8. Conclusion and future work
In this paper, we have reviewed the well-known homogeneous and heterogeneous mean field approaches to the SIR model, and we have shown how both approaches do not evaluate the state of every individual in networks. To account for this, we have presented a new individual-based SIR approach that is derived from the continuous time Markov chain model. The new approach evaluates the probability of infection of every individual separately considering the probability of infection of the individual’s neighbors. Unlike previous approaches in the literature that only consider the node degree distribution of the network, the individual-based approach considers the whole network structure. Additionally, we have reviewed the continuous time Markov chain model, and through numerical analysis and evaluation we have studied the deviation between the Markov chain model and the individual-based approach. We have also performed Monte Carlo simulations to show the accuracy of the new approach. Moreover, we have derived the epidemic threshold above which an epidemic prevails in the network. We found that the reciprocal of the spectral radius of the contact network is the epidemic threshold showing the role of network characteristics in the spread of an epidemic. Furthermore, we have shown the condition for the existence of a maximum number of new infected individuals, and how it is related to the epidemic threshold. We have also addressed the role of the spectrum of the contact network and the effective infection rate in determining the maximum number of the new infected individuals. Finally, through the numerical evaluations, we have compared the individual-based approach with the heterogeneous mean field approach.
8. Conclusion and future workIn this paper, we have reviewed the well-known homogeneous and heterogeneous mean field approaches to the SIR model, and we have shown how both approaches do not evaluate the state of every individual in networks. To account for this, we have presented a new individual-based SIR approach that is derived from the continuous time Markov chain model. The new approach evaluates the probability of infection of every individual separately considering the probability of infection of the individual’s neighbors. Unlike previous approaches in the literature that only consider the node degree distribution of the network, the individual-based approach considers the whole network structure. Additionally, we have reviewed the continuous time Markov chain model, and through numerical analysis and evaluation we have studied the deviation between the Markov chain model and the individual-based approach. We have also performed Monte Carlo simulations to show the accuracy of the new approach. Moreover, we have derived the epidemic threshold above which an epidemic prevails in the network. We found that the reciprocal of the spectral radius of the contact network is the epidemic threshold showing the role of network characteristics in the spread of an epidemic. Furthermore, we have shown the condition for the existence of a maximum number of new infected individuals, and how it is related to the epidemic threshold. We have also addressed the role of the spectrum of the contact network and the effective infection rate in determining the maximum number of the new infected individuals. Finally, through the numerical evaluations, we have compared the individual-based approach with the heterogeneous mean field approach.
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