Final size of a structured SIRD Model with active-population force of infection

TL;DR

This paper addresses the final epidemic size in a two-group SIRD model where the force of infection is scaled by the active population. It develops a fixed-point framework: a logarithmic first integral and a nonlinear map $T$ yield a fixed-point equation $S^\infty = T(S^\infty)$ that avoids solving the full ODE system. The authors establish positivity of $S^\infty$, prove existence and, under hypotheses H1 and H2, uniqueness and monotone convergence of the fixed-point iteration, and relate $S^\infty$ to the basic reproduction number $\mathcal{R}_0$ via a next-generation matrix $K_1 = B (\Gamma + M)^{-1}$ and a logarithmic identity. Simulations compare the active-population and classical two-group SIRD models, revealing qualitative differences such as multiple waves and shifts in the drivers of infection, and demonstrate convergence of the fixed-point method and Newton's method for computing $S^\infty$.

Abstract

We consider a SIRD epidemic model for a population composed of two groups of individuals with asymmetric interactions, where the force of infection depends on the active (alive) population in each group, rather than on the total population, as in the classical formulation. We prove that the final state for susceptible individuals is always positive and characterize it as the unique fixed point of a map. We also relate the final size to the basic reproduction number and show that the final number of susceptibles decreases when transmission rates increase. Numerical simulations compare the active-population and classical two-group SIRD models, showing differences in final size and the occurrence of multiple epidemic waves. The convergence of the fixed point approach is also illustrated.

Figures (3)

• Figure 1: Comparison of $S_{i,c}^\infty$ and $S_{i,a}^\infty$ vs $\mu$ (classical vs active)
• Figure 2:
• Figure :

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Definition 3.1