Probability of a Disease Outbreak in Stochastic Multipatch Epidemic Models

Lahodny, Glenn; Allen, Linda J. S.

doi:10.1007/s11538-013-9848-z

Cited by 62 publications

(59 citation statements)

References 39 publications

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“…Consequently, the infection rate of a susceptible individual at patch j and time t is β j m j /N, where β j = κ j p > 0 is the disease transmission rate. In contrast to the above assumption on the contact rate, the SIS model studied in [131,3] has a constant contact rate, which leads their model to have frequency-dependent transmission rates.…”

Section: The Markov Chain and Its Ode Limitmentioning

confidence: 99%

“…This is overcome for deterministic models in the studies made by Bailey [15,Section 7.33], Hethcote [98], Sattenspeil and Dietz [187], Allen et al [3], McCormack and Allen [141], van den Driessche [202], Salmani and van den Driessche [183], Arino and van den Driessche [9,10,11], Arino et al [7,8], Wang and Zhao [210], Wang and Mulone [209], Jin and Wang [107], Kuniya and Muroya [124], and Muroya et al [145]. The authors who have contributed on the stochastic side include Ball [22], Ball and Clancy [24,25], Clancy [50,52], Keeling et al [112,114], Sani et al [185], Lahodny Jr and Allen [131] and Neal [152]. In all these models the population structure is modelled by a metapopulation network [93,134,135].…”

Section: Accounting For Mobilitymentioning

confidence: 99%

“…However, Lahodny Jr and Allen [131] considered frequency-dependent SIS models which allowed for disease induced deaths. The stochastic models applied in their investigations were formulated using Markov processes and stochastic differential equations.…”

Section: Accounting For Mobilitymentioning

confidence: 99%

“…Other researchers have focused on simpler models incorporating similar population structure with the aim of gaining a deeper understanding of the factors affecting an epidemic's progress [22,24,25,52,50,52,131,152,185]. These models generally take the form of a continuous-time Markov chain with mobility of individuals modelled by transition rates between groups.…”

Section: Introductionmentioning

confidence: 99%

“…The model formulated in this study is closely related to the frequency-dependent SIS models studied in [131,3]; I will say more about the connection as I build the model.…”

Section: Introductionmentioning

confidence: 99%

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A model for the spread of an SIS epidemic in a human population

Shausan¹

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The aim of this work is to understand how infectious diseases spread through human populations.Attention is given to those diseases which follow the Susceptible-Infective-Susceptible (SIS) pattern.When modelling diseases spread in a human population, it is important to consider the social and spatial structure of the population. Humans usually live in groups such as work places, households, towns and cities. However, an individual's membership of a particular group is not fixed. Rather, it changes over time. This structure determines two paths for a disease to spread through the population.Disease is spread between individuals in the same group by contact between infected and susceptible individuals, and is spread from one group to another by the migration of infected individuals. This type of population structure can be modelled by a metapopulation network. I develop a continuoustime Markov chain (CTMC) model that describes the spread of an SIS epidemic in a metapopulation network.I establish an ordinary differential equations (ODE) and a Gaussian diffusion analogue of the stochastic process by applying, respectively, the theory of differential equation approximations for Markov chains, and the theory of density dependent Markov chains. I use the ODE model to derive analytic expressions for various epidemiological quantities of interest. In particular, I obtain expressions for two threshold quantities; the basic reproduction number, and a quantity called T 0 which is greater than the basic reproduction number. If the basic reproduction number is above 1, then the disease persists and if the basic reproduction number is below 1, then the disease-free equilibrium (DFE) is locally attractive. However, if T 0 is less than or equal to 1, then the DFE is globally attractive. Using the theory of cooperative differential equations and the theory of asymptotically autonomous differential equations, I show the existence and global stability of a unique endemic equilibrium (EE) and the global stability of the DFE in terms of the basic reproduction number, provided that the migration rates of susceptible and infected individuals are equal. Numerical examples indicate that a unique stable EE exists when the condition on the migration rates is relaxed. The approximating Gaussian diffusion shows that the distribution of the population at the endemic level has an approximate multivariate normal distribution whose mean is centered at the endemic equilibrium of the ODE model. iThe results of this study can serve as a basic framework on how to formulate and analyse a more realistic stochastic model for the spread of an SIS epidemic in a metapopulation which accounts for births, deaths, age, risk, and level of infectivities.Assuming that the model presented here accurately describes the spread of an SIS epidemic in a metapopulation, another question which I address is how to control the spread of the disease. Since most control strategies such as vaccination, treatment and public awareness require a high cost for their imp...

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Section: The Markov Chain and Its Ode Limitmentioning

confidence: 99%

Section: Accounting For Mobilitymentioning

confidence: 99%

Section: Accounting For Mobilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The model formulated in this study is closely related to the frequency-dependent SIS models studied in [131,3]; I will say more about the connection as I build the model.…”

Section: Introductionmentioning

confidence: 99%

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A model for the spread of an SIS epidemic in a human population

Shausan¹

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Applications of Single-Type Branching Processes

Allen

2015

Stochastic Population and Epidemic Models

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Probability of Disease Extinction or Outbreak in a Stochastic Epidemic Model for West Nile Virus Dynamics in Birds

Maliyoni

2020

Acta Biotheor

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Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finitetime to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes.

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Probability of a Disease Outbreak in Stochastic Multipatch Epidemic Models

Cited by 62 publications

References 39 publications

A model for the spread of an SIS epidemic in a human population

A model for the spread of an SIS epidemic in a human population

Applications of Single-Type Branching Processes

Probability of Disease Extinction or Outbreak in a Stochastic Epidemic Model for West Nile Virus Dynamics in Birds

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