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Modeling the Post-Containment Elimination of Transmission of COVID-19

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Modeling the Post-Containment Elimination of Transmission of COVID-19 medRxiv preprint doi: https://doi.org/10.1101/2020.06.15.20132050; this version posted June 17, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC-ND 4.0 International license . Modeling the Post-Containment Elimination of Transmission of COVID-19 Fl´avioCode¸coCoelhoa,b,∗, Luiz Max Carvalhoa,b, Raquel M Lanac,b, Oswaldo G Cruzc,b, Leonardo S Bastosc,b,d, Claudia T Code¸coc,b, Marcelo F C Gomesc,b, Daniel Villelac,b aEscola de Matem´atica Aplicada (EMAp), Funda¸c~aoGetulio Vargas, Rio de Janeiro, Brazil bN´ucleo de M´etodos Anal´ıticos para Vigil^anciaem Sa´udeP´ublica (MAVE), Rio de Janeiro, Brazil cPrograma de Computa¸c~aoCient´ıfica (PROCC), Fiocruz, Rio de Janeiro, Brazil dDepartment of Infectious Diseases Epidemiology, London School of Hygiene and Tropical Medicine, London, UK Abstract Roughly six months into the COVID-19 pandemic, many countries have man- aged to contain the spread of the virus by means of strict containment mea- sures 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 ag- glomerations are also important factors. We also explore the importance of population immunity even when above the herd immunity threshold. Keywords: COVID-19, Epidemiology, Mathematical modelling, pandemic, SARS-COV-2, probability of extinction ∗Corresponding author Email address: [email protected] (Fl´avioCode¸coCoelho) Preprint submitted to Elsevier 29/04/2020 NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. medRxiv preprint doi: https://doi.org/10.1101/2020.06.15.20132050; this version posted June 17, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC-ND 4.0 International license . 1 1. Introduction 2 The COVID-19 pandemic is among the top three biggest in the last on 3 hundred years, reaching levels only previously seen in Influenza pandemics[1]. 4 At the time of writing, more than 8 million confirmed cases have been re- 5 ported globally, with more than 500 thousand deaths [2]. Most affected 6 countries still observe transmission, even if number of new cases show signs 7 of reduction. Even with few cases, if cities (or countries) decide to lift quaran- 8 tine measures, transmission may increase again due to the presence of sizeable 9 portion of susceptible individuals suddenly at greater risk of infection. 10 The initial containment response varied considerably across countries, 11 with some countries displaying more success than others in avoiding infec- 12 tions and subsequent deaths[3, 4, 5]. The economic impact of the containment 13 efforts in the form of quarantines, lock-downs, suspension of international 14 travelling and other drastic measures, has been remarkable [6]. This eco- 15 nomic strain has forced many countries towards an early suspension of many 16 of the most severe containment measures such as quarantines and mobility 17 restrictions [7]. Naturally, the re-normalization of social interaction brings 18 with it may concerns about the potential for a second wave of transmissions, 19 which could potentially grow out of control [8]. 20 Thus, an important question that emerges after a period of isolation is: 21 What's the probability that local infections will be eliminated once the isolation 22 is lifted?. Evaluating scenarios for incidence evolution after these initial 23 containment efforts requires models which accommodate both biological and 24 population-level dynamics. In particular, models that can represent properly 25 the immunological aspects of COVID-19 progression as well as the impact of 26 containment mechanisms, such as quarantine and social distancing. 27 Many models have been proposed recently to deal with the temporal evo- 28 lution of the epidemic[9, 10, 11], but one key aspect that must be considered 29 is the contribution of stochastic fluctuations to the interruption of local trans- 30 mission after the number of active cases is brought close to zero. Kucharsky 31 et al. (2020, [12]), used an stochastic transmission model to estimate the 32 daily reproduction number, Rt, which is often used to predict disease ex- 33 tinction (Rt < 1), but empirical estimates of basic reproduction numbers 34 are very sensitive to noise in the testing rates as well as to changes in case 35 definition. Moreover, in real populations, Rt can move back above one quite 2 medRxiv preprint doi: https://doi.org/10.1101/2020.06.15.20132050; this version posted June 17, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC-ND 4.0 International license . 36 easily in response to changes in the population protective behavior. 37 In this paper, we approach the issue of local disease extinction by cal- 38 culating the probability of extinction of the disease as a stochastic epidemic 39 process. In the context of epidemics, the correct epidemiological terminology 40 for the stochastic extinction is the local \elimination of infections". In this 41 paper, however, we shall continue to use the term extinction throughout, 42 as it is shorter and more in line with the literature on stochastic processes. 43 The probability of extinction is greater when the number of infected is low, 44 so we calculate this probability assuming post-containment scenarios where 45 cases have been dropped to very low numbers. We start by presenting the 46 deterministic version of the model and derive its basic reproduction number. 47 Then we derive an stochastic version of the same model and use it to calcu- 48 late an analytical expression for the extinction probability. We conclude by 49 looking at scenarios of local extinction and discussing how it applies to real 50 scenarios, including also the impact of the fraction of the population already 51 immunized upon the lifting of containment. 52 2. Methods 53 2.1. The SEIAHR model 54 First, we describe the (deterministic) model used to represent COVID- 55 19 dynamics. The Susceptible-Exposed-Infectious-Removed (SEIR) model is 56 a classic model for diseases for which it is important to take into account 57 an incubation period, and variants of it have been employed in numerous 58 COVID-19 modelling studies [13, 14]. Here, we propose a variation of this 59 model with added asymptomatic, hospitalized compartments and a quaran- 60 tine mechanism (fig. 1). Ther is no explicit compartments for Quarantied an 61 dead individuals as they are pure sink states that do not influence the main 62 dynamics. 3 medRxiv preprint doi: https://doi.org/10.1101/2020.06.15.20132050; this version posted June 17, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC-ND 4.0 International license . Figure 1: Block diagram of the SEIAHR model. compartments Q and death are included for illustrative purposes only. The model is represented as system of ordinary differential equations: dS = −λ[(1 − χ)S]; (1a) dt dE = λ[(1 − χ)S] − αE; (1b) dt dI = (1 − p)αE − δI − φI; (1c) dt dA = pαE − γA; (1d) dt dH = φI − (ρ + µ)H; (1e) dt dR = δI + ρH + γA; (1f) dt 63 with λ = β(I + A) as the force of infection. State variables S; E; I; A; R; H 64 represent the fraction of the population in each of the compartments, thus 65 S(t) + E(t) + I(t) + A(t) + H(t) + R(t) = 1 at any time t. Quarantine enters 66 the model through the parameter χ which can be taken as a constant or as a 67 function of time, χ(t), that represents the modulation of the isolation policies. 68 Quarantine works by blocking a fraction χ of the susceptibles from being 4 medRxiv preprint doi: https://doi.org/10.1101/2020.06.15.20132050; this version posted June 17, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC-ND 4.0 International license . 69 exposed, i.e., taking part on disease transmission. Time-varying quarantine 70 is achieved through multiplying χ by activation (eq. 2) and deactivation (eq. 71 3) functions: 1 + tanh(t − s) A (t) = ; (2) s 2 72 and 1 − tanh(t − e) D (t) = ; (3) e 2 73 where s and e are the start and end of the isolation period (e > s), respec- 74 tively.
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