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doi: 10.3934/dcdsb.2021012
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Stochastic and deterministic SIS patch model

Ténan Yeo

Aix Marseille Univ, Marseille, France, CNRS, Centrale Marseille, I2M, Marseille, France, Univ. F. H. Boigny, UFR-MI, Abidjan, Côte d'Ivoire

Received  September 2020 Revised  November 2020 Early access December 2020

Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. We next study the equilibria of the deterministic model. Our main contribution is a stability result of the endemic equilibrium in the case $ \mathcal{R}_0>1 $. Finally we compare the equilibria with those of the homogeneous model, and with those of isolated patches.

Keywords: epidemic patch model, law of large numbers, endemic equilibrium.
Mathematics Subject Classification: Primary: 60J75, 34D23; Secondary: 60J20.
Citation: Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021012
References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[2]

H. Andersson and T. Britton, Stochastic Epidemic Models and their Statistical Analysis, Vol. 151, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1158-7.  Google Scholar

[3]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2$^{nd}$ edition, Griffin, London, 1975.  Google Scholar

[4]

M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: Numbers, densities and areas, Epidemiology & Infection, 129 (2002), 147-153.  doi: 10.1017/S0950268802007148.  Google Scholar

[5]

J. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[6]

D. Bichara, Y. Kang, C. Castillo-Chavez, R. Horan and C. Perrings, SIS and SIR epidemic models under virtual dispersal, Bulletin of Mathematical Biology, 77 (2015), 2004-2034.  doi: 10.1007/s11538-015-0113-5.  Google Scholar

[7]

T. Britton and E. Pardoux, Stochastic epidemic models with inference, in Lecture Notes in Math. (eds. F. Ball, C. Larédo, D. Sirl and V. C. Tran), 2255, Springer, (2019), 1–120. doi: 10.1007/978-3-030-30900-8.  Google Scholar

[8]

P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[9]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley & Sons, 1986. doi: 10.1002/9780470316658.  Google Scholar

[10]

A. Fall, Epidemiological models and Lyapunov functions, Mathematical Modelling of Natural Phenomena, 2 (2007), 62-83.  doi: 10.1051/mmnp:2008011.  Google Scholar

[11]

M. W. Hirsch, The dynamical systems approach to differential equations, Bulletin of the American Mathematical Society, 11 (1984), 1-64.  doi: 10.1090/S0273-0979-1984-15236-4.  Google Scholar

[12]

W. O. Kermack and A. G. Mckendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, 115 (1927), 700-721.   Google Scholar

[13]

L. Michael and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Applied Mathematics Quarterly, 17 (2009), 175-187.   Google Scholar

[14]

E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, J. Wiley & Sons, 2008. doi: 10.1002/9780470721872.  Google Scholar

[15]

J. Rebaza, Global stability of a multipatch disease epidemics model, Chaos, Solitons & Fractals, 120 (2019), 56-61.  doi: 10.1016/j.chaos.2019.01.020.  Google Scholar

[16]

R. Varga, Matrix Iterative Analysis, , Englewood Cliffs, NJ: Prentice-Hall, Inc, 1962.  Google Scholar

[17]

M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE Transactions Automatic Control, 25 (1980), 773-779.  doi: 10.1109/TAC.1980.1102422.  Google Scholar

[18]

G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Mathematical Biosciences, 11 (1971), 261-265.  doi: 10.1016/0025-5564(71)90087-3.  Google Scholar

[19]

S. Zhisheng and P. Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM Journal on Applied Mathematics, 73 (2013), 1513-1532.  doi: 10.1137/120876642.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[2]

H. Andersson and T. Britton, Stochastic Epidemic Models and their Statistical Analysis, Vol. 151, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1158-7.  Google Scholar

[3]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2$^{nd}$ edition, Griffin, London, 1975.  Google Scholar

[4]

M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: Numbers, densities and areas, Epidemiology & Infection, 129 (2002), 147-153.  doi: 10.1017/S0950268802007148.  Google Scholar

[5]

J. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[6]

D. Bichara, Y. Kang, C. Castillo-Chavez, R. Horan and C. Perrings, SIS and SIR epidemic models under virtual dispersal, Bulletin of Mathematical Biology, 77 (2015), 2004-2034.  doi: 10.1007/s11538-015-0113-5.  Google Scholar

[7]

T. Britton and E. Pardoux, Stochastic epidemic models with inference, in Lecture Notes in Math. (eds. F. Ball, C. Larédo, D. Sirl and V. C. Tran), 2255, Springer, (2019), 1–120. doi: 10.1007/978-3-030-30900-8.  Google Scholar

[8]

