Modeling the dengue control dynamics based on a delay stochastic differential system

Releasing Wolbachia-infected male mosquitoes into the wild to sterilize wild female mosquitoes is a bio-safe and effective strategy to block the transmission of the dengue virus. Since the spread of the dengue virus is often subject to some randomness, stochastic differential equations can be used to describe the transmission difference of dengue affected by environmental changes. Considering the latent period of virus transmission, we established a delay stochastic differential system model based on the release of Wolbachia-infected males, which has received relatively little attention. By constructing suitable Lyapunov functions, we obtain sufficient conditions for the extinction of the dengue virus and the existence of a unique stationary distribution. To study the efficacy of the releases Wolbachia-infected males, we find two threshold values \(r_{1}^{*}\) $r_1^{\ast }$ and \(r_{2}^{*}\) $r_2^{\ast }$ of the release ratio \(r\) $r$ , where \(r_{1}^{*} > r_{2}^{*}\) $r_1^{\ast }>r_2^{\ast }$ . If \(r > r_{1}^{*}\) $r>r_1^{\ast }$ , then the dengue virus is extinct. If \(r < r_{2}^{*}\) $r<r_2^{\ast }$ , a unique stationary distribution exists, which indicates that the virus is transmitted between humans and mosquitoes. We present some numerical examples to illustrate the impact of environmental noise on the virus dynamics, and we also simulate the dynamics of viruses under different release ratios.