Mathematical model of mosquito population dynamics with constants and periodic releases of Wolbachia-infected males

Abstract

In this article, we propose three mathematical models of mosquito dynamics. The first model describes the dynamics of wild mosquitoes in combination with mechanical control and larvicide. For this model, we derive a threshold parameter $\mathcal{N}$, called the basic number of offspring, and show that the trivial equilibrium is globally asymptotically stable when $\mathcal{N}\le 1$, while if $\mathcal{N}>1$, the non-trivial equilibrium is globally asymptotically stable. In the second model, we evaluated a constant release of Wolbachia-infected males in the wild mosquito population. Our analysis identified a critical threshold for releasing Wolbachia-infected male mosquitoes $M_i^{\mathrm{crit}}$ to reduce or eliminate the wild mosquito population. In the third model, we assessed wild mosquito dynamics by periodically releasing Wolbachia-infected male mosquitoes. The analysis of this model produced two thresholds, $T^{\ast }$ the threshold on the release period and ${\mathrm{\Lambda }}^{\ast }$ the threshold on the release quantity. The existence of trivial periodic solutions is obtained, and the corresponding local and global stability conditions are proved by Floquet's theory and Lyapunov's stability theorem, respectively. We also prove the existence conditions for non-trivial periodic solutions and their local stability. Finally, we perform some numerical simulations to illustrate our theoretically pertinent results.

Keywords:

• Basic offspring number
• stability
• vector control
• Wolbachia
• impulsive model
• periodic solutions

2020 Mathematics Subject Classifications:

• 65L12
• 65M20
• 65N40

Disclosure statement

No potential conflict of interest was reported by the author(s).