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November  2015, 20(9): 2859-2884. doi: 10.3934/dcdsb.2015.20.2859

## Demographic stochasticity in the SDE SIS epidemic model

 1 Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26, Richmond Street, Gasgow G1 1XH, United Kingdom 2 Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26, Richmond Street, Glasgow G1 1XH, United Kingdom, United Kingdom

Received  July 2014 Revised  July 2015 Published  September 2015

In this paper we discuss the stochastic differential equation (SDE) susceptible-infected-susceptible (SIS) epidemic model with demographic stoch-asticity. First we prove that the SDE has a unique nonnegative solution which is bounded above. Then we give conditions needed for the solution to become extinct. Next we use the Feller test to calculate the respective probabilities of the solution first hitting zero or the upper limit. We confirm our theoretical results with numerical simulations and then give simulations with realistic parameter values for two example diseases: gonorrhea and pneumococcus.
Citation: David Greenhalgh, Yanfeng Liang, Xuerong Mao. Demographic stochasticity in the SDE SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2859-2884. doi: 10.3934/dcdsb.2015.20.2859
##### References:
 [1] E. J. Allen, Modelling with Itô Stochastic Differential Equations, Springer-Verlag, Dordecht, 2007.  Google Scholar [2] L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), Lecture Notes in Biomathematics, Math. Biosci. Subser., Vol. 1945, Springer-Verlag, Berlin, (2008), 81-130. doi: 10.1007/978-3-540-78911-6_3.  Google Scholar [3] L. J. S. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoret. Popn. Biol., 64 (2003), 439-449. doi: 10.1016/S0040-5809(03)00104-7.  Google Scholar [4] L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar [5] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, Oxford, 1991. Google Scholar [6] P. Andersson and D. Lindenstrand, A stochastic SIS epidemic with demography: Initial stages and time to extinction, J. Math. Biol., 62 (2011), 333-348. doi: 10.1007/s00285-010-0336-x.  Google Scholar [7] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, $2^{nd}$ edition, Griffin, London and High Wycombe, 1975.  Google Scholar [8] N. T. J. Bailey, Some stochastic models for small epidemics in large populations, J. R. Statist. Soc. Ser. C Appl. Statist., 13 (1964), 9-19. doi: 10.2307/2985218.  Google Scholar [9] C. A. Braumann, Environmental versus demographic stochasticity in population growth, in Workshop on Branching Processes and Their Applications (eds. M. G. Velasco, I. M. Puerto, R. Martínez, M. Molina, M. Mota and A. Ramos), Lecture Notes in Statistics, Springer-Verlag, Berlin, 197 (2010), 37-52. doi: 10.1007/978-3-642-11156-3.  Google Scholar [10] T. Britton, Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.  Google Scholar [11] J. A. Cavender, Quasi-stationary distributions of birth-and-death-processes, Adv. Appl. Probab., 10 (1978), 570-586. doi: 10.2307/1426635.  Google Scholar [12] D. Clancy and P. K. Pollett, A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic, J. Appl. Probab., 40 (2003), 821-825. doi: 10.1239/jap/1059060909.  Google Scholar [13] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53. doi: 10.1016/j.jmaa.2006.01.055.  Google Scholar [14] N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behaviour of Lotka-Volterra competition systems: Non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.  Google Scholar [15] A. J. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.  Google Scholar [16] A. J. