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On a stochastic epidemic SIR model with non homogenous population: a toy model for HIV

Carles Rovira

TL;DR

The paper generalizes stochastic SIR dynamics to a nonhomogeneous population with two risk groups, a distributed infectious period, and direct S→R transitions in the high-risk group, to study HIV-like epidemics. It develops two Markovian formulations and derives infection probabilities and a bound-based $R_0$ for heterogeneous contacts, highlighting how group-specific behavior and duration of infectiousness shape spread. The toy HIV model demonstrates that PrEP uptake and increased testing can dramatically reduce infections, especially in the high-risk group, and reveals nonlinear effects from the proportion of high-risk individuals. Overall, the work provides a quantitative framework to assess prevention strategies in heterogeneous populations and emphasizes the potential impact of rapid testing and PrEP on epidemic control.

Abstract

In this paper we generalise a simple discrete time stochastic SIR type model defined by Tuckwell and Williams. The SIR model by Tuckwell and Williams assumes a homogeneous population, a fixed infectious period, and a strict transition from susceptible to infected to recovered. In contrast, our model introduces two groups, $A$ and $B$, where group $B$ has a higher risk of infection due to increased contact rates. Additionally, the duration in the infected class follows a probability distribution rather than being fixed. Finally, individuals in group $B$ can transition directly to the recovered class R, allowing us to analyze the impact of this preventive measure on disease spread. Finally, we apply this model to the spread of HIV, analyzing how risk behaviors, rapid testing, and PrEP-like therapies influence the epidemic dynamics.

On a stochastic epidemic SIR model with non homogenous population: a toy model for HIV

TL;DR

The paper generalizes stochastic SIR dynamics to a nonhomogeneous population with two risk groups, a distributed infectious period, and direct S→R transitions in the high-risk group, to study HIV-like epidemics. It develops two Markovian formulations and derives infection probabilities and a bound-based for heterogeneous contacts, highlighting how group-specific behavior and duration of infectiousness shape spread. The toy HIV model demonstrates that PrEP uptake and increased testing can dramatically reduce infections, especially in the high-risk group, and reveals nonlinear effects from the proportion of high-risk individuals. Overall, the work provides a quantitative framework to assess prevention strategies in heterogeneous populations and emphasizes the potential impact of rapid testing and PrEP on epidemic control.

Abstract

In this paper we generalise a simple discrete time stochastic SIR type model defined by Tuckwell and Williams. The SIR model by Tuckwell and Williams assumes a homogeneous population, a fixed infectious period, and a strict transition from susceptible to infected to recovered. In contrast, our model introduces two groups, and , where group has a higher risk of infection due to increased contact rates. Additionally, the duration in the infected class follows a probability distribution rather than being fixed. Finally, individuals in group can transition directly to the recovered class R, allowing us to analyze the impact of this preventive measure on disease spread. Finally, we apply this model to the spread of HIV, analyzing how risk behaviors, rapid testing, and PrEP-like therapies influence the epidemic dynamics.

Paper Structure

This paper contains 17 sections, 41 equations, 6 figures.

Table of Contents

  1. Introduction
  2. SIR type discrete time stochastic epidemic models
  3. The Reed and Frost model
  4. The SIR model by Tuckwell and Williams
  5. The nonhomogeneus-SIR model
  6. Model description
  7. First Markovian model
  8. Second Markovian model
  9. The evolution of an individual of group $A$
  10. The evolution of an individual of group $B$
  11. Following the evolution of the desease
  12. The basic reproduction number $R_0$
  13. A toy model for HIV
  14. Effect of the Population Engaging in High-Risk Practices
  15. Effect of PrEP Introduction
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Sample Sizes: Effect of Population $B$ Size on Epidemic Development.
  • Figure 2: PrEP: Effect on Group B
  • Figure 3: PrEP: Effect of a regular plan for the introduction of PrEP on the Number of Infected Individuals
  • Figure 4: Prep: Effect of an emergency plan on the Number of Infected Individuals.
  • Figure 5: Distribution $D$: Effect on the Number of Infected Individuals
  • ...and 1 more figures