P. V. D. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[9]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley & Sons, 1986. doi: 10.1002/9780470316658.  Google Scholar

[10]

A. Fall, Epidemiological models and Lyapunov functions, Mathematical Modelling of Natural Phenomena, 2 (2007), 62-83.  doi: 10.1051/mmnp:2008011.  Google Scholar

[11]

M. W. Hirsch, The dynamical systems approach to differential equations, Bulletin of the American Mathematical Society, 11 (1984), 1-64.  doi: 10.1090/S0273-0979-1984-15236-4.  Google Scholar

[12]

W. O. Kermack and A. G. Mckendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, 115 (1927), 700-721.   Google Scholar

[13]

L. Michael and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Applied Mathematics Quarterly, 17 (2009), 175-187.   Google Scholar

[14]

E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, J. Wiley & Sons, 2008. doi: 10.1002/9780470721872.  Google Scholar

[15]

J. Rebaza, Global stability of a multipatch disease epidemics model, Chaos, Solitons & Fractals, 120 (2019), 56-61.  doi: 10.1016/j.chaos.2019.01.020.  Google Scholar

[16]

R. Varga, Matrix Iterative Analysis, , Englewood Cliffs, NJ: Prentice-Hall, Inc, 1962.  Google Scholar

[17]

M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE Transactions Automatic Control, 25 (1980), 773-779.  doi: 10.1109/TAC.1980.1102422.  Google Scholar

[18]

G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Mathematical Biosciences, 11 (1971), 261-265.  doi: 10.1016/0025-5564(71)90087-3.  Google Scholar

[19]

S. Zhisheng and P. Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM Journal on Applied Mathematics, 73 (2013), 1513-1532.  doi: 10.1137/120876642.  Google Scholar

Figure 1.  Metapopulation
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Table 1.  proportion of $ \mathbf{I}_1^* $ and $ \mathbf{I}_2^* $ when patches are connected
$ \lambda_1 $ $ \lambda_2 $ $ \gamma_1 $ $ \gamma_2 $ $ \nu_I $ $ \nu_S $ $ \left(\dfrac{ \mathbf{I}_1^*}{ \mathbf{S}_1^*+ \mathbf{I}_1^*},\dfrac{ \mathbf{I}_2^*}{ \mathbf{S}_2^*+ \mathbf{I}_2^*}\right) $
1.5 2 1 1 0.0001 0.0001 (0.332, 0.507)
1.5 2 1 1 0.0001 0.0005 (0.334, 0.497)
1.5 2 1 1 0.001 0.0001 (0.333, 0.497)
1.5 2 1 1 0.0001 0.001 (0.332, 0.497)
             
3 2.5 1 1 0.0001 0.0001 (0.667, 0.598)
3 2.5 1 1 0.0007 0.0001 (0.666, 0.599)
3 2.5 1 1 0.001 0.0001 (0.666, 0.598)
3 2.5 1 1 0.0001 0.001 (0.666, 0.598)
             
1.5 1.2 01 1 0.0001 0.0001 (0.332, 0.165)
1.5 1.2 1 1 0.0001 0.0009 (0.332, 0.165)
1.5 1.2 1 1 0.001 0.0001 (0.333, 0.165)
1.5 1.2 1 1 0.0001 0.008 (0.332, 0.165)
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$ \lambda_1 $ $ \lambda_2 $ $ \gamma_1 $ $ \gamma_2 $ $ \nu_I $ $ \nu_S $ $ \left(\dfrac{ \mathbf{I}_1^*}{ \mathbf{S}_1^*+ \mathbf{I}_1^*},\dfrac{ \mathbf{I}_2^*}{ \mathbf{S}_2^*+ \mathbf{I}_2^*}\right) $
1.5 2 1 1 0.0001 0.0001 (0.332, 0.507)
1.5 2 1 1 0.0001 0.0005 (0.334, 0.497)
1.5 2 1 1 0.001 0.0001 (0.333, 0.497)
1.5 2 1 1 0.0001 0.001 (0.332, 0.497)
             
3 2.5 1 1 0.0001 0.0001 (0.667, 0.598)
3 2.5 1 1 0.0007 0.0001 (0.666, 0.599)
3 2.5 1 1 0.001 0.0001 (0.666, 0.598)
3 2.5 1 1 0.0001 0.001 (0.666, 0.598)
             
1.5 1.2 01 1 0.0001 0.0001 (0.332, 0.165)
1.5 1.2 1 1 0.0001 0.0009 (0.332, 0.165)
1.5 1.2 1 1 0.001 0.0001 (0.333, 0.165)
1.5 1.2 1 1 0.0001 0.008 (0.332, 0.165)
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