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394 (2012), 496-516. doi: 10.1016/j.jmaa.2012.05.029.  Google Scholar [17] D. Greenhalgh, K. E. Lamb and C. Robertson, A mathematical model for the spread of streptococcus pneumoniae with transmission due to sequence type, in Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. I, Disc. Cont. Dynam. Sys. (eds. W. Feng, Z. Feng, M. Grasselli, X. Lu, S. Siegmund and J. Voigt), 1 (2011), 553-567.  Google Scholar [18] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.  Google Scholar [19] H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, 56, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-662-07544-9.  Google Scholar [20] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.  Google Scholar [21] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.  Google Scholar [22] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.  Google Scholar [23] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, part I, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721. Google Scholar [24] R. J. Kryscio and C. Lefévre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Probab., 26 (1989), 685-694. doi: 10.1201/9781420030884.ch13.  Google Scholar [25] X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [26] B. A. Melbourne, Demographic stochasticity, in Encyclopedia of Theoretical Ecology (eds. A. Hastings and L. J. Gross), University of California Press, Berkeley, (2011), 706-711. Google Scholar [27] I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6.  Google Scholar [28] I. Nåsell, The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Probab., 28 (1996), 895-932. doi: 10.2307/1428186.  Google Scholar [29] I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156 (1999), 21-40. doi: 10.1016/S0025-5564(98)10059-7.  Google Scholar [30] I. Nåsell, Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model, Lecture Notes in Mathematics, 2022, Mathematical Biosciences Subseries, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-20530-9.  Google Scholar [31] I. Nåsell, On the time to extinction in recurrent epidemics, J. R. Statist. Soc. Ser. B Statist. Methodol., 61 (1999), 309-330. doi: 10.1111/1467-9868.00178.  Google Scholar [32] R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. Appl. Probab., 14 (1982), 687-708. doi: 10.2307/1427019.  Google Scholar [33] O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907. doi: 10.1239/jap/1011994180.  Google Scholar [34] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5.  Google Scholar [35] Z. Wang and C. Zhang, An analysis of stability of Milstein method for stochastic differential equations with delay, Comput. Math. Appl., 51 (2006), 1445-1452. doi: 10.1016/j.camwa.2006.01.004.  Google Scholar [36] A. Weir, Modelling the Impact of Vaccination and Competition on Pneumococcal Carriage and Disease in Scotland, Ph.D. thesis, University of Strathclyde, Glasgow, 2009. Google Scholar [37] G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265. doi: 10.1016/0025-5564(71)90087-3.  Google Scholar [38] J. A. Yorke, H. W. Hethcote and A. Nold, Dynamics and control of the transmission of gonorrhea, Sex. Trans. Dis., 5 (1978), 51-56. doi: 10.1097/00007435-197804000-00003.  Google Scholar [39] Q. Zhang, K. Arnaoutakis, C. Murdoch, R. Lakshman, G. Race, R. Burkinshaw and A. Finn, Mucosal immune responses to capsular pneumococcal polysaccharides in immunized preschool children and controls with similar nasal pneumococcal colonization rates, Ped. Inf. Dis. J., 23 (2004), 307-313. doi: 10.1097/00006454-200404000-00006.  Google Scholar

