key: cord-0759288-7ucc64uh authors: Coelho, F. C.; Carvalho, L. M.; Lana, R. M.; Cruz, O. G.; Bastos, L. S.; Codeco, C. T.; Gomes, M. F. C.; Villela, D. title: Modeling the Post-Containment Elimination of Transmission of COVID-19 date: 2020-06-17 journal: nan DOI: 10.1101/2020.06.15.20132050 sha: 437b1859b5e31cee4590084d550073722897235f doc_id: 759288 cord_uid: 7ucc64uh Roughly six months into the COVID-19 pandemic, many countries have managed to contain the spread of the virus by means of strict containment measures including quarantine, tracing and isolation of patients as well strong restrictions on population mobility. Here we propose an extended SEIR model to explore the dynamics of containment and then explore scenarios for the local extinction of the disease. We present both the deterministic and stochastic version fo the model and derive the R0 and the probability of local extinction after relaxation (elimination of transmission) of containment, P0. We show that local extinctions are possible without further interventions, with reasonable probability, as long as the number of active cases is driven to single digits and strict control of case importation is maintained. The maintenance of defensive behaviors, such as using masks and avoiding agglomerations are also important factors. We also explore the importance of population immunity even when above the herd immunity threshold. containment efforts requires models which accommodate both biological and 23 population-level dynamics. In particular, models that can represent properly 24 the immunological aspects of COVID-19 progression as well as the impact of 25 containment mechanisms, such as quarantine and social distancing. 26 Many models have been proposed recently to deal with the temporal evo-27 lution of the epidemic[9, 10, 11], but one key aspect that must be considered 28 is the contribution of stochastic fluctuations to the interruption of local trans-29 mission after the number of active cases is brought close to zero. Kucharsky 30 et al. (2020, [12] ), used an stochastic transmission model to estimate the 31 daily reproduction number, R t , which is often used to predict disease ex-32 tinction (R t < 1), but empirical estimates of basic reproduction numbers 33 are very sensitive to noise in the testing rates as well as to changes in case 34 definition. Moreover, in real populations, R t can move back above one quite 35 2 . 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) late an analytical expression for the extinction probability. We conclude by 48 looking at scenarios of local extinction and discussing how it applies to real 49 scenarios, including also the impact of the fraction of the population already 50 immunized upon the lifting of containment. 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 model is represented as system of ordinary differential equations: with λ = β(I + A) as the force of infection. State variables S, E, I, A, R, H 63 represent the fraction of the population in each of the compartments, thus 64 S(t) + E(t) + I(t) + A(t) + H(t) + R(t) = 1 at any time t. Quarantine enters the model through the parameter χ which can be taken as a constant or as a 66 function of time, χ(t), that represents the modulation of the isolation policies. Quarantine works by blocking a fraction χ of the susceptibles from being 68 4 . 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 June 17, 2020 June 17, . . https://doi.org/10.1101 June 17, /2020 doi: medRxiv preprint exposed, i.e., taking part on disease transmission. Time-varying quarantine is achieved through multiplying χ by activation (eq. 2) and deactivation (eq. 70 3) functions: where s and e are the start and end of the isolation period (e > s), respec- Figure 2 : χ(t) upon activation and deactivation of quarantine for χ = 0.3. Panel on the left shows each function separately(χA 1 (t) and χD 1 (t)) set to t = 1. The right panel shows a combination with activation on t = 1 and deactivation on t = 6 (χ(t) = χA 1 (t)D 6 (t)). 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 copyright holder for this preprint this version posted June 17, 2020. . State transition rate From equation (5) we can obtain R t = R 0 S(t) and another reproduction This model is a multivariate stochastic process . We will leave the equation of R out, because it is decoupled from the rest of the system. A joint probability function is associated with the set of random state variables, 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 copyright holder for this preprint this version posted June 17, 2020. . https://doi.org/10.1101/2020.06.15.20132050 doi: medRxiv preprint which leads to the Kolmogorov forward equation + P s,e+1,i,a−1,h pα(e + 1) + P s,e,i+1,a,h−1 φ(i + 1) + P s,e,i+1,a,h δ(i + 1) + P s,e,i,a+1,h γ(a + 1) + P s,e,i,a,h+1 ρ(h + 1) + P s,e,i,a,h+1 µ(h + 1). (6) As a continuous-time branching process, the extinction threshold for the 96 stochastic model is closely related to the corresponding one in the determin-97 istic model but depends on the initial number of infectious individuals [16] . Based of the properties of this kind of stochastic processes, Whittle (1955) 99 calculated the probability of extinction for the stochastic SIR model to be We can apply the same technique described for the stochastic SEIR model reproduction number at the time. Nevertheless, we know from equation (5) 112 that R t is a function of S(t). Therefore, we calculate P 0 for various values 113 of R t . Analytical expression for P 0 . The probability of extinction, P 0 , can be com-115 puted analytically following the derivation of probability-generating func-116 tions for the system of equations in (6), the details of which are given in Ap-117 pendix B. The probability of extinction is computed from the fixed points of 118 the PGFs, q 1 , q 2 and q 3 , which lie in (0, 1) 3 . With the fixed points in hand, 119 we arrive at where k i are the initial states k 1 = E(0), k 2 = I(0) and k 3 = A(0). Here we 121 denote the moment of relaxation of containment as t = 0. 