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Global stability in an age-structured SIRS malaria transmission model

Seraphin Djaoue, Quentin Richard, Antoine Perasso, Irépran Damakoa

Abstract

This paper proposes and analyzes a malaria transmission model structured by the chronological age of the human host population. The model couples an age-structured SIRS system for humans, incorporating waning immunity, with an SI system for mosquitoes under mass-action transmissions. Using integrated semigroup theory and spectral analysis, we establish the well-posedness of the model, derive the basic reproduction number, and prove the global asymptotic stability of the parasite-free equilibrium by using a Lyapunov functional, when $R_0\leq 1$, thereby excluding the possibility of backward bifurcation. Numerical simulations further suggest the global stability of the endemic equilibrium when $R_0>1$.

Global stability in an age-structured SIRS malaria transmission model

Abstract

This paper proposes and analyzes a malaria transmission model structured by the chronological age of the human host population. The model couples an age-structured SIRS system for humans, incorporating waning immunity, with an SI system for mosquitoes under mass-action transmissions. Using integrated semigroup theory and spectral analysis, we establish the well-posedness of the model, derive the basic reproduction number, and prove the global asymptotic stability of the parasite-free equilibrium by using a Lyapunov functional, when , thereby excluding the possibility of backward bifurcation. Numerical simulations further suggest the global stability of the endemic equilibrium when .
Paper Structure (16 sections, 18 theorems, 194 equations, 2 figures, 1 table)

This paper contains 16 sections, 18 theorems, 194 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Well-posedness and global solution
  3. Model formulation
  4. Abstract Cauchy problem
  5. Linear and nonlinear part of Cauchy problem
  6. Boundedness and global existence
  7. Equilibria and their Stability
  8. The basic reproduction number $\mathcal{R} _0$
  9. Local stability of the parasite-free equilibrium
  10. Attractiveness of the parasite-free equilibrium
  11. Global stability of the parasite-free equilibrium
  12. On the endemic equilibrium
  13. Numerical simulations
  14. Numerical scheme
  15. Parameters
  16. ...and 1 more sections

Key Result

Proposition 1

The operator $A:D(A) \subset X$ is a Hille-Yosida operator with $(-\mu_0,+\infty) \subset \rho(A)$ (which denotes the resolvent set of $A$) and for all $\lambda > -\mu_0$, $(\lambda I - A)^{-1} X_+ \subset X_+$. Moreover, for all $\omega> 0$, the operator $A-\omega I$ is also a Hille-Yosida operator

Figures (2)

  • Figure 1: Convergence to an endemic equilibrium when $\mathcal{R}_0>1$ for different initial conditions
  • Figure 2: Convergence to the parasite-free equilibrium when $\mathcal{R}_0<1$ for different initial conditions

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Proposition 4
  • Lemma 2
  • Lemma 3
  • Definition 1
  • ...and 10 more