Stochastic models for the spread of HIV in a mobile heterosexual population.

An important factor in the dynamic transmission of HIV is the mobility of the population. We formulate various stochastic models for the spread of HIV in a heterosexual mobile population, under the assumptions of constant and varying population sizes. We also derive deterministic and diffusion analogues for these models, using a convenient rescaling technique, and analyze their stability conditions and equilibrium behavior. We illustrate the dynamic behavior of the models and their approximations via a range of numerical experiments.

[1]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[2]  Valerie Isham,et al.  Mathematical modelling of the transmission dynamics of HIV infection and AIDS (a review) , 1988 .

[3]  P. K. Pollett,et al.  Approximations for the Long-Term Behavior of an Open-Population Epidemic Model , 2001 .

[4]  W. Tan,et al.  A stochastic model for the HIV epidemic in homosexual populations involving age and race , 1996 .

[5]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[6]  B Cazelles,et al.  Using the Kalman filter and dynamic models to assess the changing HIV/AIDS epidemic. , 1997, Mathematical biosciences.

[7]  T. Kurtz.  Limit theorems for sequences of jump Markov processes approximating ordinary differential processes , 1971, Journal of Applied Probability.

[8]  Roy M. Anderson,et al.  Possible demographic consequences of HIV/AIDS epidemics. I. assuming HIV infection always leads to AIDS , 1988 .

[9]  Klaus Dietz,et al.  On the transmission dynamics of HIV , 1988 .

[10]  Francesca Arrigoni,et al.  Deterministic approximation of a stochastic metapopulation model , 2003, Advances in Applied Probability.

[11]  Carlos Castillo-Chavez,et al.  Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission , 1992 .

[12]  Graeme Hugo.  Indonesia : internal and international population mobility : implications for the spread of HIV/AIDS , 2001 .

[13]  Charles J. Mode,et al.  Stochastic Processes in Epidemiology: Hiv/Aids, Other Infectious Diseases and Computers , 2000 .

[14]  W. Tan,et al.  A stochastic model of the HIV epidemic for heterosexual transmission involving married couples and prostitutes: I. The probabilities of HIV transmission and pair formation , 1996 .

[15]  M Kremer,et al.  The effect of changing sexual activity on HIV prevalence. , 1998, Mathematical biosciences.

[16]  Roy M. Anderson,et al.  Possible Demographic Consequences of HIV/AIDS Epidemics: II, Assuming HIV Infection does not Necessarily Lead to AIDS , 1989 .

[17]  K. Dietz,et al.  A structured epidemic model incorporating geographic mobility among regions. , 1995, Mathematical biosciences.

[18]  Z. Rosenberg,et al.  Microbicides urgently needed: statement by IPM CEO, Zeda Rosenberg, on new AIDS statistics. Joint United Nations Programme on HIV/AIDS (UNAIDS) releases the 2004 AIDS epidemic update. , 2004 .

[19]  Damian Clancy,et al.  A stochastic SIS infection model incorporating indirect transmission , 2005 .

[20]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[21]  J. Hyman,et al.  Using mathematical models to understand the AIDS epidemic , 1988 .

[22]  P. Pollett.  On a model for interference between searching insect parasites , 1990, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[23]  James M. Hyman,et al.  The reproductive number for an HIV model with differential infectivity and staged progression , 2005 .

[24]  J. Arino,et al.  A multi-city epidemic model , 2003 .

[25]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[26]  W. Tan,et al.  A state space model for the HIV epidemic in homosexual populations and some applications. , 1998, Mathematical biosciences.

[27]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[28]  J. Hyman,et al.  Modeling the Spread of Influenza Among Cities , 2003 .

[29]  C J Mode,et al.  A new design of stochastic partnership models for epidemics of sexually transmitted diseases with stages. , 1999, Mathematical biosciences.

[30]  J. Hyman,et al.  Modeling the impact of random screening and contact tracing in reducing the spread of HIV. , 2003, Mathematical biosciences.

[31]  Andrew D. Barbour,et al.  Quasi–stationary distributions in Markov population processes , 1976, Advances in Applied Probability.

[32]  Alun L Lloyd,et al.  Spatiotemporal dynamics of epidemics: synchrony in metapopulation models. , 2004, Mathematical biosciences.