Threshold Dynamics and Probability Density Function of a Stochastic Multi-Strain Coinfection Model with Amplification and Vaccination

Multi-strain infectious diseases, which are usually prevented from spreading widely by vaccination, have two main transmission mechanisms: competitive exclusion and co-existence. In this paper, a stochastic multi-strain coinfection model with amplification and vaccination is developed. For the deterministic model, the basic reproduction number \(R_0\) $R_0$ and fixed points are provided. For the stochastic model, we first prove the existence and uniqueness of the positive solution under any initial value. Then, a portion of those infected with the common strain will always become patients with the amplified strain, which increases the risk of death from the disease. Therefore, we verified that patients with common strains would become extinct if \(R_1^s<1\) $R_1^s<1$ . Furthermore, by constructing the Lyapunov function, we find that model (3) has a unique ergodic stationary distribution if \(R_0^S>1\) $R_0^S>1$ . Particularly, we get a concrete form of the probability density of the distribution \(\kappa (\cdot )\) $\kappa (·)$ near equilibrium \(E^*\) $E^{\ast }$ , where \(E^*\) $E^{\ast }$ is the quasi-local equilibrium of the stochastic model. Finally, the results are verified by numerical simulation. The results show that vaccination can control disease outbreaks or even eliminate them.