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

J. C. Marques, A. De Cezaro, M. J. Lazo

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_{ij} + I_{ij} + R_{ij}, remained constant (for example, individual leaves his home site (ij) 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.



Keywords


SIR; multi-population interaction; diseases dissemination.

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References


W. H. Organization, “Who coronavirus disease (covid-19) dashboard,”https://covid19.who.int 2020/7/29, 2020;

J. Govaert, “Coronavirus: Uk changes course amid death toll fears.,”https:// www. bbc. com/ news/ health-51915302, 2020;

N. Ferguson and et al., “Report 9: Impact of non-pharmaceutical interventions(npis) to reduce covid-19 mortality and healthcare demand,”Imperial CollegeCOVID-19 Response Team, pp. 1–20, 2020;

W. Wang and X. Q. Zhao, “An epidemic model with population dispersaland infection period,”SIAM Journal on Applied Mathematics, vol. 66, no. 4,pp. 1454–1472, 2006;

J. Arino and P. Vand Den Driesschet,A basic reproduction number in a multi-city compartmental epidemic model, in Positive Systems. Lecture Notes inControl and Inform. Sci. 294, Springer, 2003;

J. Arino and P. van den Driessche, “A multi-city epidemic model,”MathematicalPopulation Studies, vol. 10, no. 3, pp. 175–193, 2003;

D. J. Rodriguez and L. Torres-Sorando, “Models of infectious diseases in spa-tially heterogeneous environments,”Bulletin of Mathematical Biology, vol. 63,no. 3, pp. 547 – 571, 2001;

W. Wendi Wang and X. Q. Zhao, “An epidemic model in a patchy environment,”Mathematical Biosciences, vol. 190, no. 1, pp. 97 – 112, 2004;

R. R. Veit and M. A. Lewis, “Dispersal, population growth, and the allee effect:Dynamics of the house finch invasion of eastern north america,”The AmericanNaturalist, vol. 148, no. 2, pp. 255–274, 1996;

D. C. Mistro,Modelos para dispersão abelhas: um zoom matemático. PhDthesis, Universidade Estadual de Campinas, 1998;

I. K. Meyer and L. Bingtuan, “A spatial model of plants with an age-structuredseed bank and juvenil stage,”Society for Industrial and Applied Mathematics,vol. 73, no. 4, pp. 1676–1702, 2013;

J. C. Marques,Modelos para dispersão de javalis { Sus scrofa }. PhD thesis, Universidade Federal do Rio Grande do Sul, 2019;

M. J. Lazo and A. De Cezaro, “Why can we observe a plateau even in an outof control epidemic outbreak? a seir model with the interaction ofndistinctpopulations for covid-19 in brazil.,”TEMA, to appear 2021;

D. Bernoulli, “Essai d’une nouvelle analyse de la mortalité causée par la petitevérole, et des avantages de l’inoculation pour la prévenir,”Histoire de l’Acad.,Roy. Sci.(Paris) avec Mem, pp. 1–45, 1760;

T. De Camino-Beck and M. A. Lewis, “Invasion with stage-structured coupledmap lattices: Application to the spread of scentless chamomile,”EcologicalModelling, vol. 220, no. 23, pp. 3394–3403, 2009.




DOI: https://doi.org/10.5540/tcam.2022.023.01.00143

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Trends in Computational and Applied Mathematics

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