The Mathematics of Infectious Diseases
SIAM Review
Transmission model of endemic human malaria in a partially immune population
Mathematical and Computer Modelling: An International Journal
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This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a disease-free equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.