[Paper Review] The Dynamics of a Vertically Transmitted Disease

M. R. Razvan|ArXiv.org|Sep 26, 2000

COVID-19 epidemiological studies8 references3 citations

TL;DR

This paper develops a global analysis of an SIRS epidemiological model for vertically transmitted diseases, incorporating differential birth and death rates across susceptible, infected, and recovered classes. Using three threshold parameters—R₀, R₁, and R₂—it establishes conditions for disease extinction, endemic equilibrium, and population growth, demonstrating that removing infected newborns reduces transmission risk and stabilizes the system.

ABSTRACT

An SIRS epidemiological model for a vertically transmitted disease is discussed. We give a complete global analysis in terms of three explicit threshold parameters which respectively govern the existence and stability of an endemic proportion equilibrium, the increase of the total population and the growth of the infective population. This paper gereralize the results of Busenberg and van den Driessche.

Motivation & Objective

• To extend Busenberg and van den Driessche's SIRS model to include vertical transmission in disease dynamics.
• To analyze the global behavior of the system using proportions of susceptible, infected, and recovered individuals.
• To identify threshold parameters governing disease persistence, population growth, and stability of endemic equilibria.
• To examine the impact of removing infected newborns on epidemic control and system stability.

Proposed method

• Formulates a system of ODEs with distinct birth and death rates for susceptible, infected, and recovered individuals.
• Introduces a proportions system by normalizing S, I, R by total population N, reducing the system to a planar dynamical system on the simplex D.
• Uses the Poincaré index technique to prove uniqueness of endemic equilibria and absence of periodic orbits in the feasibility region.
• Defines three threshold parameters: R₀ (basic reproduction number), R₁ (total population growth), and R₂ (infective population growth).
• Analyzes asymptotic behavior via stability of rest points and proves convergence of all solutions to equilibrium points.
• Applies comparison theorems and integral estimates to show unbounded growth of I(t) and R(t) under certain threshold conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the total population grow or decline in the presence of a vertically transmitted disease?
• RQ2 How do distinct birth and death rates across host classes affect the stability of endemic equilibria?
• RQ3 What role does the removal of infected newborns play in controlling disease spread and population dynamics?
• RQ4 Can the proportions system be analyzed globally without periodic orbits, and what does this imply for long-term behavior?
• RQ5 How do the threshold parameters R₀, R₁, and R₂ collectively determine the fate of the disease and population?

Key findings

• If R₁ < 1, the total population N(t) → 0; if R₁ > 1, N(t) → ∞, indicating that R₁ governs population extinction or growth.
• If R₂ < 1, the infective and recovered populations (I(t), R(t)) → (0, 0); if R₂ > 1, (I(t), R(t)) → (∞, ∞), showing R₂ controls infective population dynamics.
• When R₀ ≤ 1, the system converges to the disease-free equilibrium (1, 0, 0), regardless of R₁ or R₂, implying disease extinction.
• When R₀ > 1, the system converges to a unique endemic equilibrium (s*, i*, r*), and if R₁ > 1 and R₂ > 1, N(t) → ∞ and (I(t), R(t)) → (∞, ∞).
• The parameter β₁ (proportion of infected newborns) critically affects R₀ and R₂; increasing β₂ (removal of infected newborns) reduces β₁ and improves disease control.
• The model shows that removing infected newborns (via β₂) reduces transmission risk and stabilizes the system, offering a key control strategy.