122 7 . 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 June 17, 2020. . Numerical approximation to P 0 . The expression in (7), while very useful, is derived from the assumption that the initial state is small compared to the 124 size of the population and that S(0) ≈ N . Due to the possible deviations 125 from the theoretical value of P 0 when S(0) < N , In the results, we always 126 present estimates of P 0 , obtained by simulation as well. 127 We can approximate P 0 for a given R 0 and an initial number of infected in- After a first wave of infections a fraction of the population will become im- 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 June 17, 2020. . fig. 3 . In 3 out of 10 runs the containment could eliminate the disease but the remaining 7 gave origin to a second wave. All simulations had I(0) = 2 and R 0 = 1.7, on a population of 5000. Table 3 : Probabilities of extinction for R t = 1.7. Approximate P 0 was calculated from a set of 10000 runs of the stochastic SEIAHR model with different number of initial infectious individuals(I 0 ). 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 June 17, 2020. . 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 June 17, 2020. . https://doi.org/10.1101/2020.06.15.20132050 doi: medRxiv preprint Table 4 : Probabilities of extinction for R t = 1.1. Approximate P 0 was calculated from a set of 10000 runs of the stochastic SEIAHR model with different number of initial infectious individuals(I 0 ). 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 June 17, 2020. . . 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 June 17, 2020. . In the deterministic version of SEIAHR, a second wave will always happen due to the asymptotic way that the number of infectious approach zero. Treating the epidemic process as the stochastic process that it actually is, 178 we can see that the probability of local extinction post-reactivation is quite 179 substantial ( fig 5) . In the stochastic SEIAHR, extinction events will happen 180 with slightly lower probabilities than those of a SIR model justifying using 181 more detailed model to study this problem. Figure 4 shows stochastic ex-182 tinction taking place in three out of ten runs, with R t = 1.7 and I 0 = 2. The that even if a location is still far from achieving herd immunity, any acquired 191 immunity will greatly improve the chances local extinction substantially. Communities that managed to contain the disease and bring it to the brink 193 of extinction with severe economic impact, need to know how likely they are 194 in succeeding in their fight against the disease as they return to "normal" 195 social and economic activities. Our results also reinforce the need to run 196 seroprevalence surveys previous to the reopening so that the probability to 197 eliminate local infections e properly adjusted for the context of each locality. Here we explored how to improve the chances for local transmission elimina-199 tion but it must be kept in mind that the R t must be kept low (through the After defining these we can calculate Now, if we let x = {E, I, A} and x0 be the Disease-free equilibrium (DFE), we can define 17 . 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 June 17, 2020. . https://doi.org/10. 1101 /2020 The next generation matrix is given by F V −1 : The spectral radius of K at the DFE (when S(0) ≈ 1), is the basic reproduc-271 tion number of the model, which, after simplification, gives equation (5). Appendix B. Probability generating functions 274 Following [16], we derive the probability-generating functions (PGF) for each infectious compartment. Starting with I i (0), the probability of an infected individual in state i producing offspring of type j given that I j (0) can be obtained from The desired probabilities can be obtained by differentiating the PGF with 275 respect to z i and setting all z to 1. Notice f i has a fixed point at z 1 = . . . = 276 z k = 1. Computing the relevant probabilities is straightforward by keeping track of 278 the possible transitions (given in Table 1 ) and considering that only one 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. (which was not certified by peer review) The copyright holder for this preprint this version posted June 17, 2020. . https://doi.org/10. 1101 /2020 Finally, for the case E(0) = 0, I(0) = 0 and A(0) = 1: From the PGFs, we can obtain a matrix M whose entries m ji = ∂f i ∂u i | u=1 are 284 the expected number of offspring in state j from an individual in state i. To obtain the probability of extinction, P 0 , we need to find the fixed points of the PGFs, i.e. solutions to equations of the form f i (q 1 , q 2 , q 3 ) = q i , q i ∈ (0, 1). After some tedious algebra, we arrive at with σ = ((1 − χ)β) 2 + 2(1 − χ)β(2p − 1)((φ + δ) − γ) + ((φ + δ) − γ) 2 . . 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 June 17, 2020. . https://doi.org/10. 1101 /2020 An interactive web-based dashboard to 207 track covid-19 in real time