A SIR Model with Spatially Distributed Multiple Populations Interactions for Disease Dissemination

Abstract

In this contribution we analyze a discretized SIR (Susceptible, Infectious, and Removed) compartmental model, to investigate the role of individual interactions in the spread of diseases. The compartments S_{i, j}, I_{i,j} and R_{i, j} (i, j, = 1,2, ..., n) are spatially distributed in a two-dimensional n x n network. We assume that the dynamics follow the well-known SIR-like iteration within the population in each (i,j) site. Moreover, the dynamics are enriched by considering a multi-population interaction following a Gaussian spatial distribution. Therefore, the mobility of individuals between distinct networks is measured from the width alpha of the Gaussian distribution. The interaction of individuals between distinct sites, responsible for the contagion between different populations, is assumed to occur in a time interval smaller than a fixed interval $h$ so that, the total population in each site (i, j), given by N_{i, j} =S_{i, j} + I_{i, j} + R_{i, j}, remained constant (for example, individual leaves his home site (i, j) to work at a neighboring site and returns to his home in a time interval less than h ). We numerically explore some scenarios of population interaction, based on distinct choices of width alpha, that include a hypothetically rapidly closedness and reopening of the economy. The results found show interesting dynamics in the infected population due to the interaction parameter alpha(t) between the populations. Finally, the model can be applied to evaluate the spread of diseases such as COVID-19 enabling decision making in different contexts.

On a model for the fake-news diffusion between two interacting populations

“…The ideas presented in this paper are naturally extended for populations that interact in a network [5,7].…”

Algunos efectos de la interacción poblacional en un modelo epidemiológico SIRC multipoblacional

“…In particular, restriction on V i, j and β î, ĵ (for example, β î, ĵ = 0 represents the isolation of subpopulation) and can be seen as a control strategy. In this case, if a disease starts in a subpopulation, it will become confined in such a population, insofar as there is no interaction between different subpopulations (β î, ĵ = 0), see Marques et al (2022a) and the references therein. If the network size is too small or V i, j contains as many neighborhood sites, then the model dynamics given in (2) is expected to behave as a single population model with a variable transmission rate β i,i .…”

“…However, the nonhomogeneous nature of population mixing does not mean that this is a random process; see Sattenspiel and Dietz (1995) and the references therein. Nonrandom mixing among spatially distributed subpopulations has many consequences for the outcomes of disease spread, for example Sattenspiel and Dietz (1995); Lazo and De Cezaro (2021); Rossato et al (2021); Marques et al (2022a) and references therein.…”

“…In Sattenspiel and Dietz (1995), the authors investigate the effects of migration dynamics coupling in a continuoustime multi-population SIR model. In Lazo and De Cezaro (2021) and Marques et al (2023) the emerging of a plateaulike shape of the infected population is analyzed due to the mixed interaction in a multi-population SIR model without migration. In Gomes and De Cezaro (2022), a multipopulation SIRD-type model was proposed to analyze the effects of COVID-19 on an age-distributed population as a consequence of the reopening of schools.…”

Agent-Based Modeling of COVID-19 Transmission: A Case Study of Housing Densities in Sankalitnagar, Ahmedabad

“…Spatially explicit Agent-Based Models (ABMs) are particularly useful for studying communicable diseases whose propagation is affected by the environment, such as sanitation and housing conditions [1]. While there exist other useful methods, such as expanded compartment models based on the well-regarded Susceptible-Infectious-Recovered (SIR) model [2][3][4], as well as meta-population models, the ABM approach was selected here as it (a) allowed for free mixing of individuals (as opposed to individuals allocated to specific spatially allocated grids) and (b) provided a mechanism for varying rules governing movement according to the underlying environment [5]. Moreover, ABMs are well placed to simulate the transmission of diseases as a complex process [6][7][8], as it is affected by factors such as heterogeneous interactions facilitated by network connectivity (both social and place-based) [9,10].…”