• Previous Article
    Mathematical analysis of population migration and its effects to spread of epidemics
  • DCDS-B Home
  • This Issue
  • Next Article
    Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities
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, (2007).

[2]

L. J. S. Allen, An introduction to stochastic epidemic models,, in Mathematical Epidemiology (eds. F. Brauer, (2008), 81. 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. 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. 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, (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. doi: 10.1007/s00285-010-0336-x.

[7]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases,, $2^{nd}$ edition, (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. 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, 197 (2010), 37. doi: 10.1007/978-3-642-11156-3.

[10]

T. Britton, Stochastic epidemic models: A survey,, Math. Biosci., 225 (2010), 24. 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. 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. 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. 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. 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. 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. 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, 1 (2011), 553.

[18]

H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. 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 (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. doi: 10.1137/S0036144500378302.

[21]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Publishing Co., (1981).

[22]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer-Verlag, (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.

[24]

R. J. Kryscio and C. Lefévre, On the extinction of the S-I-S stochastic logistic epidemic,, J. Appl. Probab., 26 (1989), 685. doi: 10.1201/9781420030884.ch13.

[25]

X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008). doi: 10.1533/9780857099402.

[26]

B. A. Melbourne, Demographic stochasticity,, in Encyclopedia of Theoretical Ecology (eds. A. Hastings and L. J. Gross), (2011), 706.

[27]

I. Nåsell, Stochastic models of some endemic infections,, Math. Biosci., 179 (2002), 1. 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. doi: 10.2307/1428186.

[29]

I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic,, Math. Biosci., 156 (1999), 21. 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 (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. 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. doi: 10.2307/1427019.

[33]

O. Ovaskainen, The quasistationary distribution of the stochastic logistic model,, J. Appl. Probab., 38 (2001), 898. doi: 10.1239/jap/1011994180.

[34]

M. Shaked and J. G. Shanthikumar, Stochastic Orders,, Springer, (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. 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, (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. 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. 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. doi: 10.1097/00006454-200404000-00006.

show all references

References:
[1]

E. J. Allen, Modelling with Itô Stochastic Differential Equations,, Springer-Verlag, (2007).

[2]

L. J. S. Allen, An introduction to stochastic epidemic models,, in Mathematical Epidemiology (eds. F. Brauer, (2008), 81. 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. 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. 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, (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. doi: 10.1007/s00285-010-0336-x.

[7]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases,, $2^{nd}$ edition, (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. 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, 197 (2010), 37. doi: 10.1007/978-3-642-11156-3.

[10]

T. Britton, Stochastic epidemic models: A survey,, Math. Biosci., 225 (2010), 24. 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. 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. 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. 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. 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. 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. 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, 1 (2011), 553.

[18]

H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. 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 (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. doi: 10.1137/S0036144500378302.

[21]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Publishing Co., (1981).

[22]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer-Verlag, (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.

[24]

R. J. Kryscio and C. Lefévre, On the extinction of the S-I-S stochastic logistic epidemic,, J. Appl. Probab., 26 (1989), 685. doi: 10.1201/9781420030884.ch13.

[25]

X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008). doi: 10.1533/9780857099402.

[26]

B. A. Melbourne, Demographic stochasticity,, in Encyclopedia of Theoretical Ecology (eds. A. Hastings and L. J. Gross), (2011), 706.

[27]

I. Nåsell, Stochastic models of some endemic infections,, Math. Biosci., 179 (2002), 1. 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. doi: 10.2307/1428186.

[29]

I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic,, Math. Biosci., 156 (1999), 21. 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 (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. 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. doi: 10.2307/1427019.

[33]

O. Ovaskainen, The quasistationary distribution of the stochastic logistic model,, J. Appl. Probab., 38 (2001), 898. doi: 10.1239/jap/1011994180.

[34]

M. Shaked and J. G. Shanthikumar, Stochastic Orders,, Springer, (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. 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, (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. 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. 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. doi: 10.1097/00006454-200404000-00006.

[1]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157

[2]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[3]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084

[4]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473

[5]

Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

[6]

Michael Röckner, Jiyong Shin, Gerald Trutnau. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3219-3237. doi: 10.3934/dcdsb.2016095

[7]

Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015

[8]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

[9]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[10]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255

[11]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[12]

Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291

[13]

Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure & Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037

[14]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

[15]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051

[16]

Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistence-time estimation for some stochastic SIS epidemic models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2933-2947. doi: 10.3934/dcdsb.2015.20.2933

[17]

Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826

[18]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

[19]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105

[20]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]