A DIFFUSIVE SEIR MODEL FOR COMMUNITY TRANSMISSION OF COVID-19 EPIDEMICS: APPLICATION TO BRAZIL

. A mathematical model incorporating diﬀusion is developed to describe the spatial spread of COVID-19 epidemics in geographical regions. The dynamics of the spatial spread are based on community transmission of the virus. The model is applied to the outbreak of the COVID-19 epidemic in Brazil.


Introduction
Coronavirus disease 2019 (COVID-19) firstly identified in December 2019 has infected over 20 million people worldwide as of September 2021 [11,26]. This disease is caused by a novel coronavirus named SARS-CoV-2. Major epidemics caused by viruses in the family of coronavirus include the SARS-CoV epidemic in 2003 and the MERS-CoV epidemic in 2012. A distinguished feature of COVID-19 is that a large proportion of people infected with this disease do not show symptoms [54], and the symptoms of the patients vary from mild symptoms to severe illness [26]. Moreover, infected people who do not show symptoms can also transmit the disease, and these features make the disease difficult to control [22].
In this paper, we will use a reaction-diffusion model to simulate the community transmission of COVID-19, which plays a major role in the geographical spread of the epidemics. Understanding the role of community transmission in the geographical evolution of COVID-19 epidemics can contribute to the development of measures for controlling the pandemic. Community transmission cases of COVID-19 are distinguished from sporadic cases, clusters of cases, travel-based cases, and country-wide cases [52]. Community transmission occurs with large numbers of local transmissions, without direct source of exposure identifications [51]. Community transmissions are measured in daily number of confirmed cases per specified number of inhabitants in the designated community. The World Health Organization provides specific recommendations for public health communication and engagement for reduction of COVID-19 community transmission [52]. After the outbreak of COVID-19, many governments have put in place stringent measures including: travel limitation or embargoes, lock downs or partial lock downs, requirements for home work, masking, social distancing, and the frequent disinfecting of commonly used facilities. It is reasonable to assume, when one considers the regional or national spread as opposed to the global or transnational spread, that transmission over short distances is the dominant factor. Therefore, we do not consider the long distance transport of susceptible and infected individuals, while local movement of the virus is indirectly viewed as as the movement of infectious individuals, described with diffusion processes. This motivates us to use a reaction-diffusion model with diffusion terms for infected compartments and without diffusion terms for susceptible and recovered compartments. Our approach has been adopted by recent papers [20,32]. In particular, in [32], the authors have used such a model to simulate the spatial outbreak of influenza in Puerto Rico.
The organization of this paper is as follows. In Section 2, we develop a mathematical model of COVID-19 community transmission, and state our theoretical results. In Section 3, we provide numerical simulations of our model to the spatial spread during the outbreak stage of the COVID-19 epidemic in Brazil. In Section 4, we draw a conclusion. The proofs of the mathematical results are presented in the appendix.

