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Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment
January  2020, 25(1): 99-115. doi: 10.3934/dcdsb.2019174

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

 Institute of Mathematics, Eötvös Loránd University Budapest, Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary

* Corresponding author: Noémi Nagy

Received  November 2018 Published  January 2020 Early access  July 2019

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.

Citation: Noémi Nagy, Péter L. Simon. Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 99-115. doi: 10.3934/dcdsb.2019174
##### References:
 [1] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical biosciences and engineering, 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361. [2] K. T. D. Eames and M. J. Keeling, Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases, Proceedings of the National Academy of Sciences, 99 (2002), 13330-13335.  doi: 10.1073/pnas.202244299. [3] X. C. Fu, M. Small and G. R. Chen, Propagation Dynamics on Complex Networks: Models, Methods and Stability Analysis, John Wiley & Sons, 2014. doi: 10.1002/9781118762783. [4] J. P. Gleeson, Binary-state dynamics on complex networks: Pair approximation and beyond, Physical Review X, 3 (2013), 021004. doi: 10.1103/PhysRevX.3.021004. [5] J. K. Hale, Ordinary Differential Equations, New York-London-Sydney, 1969. [6] T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179. [7] M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics, Proceedings of the Royal Society of London. Series B: Biological Sciences, 264 (1997), 1149-1156.  doi: 10.1098/rspb.1997.0159. [8] I. Z. Kiss, J. C. Miller and P.L. Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models, Springer, 2017. doi: 10.1007/978-3-319-50806-1. [9] H. Matsuda, N. Ogita, A. Sasaki and K. Sato, Statistical mechanics of population: The lattice Lotka-Volterra model, Progress of theoretical Physics, 88 (1992), 1035-1049.  doi: 10.1143/ptp/88.6.1035. [10] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Physical Review E, 63 (2001), 066117. doi: 10.1103/PhysRevE.63.066117. [11] M. A. Porter and J. P. Gleeson, Dynamical Systems on Networks: A Tutorial, A tutorial. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 4. Springer, Cham, 2016, arXiv: 1403.7663. doi: 10.1007/978-3-319-26641-1. [12] P. L. Simon, M. Taylor and I. Z. Kiss, Exact epidemic models on graphs using graph automorphism driven lumping, Journal of Mathematical Biology, 62 (2011), 479-508.  doi: 10.1007/s00285-010-0344-x. [13] M. Taylor, P. L. Simon, D. M. Green, T. House and I. Z. Kiss, From Markovian to pairwise epidemic models and the performance of moment closure approximations, Journal of Mathematical Biology, 64 (2012), 1021-1042.  doi: 10.1007/s00285-011-0443-3.

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##### References:
 [1] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical biosciences and engineering, 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361. [2] K. T. D. Eames and M. J. Keeling, Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases, Proceedings of the National Academy of Sciences, 99 (2002), 13330-13335.  doi: 10.1073/pnas.202244299. [3] X. C. Fu, M. Small and G. R. Chen, Propagation Dynamics on Complex Networks: Models, Methods and Stability Analysis, John Wiley & Sons, 2014. doi: 10.1002/9781118762783. [4] J. P. Gleeson, Binary-state dynamics on complex networks: Pair approximation and beyond, Physical Review X, 3 (2013), 021004. doi: 10.1103/PhysRevX.3.021004. [5] J. K. Hale, Ordinary Differential Equations, New York-London-Sydney, 1969. [6] T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 67-73.  doi: 10.1098/rsif.2010.0179. [7] M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics, Proceedings of the Royal Society of London. Series B: Biological Sciences, 264 (1997), 1149-1156.  doi: 10.1098/rspb.1997.0159. [8] I. Z. Kiss, J. C. Miller and P.L. Simon, Mathematics of Epidemics on Networks: From Exact to Approximate Models, Springer, 2017. doi: 10.1007/978-3-319-50806-1. [9] H. Matsuda, N. Ogita, A. Sasaki and K. Sato, Statistical mechanics of population: The lattice Lotka-Volterra model, Progress of theoretical Physics, 88 (1992), 1035-1049.  doi: 10.1143/ptp/88.6.1035. [10] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Physical Review E, 63 (2001), 066117. doi: 10.1103/PhysRevE.63.066117. [11] M. A. Porter and J. P. Gleeson, Dynamical Systems on Networks: A Tutorial, A tutorial. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 4. Springer, Cham, 2016, arXiv: 1403.7663. doi: 10.1007/978-3-319-26641-1. [12] P. L. Simon, M. Taylor and I. Z. Kiss, Exact epidemic models on graphs using graph automorphism driven lumping, Journal of Mathematical Biology, 62 (2011), 479-508.  doi: 10.1007/s00285-010-0344-x. [13] M. Taylor, P. L. Simon, D. M. Green, T. House and I. Z. Kiss, From Markovian to pairwise epidemic models and the performance of moment closure approximations, Journal of Mathematical Biology, 64 (2012), 1021-1042.  doi: 10.1007/s00285-011-0443-3.
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: $N = 1000$, $N_1 = 850$, $N_2 = 100$, $N_3 = 50$, $\gamma = 1$, $\tau = 0.5$, $\tau_c = 0.7586$
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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