 0, 10 −3 and 10 −2 . Note that the first case is actually a situation without vaccination process. Furthermore, the transmission rate employed is the mean value from β(n) of the last 90 days of the period range (see Fig. 14) which yields: β ger = 0.227, β ita = 0.230, β bra = 0.224. Fig. 18 presents the effect of vaccination 22 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. on the evolution of the epidemics for each country, showing that vaccination has a huge impact on the Covid-19 dynamics. The higher the vaccination rate is, the lower is the peak reached by active cases and the lower is the total infected population after the epidemic period. Moreover, it should be noticed that the time required to achieve, for instance, I = 10 −4 -one infected individual for each ten thousand inhabitants, takes place sooner for higher vaccination rates. As expected, the absence of vaccination results in the worst scenario. These conclusions can be drawn for all the three countries. Fig. 19 evaluates the vaccination effect by the map I-(C + V ). The curve standing for infectious rate r = 1 is also plotted, yielding the peak reached in active cases. An important conclusion is that vaccination is the only possibility to make P h increase without increasing C. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) An interesting point that can be evaluated is the influence of vaccination rate υ on the total infected population C(t → ∞). Fig. 20 presents this analysis where one can see that the higher number of cases is achieved with the absence of vaccination (υ = 0). Moreover, the sensitivity of total infected with respect to vaccination rate is given by the curve slope. The higher sensitivity occurs for lower vaccination rates. The effect of vaccination is summarized in Table 1 showing the total infected population, total deaths and the date when the infected population reaches I = 10 −4 for the three vaccination rates. The death rate µ is adopted based on the average of the final 90 days: 1.86% for Germany, 3.95% for Italy, and 2.68% for Brazil. Without vaccination, the total infected population reaches 24.75%, 29.27% and 22.83% in Germany, Italy and Brazil, respectively. If the vaccination is implemented with 10 −2 rate, the number of total deaths can drop to 35%, 67% and 59% in these three countries, respectively. The estimated date to reach I = 10 −4 shows that the vaccination shortened this period in several months. Therefore, the vaccination drastically anticipates the end of a huge crisis. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted March 12, 2021. ; https://doi.org/10.1101/2021.03.10.21253322 doi: medRxiv preprint . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 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