The Model
We assume that the host population is confined to a bounded region Ω ⊆ R N with smooth boundary ∂Ω that lies locally on one side of the boundary. The spatial density of the susceptible class at location x ∈ Ω and time t ≥ 0 is represented by the variable S(x, t). The asymptomatic infected class is represented by E(x, t), while the infected and recovered classes are represented by the variables I(x, t) and R(x, t), respectively. Individuals are recruited to the exposed class from the susceptible class via contact with either the asymptomatic infected class or the symptomatic class. We model this process with standard incidence mechanisms where σ 1 (x) and σ 2 (x) are transmission coefficients. We assume that asymptomatic infected individuals leave the class at a rate constant λ, which means that individuals spend an average time 1/λ in this class. For simplicity, we assume that all the individuals leaving the asymptomatic infected class will join the symptomatic infected class, although many COVID-19 patients have mild or no symptoms. We assume that the infected individuals recover or die at a constant rate γ, which means that infected individuals spend an average time 1/γ in this class (if an infected individual does not show any symptoms before recovery, then that individual spends zero time in this class). Our model differs from the standard diffusive SEIR model by virtue of the fact that the exposed class E is also infectious. This represents the asymptomatic phase of COVID-19.
The forgoing considerations lead to a degenerate reaction-diffusion system of the form where D E and D I are positive diffusion coefficients. On ∂Ω, we impose homogeneous Neumann boundary conditions: which means that the individuals cannot cross the boundary ∂Ω. Finally, we impose initial conditions: To include mass action infection mechanism f (x, S, E, I, R) = σ 1 (x)SE + σ 2 SI into our model, we make the following assumptions: We have following result about the asymptotic behavior of solutions of the model: Theorem 2.2. Suppose that (A1)-(A4) hold and there exists ∈ (0, 1) such that S 0 , E 0 , I 0 , R 0 ∈ C (Ω) are nonnegative. Let (S, E, I, R) be the solution of (2.1)-(2.3). Then, we have uniformly for x ∈Ω, and there exists a function S ∞ onΩ such that 0 ≤ S ∞ (x) ≤ S 0 (x) and Furthermore, S ∞ > 0 a.e. on any open subset of Ω on which S 0 > 0.
The proofs of these results are similar to those in [20], and we have attached them in the appendix.

Numerical Simulations
We apply our model to the spatial spread of COVID-19 in Brazil during the early outbreak stage. The outbreak in Brazil began in Sao Paolo in late February and early March, 2020. The first cases arose from international airline travelers to Sao Paolo, primarily from Italy [5]. The epidemic propagated rapidly from the Sao Paolo epi-center to other areas of Brazil. The states of Sao Paulo and Rio de Janeiro both identified community transmission on March 13, 2020 [10]. On March 20 the Brazilian Congress declared a Public Calamity State, on March 19 the Federal District of Brasilia decreed mandatory social restrictions, and on March 23 Sao Paolo State decreed mandatory quarantine measures [46]. Other cities and states soon followed, including Santa Catarina, Bahia, Esprito Santo, and Minas Gerais [42,45].
COVID-19 transmission is dependent on two categories of travel: (1) inter-state travel (long range) and (2) community travel (short range). Inter-state travel between regions of Brazil depends on commercial interactions, transport, social gatherings, and other socio-economic factors. Inter-state mobility between major urban centers in composed primarily of airline travel, and consequently, inter-state COVID-19 spread is primarily by air-travel [10]. Community travel is based primarily on daily activities and local public transportation. In our model, the diffusion terms represent the local spread of infectious agents. The parameters of transmission in our model are designed to account for the regional characteristics of community transmission based on this community travel. In our simulations, the initial date is set to be March 17, 2020 when social strict measures were implemented to limit inter-state transmissions.  .3) with standard incidence mechanism, and the parameters used are listed in Table 3. If D E = D I = 0, at each local point x ∈ Ω, the basic reproduction number is We use the finite element solver in the partial differential equation package of Matlab to solve (2.1)-(2.3). The boundary ∂Ω is constructed using the geographic boundary data of Brazil. Figure 1 shows a mesh with 15908 nodes and 7723 triangles for the finite element scheme while more modes and triangles are used in simulations.
At the beginning of the epidemics, we assume that the whole population is susceptible, and therefore the initial value of the susceptible population, S(x, 0), can be described by the distribution of the Brazilian population. The distribution of the Brazilian population data is obtained from [8]. The initial value S(x, 0) is plotted in Figure 2, where we can see that the Brazilian population are mainly concentrated in the Eastern and Southern coast regions.
We obtain the reported COVID-19 case data for each 27 states of Brazil from [23]. We assume that only a fraction, f = 0.2, of newly infected cases are reported [16,39]. Therefore, the expression f t 0 Ωi λI(x, t)dx gives the accumulated reported cases of state i, i = 1, 2, .., 27, at day t, where Ω i ⊂ Ω is the domain of state i (we do not take account for the reported cases before t = 0 for simplicity). The starting point t = 0 is set to be March 17, 2020 in the simulations. The initial distribution of asymptotically infected people, E(x, 0), of each state is assumed to be proportional to the susceptible population distribution, where the total initial asymptotically infected people in each state is estimated. The total number of initial infected people Ωi I(x, 0)dx for each state i is based on data from [23], while the distribution I(x, 0) is taken to be proportional to the susceptible population distribution. We solve the model with parameters and initial values described as above. In Figure 3, we plot the simulated total accumulated COVID-19 cases in Brazil, which match well with the data. We are also able to compute the accumulated reported cases in each 27 states of Brazil and compare the simulations with data. For most of the 27 states, the simulated accumulated cases match well with the data from [23]. In Figure 4, we plot the simulated accumulated reported COVID-19 cases and compare with data for six most populated states: Bahia, Minas Gerais, Parana, Rio de Janeiro, Rio Grande do Sul and Sao Paulo. There are few states that the simulation do not match the data well, which are expected as we assumed uniform in space and time constants for simplicity.
In Figure 5, we graph the simulated distribution of infected COVID-19 cases, I(x, t), in Brazil at March 18 and March 28. From the figure, we can see that the infected cases are concentrated at the most populated areas. To compare with Figure 5, Figure 6 shows the actual infected cases data.
In Figure 7, we plot the simulated number of active cases per 1,000,000 people in each state at different days. In Figure 8, we plot the number of active cases per 1,000,000 people in each state at different days with reported case data from [23]. The simulation values in Figure 7 and the reported data values in Figure 8 are in essential agreement.
Data accessibility: The MATLAB code associated with this article is available at https://github.com/YixiangMath/SpatialSimulations.git

