key: cord-0970017-0imdy5v4
authors: Monteiro, L.H.A.
title: Short Communication An epidemiological model for SARS-CoV-2
date: 2020-05-29
journal: Ecological Complexity
DOI: 10.1016/j.ecocom.2020.100836
sha: ff66629cff37e5d42a80a0c136782184e7580dbb
doc_id: 970017
cord_uid: 0imdy5v4

Abstract The spread of SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) is here investigated from an epidemic model considering four pathways of person-to-person transmission. These pathways represent the propagation of this novel coronavirus by asymptomatic and symptomatic infected individuals. In this work, analytical expressions for the disease-free and endemic steady-states are derived. Also, the conditions for eradication of this contagious disease are determined. By taking into account realistic parameter values, the proposed model shows an oscillatory convergence to the endemic steady-state, which means the occurrence of a sequence of peaks in the number of sick individuals as time passes. These results are discussed from a public health standpoint.

The ongoing pandemic of coronavirus disease 2019 (COVID-19) has been responsible for countless deaths, many of them due to the lack of adequate medical treatment, even in developed countries (Singhal, 2020 This paper is organized as follows. In Section 2, a deterministic compartmental model written in terms of differential equations is introduced and analyzed. Recall that a compartment is a homogeneous subpopulation. In Section 3, numerical sim-ulations are presented to illustrate the spread of the infectious agent. In Section 4, the possible relevance of this study is stressed.

Let S(t), A(t), I(t), and R(t) be the numbers of S, A, I, and R-individuals in a given geographic region at the instant t, respectively. By taking into consideration the homogeneous mixing assumption (Turnes Jr and Monteiro, 2014) , the proposed model is described by the following set of first-order differential equations:

The nine parameters α 1 , α 2 , β, γ, a 1 , a 2 , b, c, and d are positive numbers. The rate constants α 1 and a 1 respectively express the transmission by A and I-individuals

to S-individuals leading to A-individuals; α 2 and a 2 respectively express the transmission by A and I-individuals to S-individuals leading to I-individuals. Thus, the rate constants α = α 1 + α 2 and a = a 1 + a 2 are related to the infections caused by A and I individuals, respectively. Also, β and b are the recovery rate constants of A and I-individuals, respectively; γ and c are the death rate constants of A and I-individuals, respectively; and d is the death rate constant of R-individuals. In addition, R-individuals are supposed to be fully protected from reinfections. If this is not true, then d also includes the immunity-loss rate constant.

Note that dS(t)/dt + dA(t)/dt + dI(t)/dt + dR(t)/dt = 0, because the deaths of A, I, and R-individuals are balanced by the births of S-individuals. Therefore, S(t) + A(t) + I(t) + R(t) = N ; that is, the total number of individuals N remains constant. Since R(t) = N − S(t) − A(t) − I(t), the model can be rewritten as:

This third-order system is analyzed from a dynamical systems theory perspective (Guckenheimer and Holmes, 2002) . A stationary solution (S * , A * , I * ), corresponding to an equilibrium point in the space state S × A × I, is obtained from dS/dt = 0, dA/dt = 0, and dI/dt = 0. In this model, there are a disease-free stationary solution given by:

and an endemic stationary solution given by:

with:

If m = 0, then S * 2 = p/n.

The local stability of an equilibrium point can be inferred from the eigenvalues of the Jacobian matrix J, which is obtained from the linearization of the set of non-linear differential equations around such a point. Let λ be the eigenvalues of J, which are determined from det(J−λI) = 0 (I is the identity matrix). The Hartman-Grobman theorem (Guckenheimer and Holmes, 2002) says that an equilibrium point is locally asymptotically stable if all its eigenvalues have negative real parts; if at least one eigenvalue has positive real part, then this point is unstable.

Consider the parameters ρ 1 and ρ 2 defined as:

Stability analysis of the SAIR model reveals that its disease-free solution is asymptotically stable if ρ 1 < 1 and ρ 2 < 1, and it is unstable if ρ 1 > 1 and/or ρ 2 > 1.

In epidemiology, the basic reproduction number R 0 is defined as the average number of secondary infections caused by a single infectious individual inserted into a completely susceptible population (Anderson and May, 1992) . Therefore, if R 0 > 1, the corresponding pathogen can invade and/or chronically persist in the host population; if R 0 < 1, it cannot invade and/or it will be naturally eradicated.

A formula for R 0 can be derived from a method based on the next generation matrix Therefore, the disease eradication requires ρ 1 < 1 and ρ 2 < 1 (from the Jacobian matrix J); alternatively, R 0 < 1 (from the next generation matrix FV −1 ).

The SAIR model was numerically solved by using the 4th-order Runge-Kutta in- Also, ρ 1 1.8, ρ 2 1.6, R 0 2.6, and q 6.3. Observe that the convergence is also oscillatory to the endemic steady-state; however, this convergence is faster and smoother as compared to Figs. 1-6.

Usually, the COVID-19 propagation is theoretically investigated by considering Eindividuals, which are those who were exposed to the pathogen and are in the incu- From the proposed model, formulas were derived for R 0 (Eq. (16)), the endemic steady-state (S * 2 , A * 2 , I * 2 ) (Eq. (9)), the proportion q = I * 2 /A * 2 (Eq. (13)), and the stability of the disease-free steady-state (Eqs. (14) and (15)). These expressions can be employed to evaluate the effects of public health actions on the disease spread.

In the early phase of this pandemic, studies estimated R 0 ≈ 2 − 6. Also, from the current knowledge of this illness, q ≈ 0.25 − 9. From assumed values for the contagion rate constants a 1 , a 2 , α 1 , and α 2 , computer simulations were performed. It is relevant to stress that the proportion of the infected population in steady state, given by A * 2 + I * 2 , can be very very small. In fact, from Eqs. (5)- (7), the following relation can be obtained:

by taking N = 1. For the vast majority of viral infections, β d and b d, because the recovery time (typically, one or two weeks) is much shorter than the duration of acquired immunity (typically, years or decades) and the average life expectancy (typically, six to eight decades, depending on the country). Therefore:

or:

Thus, in steady state, the population is composed almost exclusively of S and R- As in Fig. 1 , the disease apparently tends to disappear. In this figure, the convergence to the endemic steady-state is faster and smoother as compared to Fig. 1 .

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