Research article Special Issues

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

  • 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.

    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

    Related Papers:

  • 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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