### Mathematical Biosciences and Engineering

2019, Issue 4: 2852-2874. doi: 10.3934/mbe.2019141
Research article

# The SIS model with diffusion of virus in the environment

• Received: 09 January 2019 Accepted: 20 March 2019 Published: 02 April 2019
• In this paper, we propose an SIS-type reaction-diffusion equations, which contains both direct transmission and indirect transmission via free-living and spatially diffusive bacteria/virus in the contaminated environment, motivated by the dynamics of hospital infections. We establish the basic reproduction number $R_0$ which can act as threshold level to determine whether the disease persists or not. In particular, if $R_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, whereas the system is uniformly persistent for $R_0>1$. For the spatially homogeneous system, we investigate the traveling wave solutions and obtain that there exists a critical wave speed, below which there has no traveling waves, above which the traveling wave solutions may exist for small diffusion coefficient by the geometric singular perturbation method. The finding implies that great spatial transmission leads to an increase in new infection, while large diffusion of bacteria/virus results in the new infection decline for spatially heterogeneous environment.

Citation: Danfeng Pang, Yanni Xiao. The SIS model with diffusion of virus in the environment[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2852-2874. doi: 10.3934/mbe.2019141

### Related Papers:

• In this paper, we propose an SIS-type reaction-diffusion equations, which contains both direct transmission and indirect transmission via free-living and spatially diffusive bacteria/virus in the contaminated environment, motivated by the dynamics of hospital infections. We establish the basic reproduction number $R_0$ which can act as threshold level to determine whether the disease persists or not. In particular, if $R_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, whereas the system is uniformly persistent for $R_0>1$. For the spatially homogeneous system, we investigate the traveling wave solutions and obtain that there exists a critical wave speed, below which there has no traveling waves, above which the traveling wave solutions may exist for small diffusion coefficient by the geometric singular perturbation method. The finding implies that great spatial transmission leads to an increase in new infection, while large diffusion of bacteria/virus results in the new infection decline for spatially heterogeneous environment.

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• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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