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December  2021, 26(12): 6173-6184. doi: 10.3934/dcdsb.2021012

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 Published  December 2021 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 and Continuous Dynamical Systems - B, 2021, 26 (12) : 6173-6184. 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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. 

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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. 

[13]

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

[14]

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

[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.

[16]

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

[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.

[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.

[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.

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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