TY - JOUR

T1 - Modeling a SI epidemic with stochastic transmission

T2 - hyperbolic incidence rate

AU - Christen, Alejandra

AU - Maulén-Yañez, M. Angélica

AU - González-Olivares, Eduardo

AU - Curé, Michel

N1 - Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - In this paper a stochastic susceptible-infectious (SI) epidemic model is analysed, which is based on the model proposed by Roberts and Saha (Appl Math Lett 12: 37–41, 1999), considering a hyperbolic type nonlinear incidence rate. Assuming the proportion of infected population varies with time, our new model is described by an ordinary differential equation, which is analogous to the equation that describes the double Allee effect. The limit of the solution of this equation (deterministic model) is found when time tends to infinity. Then, the asymptotic behaviour of a stochastic fluctuation due to the environmental variation in the coefficient of disease transmission is studied. Thus a stochastic differential equation (SDE) is obtained and the existence of a unique solution is proved. Moreover, the SDE is analysed through the associated Fokker–Planck equation to obtain the invariant measure when the proportion of the infected population reaches steady state. An explicit expression for invariant measure is found and we study some of its properties. The long time behaviour of deterministic and stochastic models are compared by simulations. According to our knowledge this incidence rate has not been previously used for this type of epidemic models.

AB - In this paper a stochastic susceptible-infectious (SI) epidemic model is analysed, which is based on the model proposed by Roberts and Saha (Appl Math Lett 12: 37–41, 1999), considering a hyperbolic type nonlinear incidence rate. Assuming the proportion of infected population varies with time, our new model is described by an ordinary differential equation, which is analogous to the equation that describes the double Allee effect. The limit of the solution of this equation (deterministic model) is found when time tends to infinity. Then, the asymptotic behaviour of a stochastic fluctuation due to the environmental variation in the coefficient of disease transmission is studied. Thus a stochastic differential equation (SDE) is obtained and the existence of a unique solution is proved. Moreover, the SDE is analysed through the associated Fokker–Planck equation to obtain the invariant measure when the proportion of the infected population reaches steady state. An explicit expression for invariant measure is found and we study some of its properties. The long time behaviour of deterministic and stochastic models are compared by simulations. According to our knowledge this incidence rate has not been previously used for this type of epidemic models.

KW - Asymptotic behaviour

KW - Epidemic model

KW - Non linear incidence rates

KW - Stochastic differential equations

KW - Stochastic transmission

UR - http://www.scopus.com/inward/record.url?scp=85026919039&partnerID=8YFLogxK

U2 - 10.1007/s00285-017-1162-1

DO - 10.1007/s00285-017-1162-1

M3 - Article

C2 - 28752421

AN - SCOPUS:85026919039

SN - 0303-6812

VL - 76

SP - 1005

EP - 1026

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

IS - 4

ER -