This paper proposes a stochastic intestinal chemostat model considering microbial migration, intraspecific competition and stochastic perturbation. First, the extinction and persistence in mean of the intestinal microbe of the chemostat model are investigated by constructing the appropriate Lyapunov functions. Second, we explore and obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the model by using ergodic theory. The results show stochastic interference has a critical impact on the extinction and sustainable survival of the intestinal microbe. Eventually, numerical simulations are carried out to verify the theoretical results.

We investigated the dynamic effect of stochastic environmental fluctuations on the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus infection system with time delay and mediations by the angiotensin-converting enzyme 2 (ACE2) receptor protein. First, we discussed the existence and uniqueness of global positive solutions as well as the stochastic ultimate boundedness of the stochastic SARS-CoV-2 model. Second, the asymptotic properties of stochastic time-delay system were investigated by constructing a number of appropriate Lyapunov functions and applying differential inequality techniques. These properties indicated a positive relationship between the strength of oscillations and the intensity of environmental fluctuations, and this launched the properties of a deterministic system. When the random disturbance was relatively large, the disease went extinct. When the random disturbance was relatively small and $R_0<1$, the disease could become extinct. Conversely, when the random disturbance was smaller and $R_0>1$, then it would oscillate around the disease enduring equilibrium. At last, a series of numerical simulations were carried out to show how the SARS-CoV-2 system was affected by the intensity of environmental fluctuations and time delay.

The spread of viruses can be effectively reduced by the publicity of epidemic prevention. Additionally, the interaction between heterogeneous strains has a significant effect on virus evolution. Thus, we first establish an evolutionary dynamics Susceptible-Infected- Recovered (SIR) model which considers the interaction between heterogeneous strains. We utilize adaptive dynamics to investigate the evolutionary outcomes of the trade-off between transmission and virulence. Second, we perform a critical function analysis to generalize the results independent of specific trade-off assumptions and to determine the conditions for evolutionary stability and convergence stability. Last, we investigate the effects of different publicity measures on virulence evolution under two types of interactions, including the case of excess mortality alone and the coexistence of excess mortality and superinfection. Based on the general hypothesis of transmission virulence trade-off, we introduce the cost of host mobility caused by the scope and intensity of publicity. Numerical simulations present a set of evolutionary results, including continuously stable strategies, evolutionary branching points, repellers, and the Garden of Eden. Our results indicate that an excessive publicity scope and intensity can drive the epidemic evolution towards higher virulence. Both types of interactions suggest that continuously increasing the publicity scope under a low publicity intensity can effectively reduce virulence. Furthermore, the concurrent presence of excess mortality and superinfection induces the emergence of a higher virulence.