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# Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks

• * Corresponding author: Noémi Nagy
• The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at $\tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n}$, where $\tau$ and $\gamma$ are infection and recovery rates, respectively, $n$ is the average degree of the network and $\langle n^{2}\rangle$ is the second moment of the degree distribution. For subcritical values of $\tau$ the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of $\tau$ the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.

Mathematics Subject Classification: Primary: 34C23, 34D23, 92C42.

 Citation: • • Figure 1.  Case of the globally stable disease-free equilibrium: Time evolution of the expected number of the infected nodes $[I_1]$, $[I_2]$, $[I_3]$ of degree $n_1 = 2$, $n_2 = 3$, $n_3 = 4$ respectively, started with $900$, $500$ randomly chosen infected nodes (i.e. firstly $765$, $90$, $45$ infected nodes of degree 2, 3, 4 respectively (continuous curves), secondly $425$, $50$, $25$ infected nodes of degree $2$, $3$, $4$ respectively (dashed curves)). The parameters are: $N = 1000$, $N_1 = 850$, $N_2 = 100$, $N_3 = 50$, $\gamma = 1$, $\tau = 0.5$, $\tau_c = 0.7586$

Figure 2.  Case of the globally stable endemic equilibrium: Time evolution of the expected number of the infected nodes $[I_1]$, $[I_2]$, $[I_3]$ of degree $n_1 = 2$, $n_2 = 3$, $n_3 = 4$ respectively, started with $900$, $500$ randomly chosen infected nodes (i.e. firstly $765$, $90$, $45$ infected nodes of degree 2, 3, 4 respectively (continuous curves), secondly $425$, $50$, $25$ infected nodes of degree $2$, $3$, $4$ respectively (dashed curves)). The parameters are: $N = 1000$, $N_1 = 850$, $N_2 = 100$, $N_3 = 50$, $\gamma = 1$, $\tau = 1$, $\tau_c = 0.7586$

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