key: cord-0926271-6vaqlhtj authors: Webb, G. F.; Fitzgibbon, W. E.; Morgan, J. J.; Wu, Y. title: Predicting the end-stage of the COVID-19 epidemic in Brazil date: 2020-05-29 journal: nan DOI: 10.1101/2020.05.28.20116103 sha: 9fa42de425f47a91d240ef6f3a95a80ef62614b6 doc_id: 926271 cord_uid: 6vaqlhtj We develop a dynamic model of a COVID-19 epidemic as a system of differential equations. The model incorporates an asymptomatic infectious stage and a symptomatic infectious stage. We apply the model to the current COVID-19 epidemic in Brazil. We compare the model output to current epidemic data, and project forward in time possible end-stages of the epidemic in Brazil. The model emphasizes the importance of reducing asymptomatic infections in controlling the epidemic. The epidemic population is divided into three classes: a susceptible class S, and two infected classes I 1 and I 2 , representing the asymptomatic (or low level symptomatic) infectious individuals, and symptomatic (or high level symptomatic) infectious individuals, respectively. Susceptible individuals become asymptomatically infected via contact with either asymptomatically infectious or symptomatically infectious individuals. The transmission process is modelled by the force of infection term S (τ 1 I 1 + τ 2 I 2 ), which constitutes a loss rate for the susceptible class and a gain rate for the asymptomatic infected class. Asymptomatic individuals become symptomatic at rate λ. Symptomatic individuals are removed at rate γ, due to recovery, isolation, mortality, or other reasons. These elements lead to the following system of ordinary differential equations: S (t) = −S(t) (τ 1 I 1 (t) + τ 2 I 2 (t)) , t ≥ 0, I 1 (t) = S(t) (τ 1 I 1 (t) + τ 2 I 2 (t)) − λI 1 (t), t ≥ 0, (1.1) I 2 (t) = λI 1 (t) − γI 2 (t), t ≥ 0, S(0) > 0, I 1 (0), I 2 (0) ≥ 0. A flow diagram of the model is given in Figure 1 . We remark that a model similar to (1.1) is developed in [10, 11, 12, 13] , where I 2 is further differentiated into reported and unreported classes. The global dynamics of (1.1) are given in the following theorem: This result is proved in greater generality in [9] , in a setting with S, I 1 and I 2 spatially dependent, and their equations replaced by reaction-diffusion equations over a geographical domain. For the spatially homogeneous case in Theorem 2.1, we can express S ∞ in terms of the initial conditions and parameters in the model (1.1): Corollary 1.2 S ∞ satisfies the following equation: Proof. From the first equation in (1.1) we obtain From the sum of the first, second, and third equation in (1.1) we obtain From the third equation in (1.1) we obtain (1. We apply this result to the COVID-19 epidemic in Brazil. Other models of the COVID-19 epidemic in Brazil are in [1, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16, 17, 18] . Currently, the epidemic in Brazil is in a rapid growth phase, with limited social distancing measures in effect, and limited compliance with these measures. We use the cumulative daily reported cases data for this epidemic from the Ministério da Saúde of Brazil (https://coronavirus.saude.gov.br/): 17 We set the time units to days. We view I 1 (t) as the population of asymptomatic or low level symptomatic infectious individuals at time t. We view I 2 (t) as the population of high level symptomatic infectious individuals at time t. We set S(0) = 210, 000, 000, the current population of Brazil. We set the loss rate λ of I 1 (t) to 1/7 per day, which means I 1 infectiousness lasts 7 days on average. We set the loss rate γ of I 2 (t) to 1/7 per day, which means I 2 infectiousness lasts 7 days on average. We identify an interval of time 2 . CC-BY 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 May 29, 2020. . https://doi.org/10.1101/2020.05.28.20116103 doi: medRxiv preprint 1 2 3 4 5 6 7 8 9 91299 96396 101147 107780 114715 125218 135106 145328 155939 10 11 12 13 14 15 16 17 18 162699 168331 177589 188974 202918 218223 233142 241080 254220 19 20 21 22 23 24 25 26 27 271628 291579 310087 330890 347398 363211 374898 391222 411821 Table 3 : May -cumulative reported cases in Brazil. [t 0 , t 1 ] on which the cumulative daily reported cases data is growing: t 0 = 1, corresponding to March 17 and t 1 = 72, corresponding to May 27. The cumulative number I 1 (t) of low level infectious cases as a function of time t is t t0 S(s) (τ 1 I 1 (s) + τ 2 I 2 (s))ds. The cumulative number I 2 (t) of high level infectious cases as a function of time t is λ t t0 I 1 (s)ds. The high level infectious cases I 2 are removed at the rate γ per day. We assume that a fraction f = 0.3 of these removed cases are reported, isolated, and cause no further transmissions. Thus, the cumulative number of reported cases at time t is f γ t t0 I 2 (s)ds, with f = 0.3 and γ = 1/7. Other fractional values f could be assumed, but currently this fractional value is not known. We fit the parameters τ 1 , τ 2 , and initial values I 1 (t 0 ), I 2 (t 0 ), so that the solutions of (1.1) align with the cumulative reported cases data on the time interval [t 0 , t 1 ]. For this time interval, we estimate τ 1 = 2.5 × 10 −10 , τ 2 = 1.0 × 10 −9 , I 1 (t 0 ) = 13, 000, I 2 (t 0 ) = 8, 500, by fitting the solutions of (1.1) to the cumulative reported cases data between March 17 and May 27. This means that 25% of transmissions are due to I 1 asymptomatic or low