[Paper Review] Backward bifurcation in SIRS malaria model

Miliyon Tilahun|arXiv (Cornell University)|Jul 4, 2017

Mathematical and Theoretical Epidemiology and Ecology Models9 references6 citations

TL;DR

This study develops an SIRS malaria transmission model with waning immunity, demonstrating backward bifurcation where the disease-free equilibrium coexists with a stable endemic equilibrium even when the basic reproduction number $R_0 < 1$. The key finding is that reducing the malaria-induced death rate below a critical threshold ($\rho_h^*$) can eliminate backward bifurcation and thus enable disease eradication despite $R_0 < 1$, highlighting the importance of effective treatment in control strategies.

ABSTRACT

We present a deterministic mathematical model for malaria transmission with waning immunity. The model consists of five non-linear system of differential equations. We used next generation matrix to derive the basic reproduction number $R_0$. The disease free equilibrium was computed and its local stability has been shown by the virtue of the Jacobean matrix. Moreover, using Lyapunov function theory and LaSalle Invariance Principle we have proved that the disease free equilibrium is globally asymptotically stable. Conditions for existence of endemic equilibrium point have been established. A qualitative study based on bifurcation theory reveals that backward bifurcation occur in the model. The stable disease free equilibrium of the model coexists with the stable endemic equilibrium when $R_0&lt;1$. Furthermore, we have shown that bringing the number of disease (malaria) induced death rate below some threshold is sufficient enough to eliminate backward bifurcation in the model.

Motivation & Objective

• To develop a deterministic SIRS model for malaria transmission that incorporates waning immunity in humans and SI dynamics in mosquitoes.
• To analyze the stability of the disease-free equilibrium (DFE) and establish conditions for its global asymptotic stability.
• To investigate the existence and stability of endemic equilibria and explore the possibility of backward bifurcation.
• To identify conditions under which backward bifurcation can be eliminated, particularly through reduction of malaria-induced death rate.
• To evaluate the implications of these dynamics for malaria control and eradication strategies.

Proposed method

• Formulated a five-dimensional nonlinear system of ODEs representing SIRS human and SI mosquito compartments with waning immunity.
• Used the next-generation matrix method to derive the basic reproduction number $R_0$.
• Applied Lyapunov function theory and the LaSalle Invariance Principle to prove global asymptotic stability of the disease-free equilibrium when $R_0 < 1$.
• Conducted a qualitative bifurcation analysis using the Sotomayor theorem to establish conditions for backward bifurcation at $R_0 = 1$.
• Derived a threshold value $\rho_h^*$ for the disease-induced death rate below which backward bifurcation is eliminated.
• Performed numerical simulations using the 4th-order Runge-Kutta method in MATLAB to validate analytical results.

Experimental results

Research questions

• RQ1 Under what conditions does backward bifurcation occur in an SIRS malaria model with waning immunity?
• RQ2 Can the disease-free equilibrium remain stable when $R_0 < 1$ while a stable endemic equilibrium also exists?
• RQ3 What role does the malaria-induced death rate play in the occurrence or elimination of backward bifurcation?
• RQ4 Is there a threshold value of the disease-induced death rate that can prevent backward bifurcation and thus enable disease eradication?
• RQ5 How do treatment and intervention strategies targeting the death rate affect the long-term dynamics of malaria transmission?

Key findings

• The disease-free equilibrium is globally asymptotically stable when $R_0 < 1$, as proven using Lyapunov functions and the LaSalle Invariance Principle.
• Backward bifurcation occurs at $R_0 = 1$ when the coefficient $a > 0$, which depends on model parameters including the disease-induced death rate.
• A stable disease-free equilibrium coexists with a stable endemic equilibrium when $R_0 < 1$, indicating that $R_0 < 1$ is not sufficient for disease elimination.
• Backward bifurcation can be eliminated if the disease-induced death rate $\rho_h$ is reduced below the threshold $\rho_h^* = \frac{\Lambda_h^2 \alpha_v^3 (\gamma_h + \alpha_h)}{\beta_v \Lambda_v - \Lambda_h^2 \alpha_v^3}$.
• Numerical simulations confirm the local and global stability of the disease-free equilibrium for $R_0 < 1$, and show sustained endemic transmission when $R_0 > 1$, even with high recovery and immunity loss rates.
• The model simulations with $R_0 = 47.6631$ demonstrate persistent endemic transmission, highlighting the challenge of controlling malaria when $R_0$ is high.