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# On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections

• * Corresponding author: A. Gómez-Corral
The authors are supported by the Ministry of Economy and Competitiveness (Government of Spain), Project MTM2014-58091-P.
• A Markov-modulated framework is presented to incorporate correlated inter-event times into the stochastic susceptible-infectious-recovered (SIR) epidemic model for a closed finite community. The resulting process allows us to deal with non-exponential distributional assumptions on the contact process between the compartment of infectives and the compartment of susceptible individuals, and the recovery process of infected individuals, but keeping the dimensionality of the underlying Markov chain model tractable. The variability between SIR-models with distinct level of correlation is discussed in terms of extinction times, the final size of the epidemic, and the basic reproduction number, which is defined here as a random variable rather than an expected value.

Mathematics Subject Classification: Primary: 92D30.

 Citation: • • Figure 1.  The expected number $E[S(T_{(i, s)})]$ of surviving susceptibles versus the basic reproduction number $\mathcal{R}_0$ for scenarios A, C and I, and initial state $(i, s) = (1, n)$ with $1+n = 50$

Figure 2.  The expected value $\overline{R}_{exact, 0}$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios A, E and G, and initial size $N = 1+n\in\{10, 50,100,150\}$

Figure 3.  The expected value $\overline{R}_{exact, 0}$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios B, F and H, and initial size $N = 1+n\in\{10, 50,100,150\}$

Figure 4.  The expected value $\overline{R}_{exact, 0}$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios C, D and I, and initial size $N = 1+n\in\{10, 50,100,150\}$

Figure 5.  The expected value $\overline{R}_p$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios A, E and G, and initial size $N = 1+n\in\{10, 50,100,150\}$

Figure 6.  The expected value $\overline{R}_p$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios B, F and H, and initial size $N = 1+n\in\{10, 50,100,150\}$

Figure 7.  The expected value $\overline{R}_p$ versus the basic reproduction number $\mathcal{R}_0$ for scenarios C, D and I, and initial size $N = 1+n\in\{10, 50,100,150\}$

Table 1.  Stochastic transitions, events and rates in the basic SIR-model

 Transitions Events $~$ Rates $i\to i+1, ~s\to s-1, ~r\to r, ~\hbox{for }i, s\in\mathbb{N}$ A new infection $\lambda_{i, s}$ $i\to i-1, ~s\to s, ~r\to r+1, ~\hbox{for }i\in\mathbb{N}, s\in\mathbb{N}_0$ A removal $\mu_i$

Table 2.  Nine scenarios defined in terms of the matrices ${\bf C}^*_{1, (i, s)}$ and ${\bf D}^*_{1, (i, s)}$ for the occurrence of an infection and the removal of an infective, respectively, which are related to MAPs with positive and negative correlation (with characteristic matrices $({\bf E}^+_0, {\bf E}_1^+)$ and $({\bf E}^-_0, {\bf E}_1^-)$, respectively), and Poisson streams (with rates $\lambda_{i, s}$ and $\mu_i$)

 Scenario Occurrence of an infection (Matrices ${\bf C}_{1, (i, s)}^*$) Occurrence of a removal (Matrices ${\bf D}_{1, (i, s)}^*$) A $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}^+_1$ $\mu_i$ B $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}^-_1$ $\mu_i$ C $\lambda_{i, s}$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$ D $\lambda_{i, s}$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$ E $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^+$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$ F $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^-$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$ G $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^+$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^+_1$ H $\hat{\lambda}^{-1}(1)\lambda_{i, s} {\bf E}_1^-$ $\hat{\lambda}^{-1}(2)\mu_i {\bf E}^-_1$ I $\lambda_{i, s}$ $\mu_i$
• Figures(7)

Tables(2)

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