level symptomatic cases. In Figure 2 we graph the cumulative reported cases data and the cumulative reported cases from the simulation of the model. Other choices of the parameters and initial values are possible. The daily reported cases from the model simulation, with f = 0.3, γ = 1/7, t 0 = 1 (March 17), is obtained by solving the differential equation In Figure 3 we graph the daily reported cases data and the daily reported cases from the model simulation with f = 0.3, γ = 1/7 and t 0 = 1 (March 17). We project forward in time the model simulation of the cumulative reported cases 0.3 γ t t0 I 2 (s)ds, as well as the model simulation of the total cumulative I 1 (t) cases t t0 (τ 1 I 1 (s) + τ 2 I 2 (s))S(s)ds and the model simulation of the total I 2 (t) cases t t0 λI 1 (s)ds, in Figure 4 . The final size of the epidemic is approximately 157, 000, 000 cases. The turning points of the cumulative cases are the times at which the graphs transition from concave up to concave down. The turning points are obtained by setting the second derivative of these graphs to 0. The turning points are graphed as vertical lines. The turning point of the cumulative I 2 cases is approximately one week later than the turning point of the cumulative I 1 cases, which is consistent with the assumption that the average time of I 1 infectiousness 1/λ = 7 days. The turning point of the reported cumulative I 2 cases is approximately one week later than the turning point of the cumulative I 2 cases, which is consistent with the assumption that the average time of I 2 infectiousness 1/γ = 7 days. The extremely high number of cases projected in the simulation, with an on-going unaltered transmission rate, is very unlikely. It is certain that government imposed and socially adopted distancing measures would take effect, and mitigate transmission. The formula (1.2) for S ∞ in Corollary 1.2 can be used to estimate the effects of these mitigations. The values of the initial conditions in formula (1.2) can be reset to up-dated values, with modified parameters λ, γ, τ 1 , τ 2 , to give forward time predictions of the final size S 0 − S ∞ . In Figure 5 we graph the final size S 0 − S ∞ as a function of 1/λ and 1/γ. If the average number of days low level infectious individuals I 1 remain infectious is reduced to 4, and the average number of days high level infectious individuals I 2 remain infectious is reduced to 4, then the final size S(0) − S ∞ of the epidemic is approximately 20, 000, 000 cases. In Figure 6 we graph the final size S 0 − S ∞ as a function of the I 1 transmission rate τ 1 and the I 2 transmission rate τ 2 . If the transmission rate τ 1 of low level infectious individuals I 1 is reduced to 1.5 × 10 −10 , and the transmission rate τ 2 of high level infectious individuals I 2 is reduced to 0.6 × 10 −9 , then the final size S(0) − S ∞ of the epidemic is approximately 38, 000, 000 cases. 3 . CC-BY 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 May 29, 2020. . We have developed a dynamic model of a COVID-19 epidemic outbreak in a susceptible population. Our model describes the outbreak as a growth of transmission of susceptible individuals from two classes of infectious individuals: I 1 asymptomatic (low level symptomatic), and I 2 symptomatic (high level symptomatic). The model does not assume that major government measures or social behaviour changes have been implemented to mitigate the epidemic transmission. The model also assumes that reported cases are a fraction of the total cases in the population. We apply the model to the COVID-19 current in Brazil. We identify parameters and initial conditions that give agreement of the model output with current case data from Ministério da Saúde of Brazil. We project forward in time the model solutions, to give final size predictions of the epidemic in the absence of social distancing measures. The extreme scenario that the epidemic continues, without social distancing mitigation, is extremely unlikely, given the magnitude of this final size. Intervention measures, such as isolation and contact tracing of high level symptomatic I 2 cases, and quarantining of low level symptomatic I 1 cases, can be quantified in terms of the final size formula (Corollary 2.2) of the epidemic. The reduction of 1/γ and τ 2 , corresponds to the identification and isolation of I 2 high level symptomatic cases. The reduction of 1/λ and τ 1 , corresponds to the contact tracing of I 2 high level symptomatic cases and quarantining of contact traced I 1 asymptomatic or low level symptomatic cases. Both measures have major effect in controlling the epidemic. 6 . CC-BY 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. 7 . CC-BY 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 May 29, 2020. . https://doi.org/10.1101/2020.05.28.20116103 doi: medRxiv preprint Figure 6 : Blue surface: final size S 0 − S ∞ graphed as a function of τ 1 and τ 2 . The yellow lines and red and green planes correspond to the baseline values τ 1 = 2.5 × 10 −10 , τ 2 = 1.0 × 10 −9 , S 0 − S ∞ = 157, 000, 000. The final size S(0) − S ∞ decreases greatly as τ 1 decreases and as τ 2 decreases. For τ 1 = 1.5 × 10 −10 and τ 2 = 0.6 × 10 −9 , the final size S(0) − S ∞ is approximately 38, 000, 000. 8 . CC-BY 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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