A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission

Jummy F. David; Sarafa A. Iyaniwura; Michael J. Ward; Fred Brauer; Jummy F. David; Sarafa A. Iyaniwura; Michael J. Ward; Fred Brauer

doi:10.3934/mbe.2020188

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 4: 3294-3328. doi: 10.3934/mbe.2020188

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Research article

A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission

1.
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada
2.
Interdisciplinary Studies, University of British Columbia, Vancouver, B.C., Canada

Received: 03 December 2019 Accepted: 20 April 2020 Published: 27 April 2020

We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.
- disease dynamics,
- epidemics,
- airborne disease,
- indirect transmission,
- matched asymptotic analysis,
- Greenos function,
- ODE,
- PDE
Citation: Jummy F. David, Sarafa A. Iyaniwura, Michael J. Ward, Fred Brauer. A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3294-3328. doi: 10.3934/mbe.2020188

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Abstract

We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.

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