Critical value in a SIR network model with heterogeneous infectiousness and susceptibility

Shuixian Yan; Sanling Yuan; Shuixian Yan; Sanling Yuan

doi:10.3934/mbe.2020310

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5802-5811. doi: 10.3934/mbe.2020310

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Critical value in a SIR network model with heterogeneous infectiousness and susceptibility

Shuixian Yan ^1,2,
Sanling Yuan ^{1
,
,}

1.
University of Shanghai for Science and Technology, Shanghai 200093, China
2.
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou 341000, China

Received: 16 May 2020 Accepted: 27 July 2020 Published: 01 September 2020

Using the technique of edge-based compartmental modelling (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks, in a recent paper (PloS One, 8(2013), e69162), Miller and Volz established an SIR disease network model with heterogeneous infectiousness and susceptibility. The authors provided a numerical example to demonstrate its validity but they did not perform any mathematical analysis of the model. In this paper, we resolve this problem. Using the nature of irreducible cooperative system in the theory of monotonic dynamical system, we prove that the dynamics of the model are completely determined by a critical value ρ₀: When ρ₀ > 0, the disease persists in a globally stable outbreak equilibrium; while when ρ₀ < 0, the disease dies out in the population and the disease free equilibrium is globally stable.
- edge-based SIR epidemic model,
- contact network,
- heterogeneous infectiousness and susceptibility,
- global dynamics
Citation: Shuixian Yan, Sanling Yuan. Critical value in a SIR network model with heterogeneous infectiousness and susceptibility[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5802-5811. doi: 10.3934/mbe.2020310

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Abstract

Using the technique of edge-based compartmental modelling (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks, in a recent paper (PloS One, 8(2013), e69162), Miller and Volz established an SIR disease network model with heterogeneous infectiousness and susceptibility. The authors provided a numerical example to demonstrate its validity but they did not perform any mathematical analysis of the model. In this paper, we resolve this problem. Using the nature of irreducible cooperative system in the theory of monotonic dynamical system, we prove that the dynamics of the model are completely determined by a critical value ρ₀: When ρ₀ > 0, the disease persists in a globally stable outbreak equilibrium; while when ρ₀ < 0, the disease dies out in the population and the disease free equilibrium is globally stable.

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