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A new co-infection model for the transmission dynamics of two virus hepatitis B (HBV) and coronavirus (COVID-19) is formulated to study the effect of white noise intensities. First, we present the model equilibria and basic reproduction number. The local stability of the equilibria points is proved. Moreover, the proposed stochastic model has been investigated for a non-negative solution and positively invariant region. With the help of Lyapunov function, analysis was performed and conditions for extinction and persistence of the disease based on the stochastic co-infection model were derived. Particularly, we discuss the dynamics of the stochastic model around the disease-free state. Similarly, we obtain the conditions that fluctuate at the disease endemic state holds if . Based on extinction as well as persistence some conditions are established in form of expression containing white noise intensities as well as model parameters. The numerical results have also been used to illustrate our analytical results.

In this paper, we study a multi-strain SEIR epidemic model with both bilinear and non-monotone incidence functions. Under biologically motivated assumptions, we show that the model has two basic reproduction numbers that we noted [Formula: see text] and [Formula: see text]; and four equilibrium points. Using the Lyapunov method, we prove that if [Formula: see text] and [Formula: see text] are less than one then the disease-free equilibrium is Globally Asymptotically Stable, thus the disease will be eradicated. However, if one of the two basic reproduction numbers is greater than one, then the strain that persists is that with the larger basic reproduction number. And finally if both of the two basic reproduction numbers are equal or greater than one then the total endemic equilibrium is globally asymptotically stable. A numerical simulation is also presented to illustrate the influence of the psychological effect, of people to infection, on the spread of the disease in the population. This simulation can be used to determine the status of different diseases in a region using the corresponding data and infectious disease parameters.

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