The SIS epidemic model with Markovian switching

Abstract Population systems are often subject to environmental noise. Motivated by Takeuchi et al. [7] , we will discuss in this paper the effect of telegraph noise on the well-known SIS epidemic model. We establish the explicit solution of the stochastic SIS epidemic model, which is useful in performing computer simulations. We also establish the conditions for extinction and persistence for the stochastic SIS epidemic model and compare these with the corresponding conditions for the deterministic SIS epidemic model. We first prove these results for a two-state Markov chain and then generalise them to a finite state space Markov chain. Computer simulations based on the explicit solution and the Euler–Maruyama scheme are performed to illustrate our theory. We include a more realistic example using appropriate parameter values for the spread of S t r e p t o c o c c u s p n e u m o n i a e in children.

[1]  H. Hethcote.  Qualitative analyses of communicable disease models , 1976 .

[2]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[3]  H. Caswell,et al.  Red, white and blue: environmental variance spectra and coexistence in metapopulations , 1995 .

[4]  X. Mao,et al.  Stability of Stochastic Differential Equations With Respect to Semimartingales , 1991 .

[5]  Liangjian Hu,et al.  A Stochastic Differential Equation SIS Epidemic Model , 2011, SIAM J. Appl. Math..

[6]  C. Murdoch,et al.  Mucosal immune responses to capsular pneumococcal polysaccharides in immunized preschool children and controls with similar nasal pneumococcal colonization rates , 2004, The Pediatric infectious disease journal.

[7]  Patrik Andersson,et al.  A stochastic SIS epidemic with demography: initial stages and time to extinction , 2011, Journal of mathematical biology.

[8]  O. Diekmann,et al.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation , 2000 .

[9]  Hong-ke Wang,et al.  On the Exponential Stability of Stochastic Differential Equations , 2009, ICFIE.

[10]  S. Adolph,et al.  Plastic inducible morphologies are not always adaptive: The importance of time delays in a stochastic environment , 2005, Evolutionary Ecology.

[11]  B. Spratt,et al.  Recombinational exchanges at the capsular polysaccharide biosynthetic locus lead to frequent serotype changes among natural isolates of Streptococcus pneumoniae , 1998, Molecular microbiology.

[12]  Y. Takeuchi,et al.  Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise , 2004 .

[13]  M. Iannelli,et al.  Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission , 1992 .

[14]  P. Neal.  Stochastic and deterministic analysis of SIS household epidemics , 2006, Advances in Applied Probability.

[15]  Xinzhi Liu,et al.  Pulse and constant control schemes for epidemic models with seasonality , 2011 .

[16]  Daqing Jiang,et al.  The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence , 2012 .

[17]  David R. Anderson.  Optimal Exploitation Strategies for an Animal Population in a Markovian Environment: A Theory and an Example , 1975 .

[18]  K. Mühlemann,et al.  Detection of Streptococcus pneumoniae Strain Cocolonization in the Nasopharynx , 2009, Journal of Clinical Microbiology.

[19]  J. Yorke,et al.  Gonorrhea Transmission Dynamics and Control , 1984 .

[20]  Montgomery Slatkin,et al.  The Dynamics of a Population in a Markovian Environment , 1978 .

[21]  B. Mukhopadhyay,et al.  On an eco-epidemiological model with prey harvesting and predator switching: Local and global perspectives , 2010 .

[22]  James Watmough,et al.  A simple SIS epidemic model with a backward bifurcation , 2000, Journal of mathematical biology.

[23]  J. Peccoud,et al.  Markovian Modeling of Gene-Product Synthesis , 1995 .

[24]  X. Mao.  Stability of stochastic differential equations with Markovian switching , 1999 .

[25]  K. Sato,et al.  Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment , 2006 .

[26]  W. J. Anderson.  Continuous-Time Markov Chains , 1991 .

[27]  Kari Auranen,et al.  Bmc Infectious Diseases Outbreaks of Streptococcus Pneumoniae Carriage in Day Care Cohorts in Finland – Implications for Elimination of Transmission , 2022 .

[28]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .

[29]  Zhien Ma,et al.  GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA , 2006 .

[30]  David Greenhalgh,et al.  A simple mathematical model for genetic effects in pneumococcal carriage and transmission , 2011, J. Comput. Appl. Math..

[31]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[32]  Carlos Castillo-Chavez,et al.  Global behavior of a multi-group SIS epidemic model with age structure , 2005, Journal of Differential Equations.

[33]  Y. Takeuchi.  Global Dynamical Properties of Lotka-Volterra Systems , 1996 .

[34]  M. Gilpin.  Group selection in predator-prey communities. , 1975, Monographs in population biology.

[35]  Peter Neal,et al.  The SIS Great Circle Epidemic Model , 2008, Journal of Applied Probability.

[36]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[37]  M. Lipsitch,et al.  Vaccination against colonizing bacteria with multiple serotypes. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[38]  J R Artalejo,et al.  On the number of recovered individuals in the SIS and SIR stochastic epidemic models. , 2010, Mathematical biosciences.