Qualitative study of transmission dynamics of drug-resistant malaria

Lourdes Esteva, Abba Gumel, Cruz Vargas de León

Research output: Contribution to journalArticle

34 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)611-630
Number of pages20
JournalMathematical and Computer Modelling
Volume50
Issue number3-4
DOIs
StatePublished - Aug 2009
Externally publishedYes

Fingerprint

Malaria
Drugs
Reproduction number
Coexistence
Globally Asymptotically Stable
Exceed
Lyapunov functions
Invariance
Model
Competitive Exclusion
LaSalle's Invariance Principle
Drug Resistance
Deterministic Model
Asymptotically Stable
Nonlinear Function
Linearity
Lyapunov Function
Monitoring
Continuum
Proportion

Keywords

  • Antimalarial drugs
  • Equilibria
  • Malaria
  • Reproduction number
  • Stability
  • Wild/resistant strain

ASJC Scopus subject areas

  • Computer Science Applications
  • Modeling and Simulation

Cite this

Qualitative study of transmission dynamics of drug-resistant malaria. / Esteva, Lourdes; Gumel, Abba; de León, Cruz Vargas.

In: Mathematical and Computer Modelling, Vol. 50, No. 3-4, 08.2009, p. 611-630.

Research output: Contribution to journalArticle

Esteva, Lourdes ; Gumel, Abba ; de León, Cruz Vargas. / Qualitative study of transmission dynamics of drug-resistant malaria. In: Mathematical and Computer Modelling. 2009 ; Vol. 50, No. 3-4. pp. 611-630.
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