The dynamics of a simple, risk-structured HIV model

Mark Kot; Dobromir T. Dimitrov; Mark Kot; Dobromir T. Dimitrov

doi:10.3934/mbe.2020232

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 4: 4184-4209. doi: 10.3934/mbe.2020232

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The dynamics of a simple, risk-structured HIV model

Mark Kot ^{1
,
,},
Dobromir T. Dimitrov ²

1.
Department of Applied Mathematics, Box 353925, University of Washington, Seattle, WA 98195-3925, USA
2.
Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, MC-C200, P. O. Box 19024, 1100 Fairview Ave. N., Seattle, WA 98109-1024, USA

Received: 30 April 2020 Accepted: 09 June 2020 Published: 15 June 2020

Many diseases, such as HIV, are heterogeneous for risk. In this paper, we study an infectious-disease model for a population with demography, mass-action incidence, an arbitrary number of risk classes, and separable mixing. We complement our general analyses with two specific examples. In the first example, the mean of the components of the transmission coefficients decreases as we add more risk classes. In the second example, the mean stays constant but the variance decreases. For each example, we determine the disease-free equilibrium, the basic reproduction number, and the endemic equilibrium. We also characterize the spectrum of eigenvalues that determine the stability of the endemic equilibrium. For both examples, the basic reproduction number decreases as we add more risk classes. The endemic equilibrium, when present, is asymptotically stable. Our analyses suggest that risk structure must be modeled correctly, since different risk structures, with similar mean properties, can produce different dynamics.
- multi-compartment disease model,
- separable mixing,
- basic reproduction number
Citation: Mark Kot, Dobromir T. Dimitrov. The dynamics of a simple, risk-structured HIV model[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 4184-4209. doi: 10.3934/mbe.2020232

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Abstract

Many diseases, such as HIV, are heterogeneous for risk. In this paper, we study an infectious-disease model for a population with demography, mass-action incidence, an arbitrary number of risk classes, and separable mixing. We complement our general analyses with two specific examples. In the first example, the mean of the components of the transmission coefficients decreases as we add more risk classes. In the second example, the mean stays constant but the variance decreases. For each example, we determine the disease-free equilibrium, the basic reproduction number, and the endemic equilibrium. We also characterize the spectrum of eigenvalues that determine the stability of the endemic equilibrium. For both examples, the basic reproduction number decreases as we add more risk classes. The endemic equilibrium, when present, is asymptotically stable. Our analyses suggest that risk structure must be modeled correctly, since different risk structures, with similar mean properties, can produce different dynamics.

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