A Reduced Multiscale SIR Model with Viral-Load-Dependent Transmission and Adaptive Isolation

Abstract

Multiscale epidemic models are used to connect processes acting inside infected hosts with transmission at the population level. Many fully coupled models, however, are too detailed for direct qualitative analysis. This paper proposes and studies a reduced deterministic SIR model in which a scalaraggregate viral-load variable modifies both the transmission rate and the rate of additional isolation. The model is formulated as a four-dimensional ordinary differential system with vital dynamics. We prove nonnegativity, positive invariance, boundedness, and reduction to a three-dimensional system on the constant-population simplex. The basic reproduction number is obtained by the next-generation matrix method and is used to prove local stability of the disease-free equilibrium. A stronger sufficient condition for global elimination is also given. Endemic equilibria are characterized by a scalar algebraic equation; a monotonicity condition ensuring uniqueness is stated and proved. Finally, a Routh-Hurwitz criterion for local stability of an endemic equilibrium is derived from the reduced Jacobian. Numerical experiments illustrate how viral-load-dependent transmission and adaptive isolation affect peak prevalence and peak timing. The model is intended as a mathematically tractable bridge between classical compartmental models and detailed within-host–between-host simulations.

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