Conclusion
In this paper, we propose and analyze a diffusive epidemic model for the community transmission of COVID-19. The major feature of the model is that there is no diffusion term for the equation of susceptible compartment, while diffusion term is used to describe the community spread of the virus in the equations of asymptomatic infected and infected compartments. The model has been used to simulate the outbreak of COVID-19 in Brazil. In the simulation, the geographical boundary data of Brazil is used to construct the domain of the model, and the population data of Brazil is used to set up the initial conditions of the model. We use the partial differential equation toolbox in MATLAB to solve the model and use the mapping toolbox to plot the solutions. Using the geographical boundary data of the states of Brazil, we are able to determine and plot infected cases for different states and compare with real data.
Our simulations are also limited. We use spatial uniform parameters in the simulations, and the spatial heterogeneity is reflected mainly in the distribution of the susceptible population. It is desirable to come up with methods to design or estimate spatially heterogeneous parameters for the model. Though the simulated total cases fit the data well at the first thirty days, the simulation will overestimate the epidemic size in the following days. Factors such as quarantine measures, social distancing and masking will affect the transmission of the disease, so it is unlikely to use a model with time-independent parameters to fit the epidemic data over a large time interval.
Multiplying the first two equations in (5.4) by θ, adding up them, and integrating on Ω × (0, 1), we obtain for τ ≥ 0 and p > (n + 2)/n, where the right hand side of (5.22) is independent of N .
Finally, we consider the asymptotic behavior of S(x, t) as t → ∞. Since ∂ t S(x, t) ≤ 0 for all x ∈Ω and t > 0, there exists a function S ∞ (x) ≥ 0 such that S(x, t) → S ∞ (x) as t → ∞ for each x. Multiplying the first equation of (2.1) by −1/S(x, t), we obtain d dt − ln(S(x, t)) = f (x, S(x, t), E(x, t), I(x, t), R(x, t)) S(x, t) ≤ g(S(x, t))P (E(x, t), I(x, t)) S(x, t) , (5.27) where g and P are defined in (A4). Since g is continuously differentiable, g(0) = 0 and S(x, t) is bounded, g(S)/S is bounded. Since P (E, I) is a polynomial, P (0, 0) = 0, and E and I are bounded, there exists K > 0 such that g(S)P (E, I) S ≤ K(E + I).