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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 and 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. [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. [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. [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. [5] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, Oxford, 1991. [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. [7] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, $2^{nd}$ edition, Griffin, London and High Wycombe, 1975. [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. [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. [10] T. Britton, Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006. [11] J. A. Cavender, Quasi-stationary distributions of birth-and-death-processes, Adv. Appl. Probab., 10 (1978), 570-586. doi: 10.2307/1426635. [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. [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. [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. [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. [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. [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. [18] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2. [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. [20] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [21] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. [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. [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. [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. [25] X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402. [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. [27] I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6. [28] I. Nåsell, The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Probab., 28 (1996), 895-932. doi: 10.2307/1428186. [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. [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. [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. [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. [33] O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907. doi: 10.1239/jap/1011994180. [34] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5. [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. [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. [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. [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. [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.

show all references

##### References:
 [1] E. J. Allen, Modelling with Itô Stochastic Differential Equations, Springer-Verlag, Dordecht, 2007. [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. [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. [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. [5] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, Oxford, 1991. [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. [7] N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, $2^{nd}$ edition, Griffin, London and High Wycombe, 1975. [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. [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. [10] T. Britton, Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006. [11] J. A. Cavender, Quasi-stationary distributions of birth-and-death-processes, Adv. Appl. Probab., 10 (1978), 570-586. doi: 10.2307/1426635. [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. [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. [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. [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. [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. [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. [18] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2. [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. [20] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [21] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. [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. [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. [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. [25] X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, 2008. doi: 10.1533/9780857099402. [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. [27] I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6. [28] I. Nåsell, The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Probab., 28 (1996), 895-932. doi: 10.2307/1428186. [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. [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. [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. [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. [33] O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907. doi: 10.1239/jap/1011994180. [34] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5. [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. [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. [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. [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. [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.
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