show all references

##### References:
 [1] E. J. Allen, Modelling with Itô Stochastic Differential Equations, Springer-Verlag, Dordecht, 2007.  Google Scholar [2] L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), Lecture Notes in Biomathematics, Math. Biosci. Subser., Vol. 1945, Springer-Verlag, Berlin, (2008), 81-130. doi: 10.1007/978-3-540-78911-6_3.  Google Scholar [3] L. J. S. Allen and E. J. Allen, A comparison of three different stochastic population models with regard to persistence time, Theoret. Popn. Biol., 64 (2003), 439-449. doi: 10.1016/S0040-5809(03)00104-7.  Google Scholar [4] L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci., 163 (2000), 1-33. doi: 10.1016/S0025-5564(99)00047-4.  Google Scholar [5] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, Oxford, 1991. Google Scholar [6] P. Andersson and D. Lindenstrand, A stochastic SIS epidemic with demography: Initial stages and time to extinction, J. Math. Biol., 62 (2011), 333-348. doi: 10.1007/s00285-010-0336-x.  Google Scholar [7] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, $2^{nd}$ edition, Griffin, London and High Wycombe, 1975.  Google Scholar [8] N. T. J. Bailey, Some stochastic models for small epidemics in large populations, J. R. Statist. Soc. Ser. C Appl. Statist., 13 (1964), 9-19. doi: 10.2307/2985218.  Google Scholar [9] C. A. Braumann, Environmental versus demographic stochasticity in population growth, in Workshop on Branching Processes and Their Applications (eds. M. G. Velasco, I. M. Puerto, R. Martínez, M. Molina, M. Mota and A. Ramos), Lecture Notes in Statistics, Springer-Verlag, Berlin, 197 (2010), 37-52. doi: 10.1007/978-3-642-11156-3.  Google Scholar [10] T. Britton, Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.  Google Scholar [11] J. A. Cavender, Quasi-stationary distributions of birth-and-death-processes, Adv. Appl. Probab., 10 (1978), 570-586. doi: 10.2307/1426635.  Google Scholar [12] D. Clancy and P. K. Pollett, A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic, J. Appl. Probab., 40 (2003), 821-825. doi: 10.1239/jap/1059060909.  Google Scholar [13] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53. doi: 10.1016/j.jmaa.2006.01.055.  Google Scholar [14] N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behaviour of Lotka-Volterra competition systems: Non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422. doi: 10.1016/j.cam.2004.02.001.  Google Scholar [15] A. J. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X.  Google Scholar [16] A. J. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394 (2012), 496-516. doi: 10.1016/j.jmaa.2012.05.029.  Google Scholar [17] D. Greenhalgh, K. E. Lamb and C. Robertson, A mathematical model for the spread of streptococcus pneumoniae with transmission due to sequence type, in Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl. Vol. I, Disc. Cont. Dynam. Sys. (eds. W. Feng, Z. Feng, M. Grasselli, X. Lu, S. Siegmund and J. Voigt), 1 (2011), 553-567.  Google Scholar [18] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.  Google Scholar [19] H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, 56, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-662-07544-9.  Google Scholar [20] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.  Google Scholar [21] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981.  Google Scholar [22] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.  Google Scholar [23] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, part I, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700-721. Google Scholar [24] R. J. Kryscio and C. Lefévre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Probab., 26 (1989), 685-694. doi: 10.1201/9781420030884.ch13.  Google Scholar [25] X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar [26] B. A. Melbourne, Demographic stochasticity, in Encyclopedia of Theoretical Ecology (eds. A. Hastings and L. J. Gross), University of California Press, Berkeley, (2011), 706-711. Google Scholar [27] I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6.  Google Scholar [28] I. Nåsell, The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Probab., 28 (1996), 895-932. doi: 10.2307/1428186.  Google Scholar [29] I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156 (1999), 21-40. doi: 10.1016/S0025-5564(98)10059-7.  Google Scholar [30] I. Nåsell, Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model, Lecture Notes in Mathematics, 2022, Mathematical Biosciences Subseries, Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-642-20530-9.  Google Scholar [31] I. Nåsell, On the time to extinction in recurrent epidemics, J. R. Statist. Soc. Ser. B Statist. Methodol., 61 (1999), 309-330. doi: 10.1111/1467-9868.00178.  Google Scholar [32] R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. Appl. Probab., 14 (1982), 687-708. doi: 10.2307/1427019.  Google Scholar [33] O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907. doi: 10.1239/jap/1011994180.  Google Scholar [34] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5.  Google Scholar [35] Z. Wang and C. Zhang, An analysis of stability of Milstein method for stochastic differential equations with delay, Comput. Math. Appl., 51 (2006), 1445-1452. doi: 10.1016/j.camwa.2006.01.004.  Google Scholar [36] A. Weir, Modelling the Impact of Vaccination and Competition on Pneumococcal Carriage and Disease in Scotland, Ph.D. thesis, University of Strathclyde, Glasgow, 2009. Google Scholar [37] G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265. doi: 10.1016/0025-5564(71)90087-3.  Google Scholar [38] J. A. Yorke, H. W. Hethcote and A. Nold, Dynamics and control of the transmission of gonorrhea, Sex. Trans. Dis., 5 (1978), 51-56. doi: 10.1097/00007435-197804000-00003.  Google Scholar [39] Q. Zhang, K. Arnaoutakis, C. Murdoch, R. Lakshman, G. Race, R. Burkinshaw and A. Finn, Mucosal immune responses to capsular pneumococcal polysaccharides in immunized preschool children and controls with similar nasal pneumococcal colonization rates, Ped. Inf. Dis. J., 23 (2004), 307-313. doi: 10.1097/00006454-200404000-00006.  Google Scholar
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