A delayed differential equation model for mosquito population suppression with sterile mosquitoes

Abstract

The technique of sterile mosquitoes plays an important role in the control of mosquito-borne diseases such as malaria, dengue, yellow fever, west Nile, and Zika. To explore the interactive dynamics between the wild and sterile mosquitoes, we formulate a delayed mosquito population suppression model with constant releases of sterile mosquitoes. Through the analysis of global dynamics of solutions of the model, we determine a threshold value of the release rate such that if the release threshold is exceeded, then the wild mosquito population will be eventually suppressed, whereas when the release rate is less than the threshold, the wild and sterile mosquitoes coexist and the model exhibits a complicated feature. We also obtain theoretical results including a sufficient and necessary condition for the global asymptotic stability of the zero solution. We provide numerical examples to demonstrate our results and give brief discussions about our findings.

Stability analysis in a mosquito population suppression model

“…The dynamics of (4). For t 0 = 0 and parameters given in (16), (4) with s g = 8000 displays bistability as shown in (A), while in (B), the unique nonnegative equilibrium s 0 = 0 of (4) with s g = 15000 is globally asymptotically stable.…”

“…Next, we consider the dynamics of (3) when the number of sterile mosquitoes released is assumed as a given nonnegative continuous function S g (t). Assume the model parameters of (3) are given in (16) and t 0 = 0. If we take S g (t) = 1000(7 + 3 sin(π t/4)), t ≥ 0,…”

“…The dynamics of (3). For t 0 = 0 and the model parameters given in (16), as shown in (A), the trivial equilibrium s 0 = 0 with S g (t) defined in (17) is locally but not globally asymptotically stable, while in (B), it is globally asymptotically stable with S g (t) given in (18). For the case when S g (t) passes through s * g , we find in (C) that s 0 = 0 with S g (t) defined in (19) is locally but not globally asymptotically stable, while in (D), it is globally asymptotically stable with S g (t) given in (20).…”

“…Finally, we further check the global dynamics of (4) where the number of sterile mosquitoes released in the field keeps at a constant level. Assume that t 0 = 0 and the model parameters of (4) are given in (16). In Figure 2(A), let s g = 8000 < s * g ≈ 12005.…”

“…In recent years, mathematical models have been developed to understand the interplays between wild and sterile mosquitoes [4,5,7,11,15,16,18,19,21,25,31,32], or Wolbachiainfected mosquitoes [8][9][10][12][13][14]30,33,[35][36][37], and thus to provide feasible releasing policies in mosquito control. Among these works, an ordinary differential equation model for the competition of wild and sterile mosquitoes was proposed in [4]…”