Multi-patch multi-group epidemic model with varying infectivity

Raphaël Forien; Guodong Pang; Étienne Pardoux

doi:10.3934/puqr.2022019

Article Contents

2022, Volume 7, Issue 4: 333-364. Doi: 10.3934/puqr.2022019

This issue Previous Article Backward stochastic differential equations and backward stochastic Volterra integral equations with anticipating generators Next Article Optimal control of SDEs with expected path constraints and related constrained FBSDEs

Multi-patch multi-group epidemic model with varying infectivity

1.
INRAE, Centre INRAE PACA, Domaine St-Paul-Site Agroparc, 84914 Avignon Cedex, France
2.
Department of Computational Applied Mathematics and Operations Research, George R. Brown College of Engineering, Rice University, Houston, TX 77005, US
3.
Aix Marseille University, CNRS, I2M, 13453 Marseille, France

Corresponding author: Étienne Pardoux
Corresponding author: Étienne Pardoux

Received: June 24, 2022

Accepted: July 22, 2022

Early access: October 14, 2022

Published online: October 14, 2022

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Abstract

This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $ L $ distinct patches (with migrations between them) and $ K $ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d.

The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $ {\bf {D}} $, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.

Keywords:

Stochastic epidemic model,
Multi-patch multi-group,
Varying infectivity,
SIR model,
Law of large numbers,
Feynman-Kac formula,
Poisson-driven (McKean-Vlasov type) stochastic equations,
Propagation of chaos.

Mathematics Subject Classification: 60G55, 60H20, 92D30.

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References

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