Stochastic Epidemics Research Articles

Stochastic multi-type SIR epidemics among a population partitioned into households

We consider a stochastic model for the spread of an SIR (susceptible → infective → removed) epidemic among a closed, finite population that contains several types of individuals and is partitioned into households. The infection rate between two individuals depends on the types of the transmitting and receiving individuals and also on whether the infection is local (i.e., within a household) or global (i.e., between households). The exact distribution of the final outcome of the epidemic is outlined. A branching process approximation for the early stages of the epidemic is described and made fully rigorous, by considering a sequence of epidemics in which the number of households tends to infinity and using a coupling argument. This leads to a threshold theorem for the epidemic model. A central limit theorem for the final outcome of epidemics which take off is derived, by exploiting an embedding representation.

Stochastic epidemics in dynamic populations: quasi-stationarity and extinction.

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for tau, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how tau depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours.

A computational method for the study of stochastic epidemics

Compartmental models for stochastic epidemics are typically intractable mathematically, and extremely demanding to intractable, when studied by traditional numerical methods. Researchers have consequently resorted to deterministic approximations or simulation to investigate them. This paper describes an alternative numerical method, called here the Probability Vector Method (PVM), for analyzing such models. It has the potential of estimating the first few moments of a compartmentalized epidemic model over a sequence of times. For compartmental models with a constant, homogeneous population, this method can be relatively efficient in computational resources compared to simulation, and the error bounds have greater analytic justification. The methods proposed here provide an effective alternative to simulation, and since they are so radically different, the PVM and simulation constitute checks on one another. Computational studies of a stochastic Susceptible/Infective model and a highly-compartmentalized HIV/AIDS model of Bailey illustrate the methodology.

Bayesian Inference for Partially Observed Stochastic Epidemics

Summary The analysis of infectious disease data is usually complicated by the fact that real life epidemics are only partially observed. In particular, data concerning the process of infection are seldom available. Consequently, standard statistical techniques can become too complicated to implement effectively. In this paper Markov chain Monte Carlo methods are used to make inferences about the missing data as well as the unknown parameters of interest in a Bayesian framework. The methods are applied to real life data from disease outbreaks.

Stochastic epidemics: the expected duration of the endemic period in higher dimensional models.

A method is presented to approximate the long-term stochastic dynamics of an epidemic modelled by state variables denoting the various classes of the population such as in SIR and SEIR model. The modelling includes epidemics in populations at different locations with migration between these populations. A logistic stochastic process for the total infectious population is formulated; it fits the long-term stochastic behaviour of the total infectious population in the full model. A good approximation is obtained if only the dynamics near the equilibria is fit.

Limit theorems for the total size of a spatial epidemic

We study the long-term behaviour of a sequence of multitype general stochastic epidemics, converging in probability to a deterministic spatial epidemic model, proposed by D. G. Kendall. More precisely, we use branching and deterministic approximations in order to study the asymptotic behaviour of the total size of the epidemics as the number of types and the number of individuals of each type both grow to infinity.

Limit theorems for the total size of a spatial epidemic

We study the long-term behaviour of a sequence of multitype general stochastic epidemics, converging in probability to a deterministic spatial epidemic model, proposed by D. G. Kendall. More precisely, we use branching and deterministic approximations in order to study the asymptotic behaviour of the total size of the epidemics as the number of types and the number of individuals of each type both grow to infinity.

Stochastic epidemics: the probability of extinction of an infectious disease at the end of a major outbreak.

The aim of this study is to derive an asymptotic expression for the probability that an infectious disease will disappear from a population at the end of a major outbreak ('fade-out'). The study deals with a stochastic SIR-model. Local asymptotic expansions are constructed for the deterministic trajectories of the corresponding deterministic system, in particular for the deterministic trajectory starting in the saddle point. The analytical expression for the probability of extinction is derived by asymptotically solving a boundary value problem based on the Fokker-Planck equation for the stochastic system. The asymptotic results are compared with results obtained by random walk simulations.

Computational methods for Markov series with large state spaces, with application to AIDS modeling

AIDS models, and epidemiological models generally, are almost exclusively either differential equations or Markov processes. Of course, the phenomena are fundamentally random, so at best differential equations track the expectation of the process and variability is masked. There are few techniques in the literature for numerical analysis of Markov chains of any size, and so those wishing to analyze stochastic epidemics presently have little alternative to simulation. The contribution of the present paper is to propose numerical techniques capable of finding marginal probabilities of Markov chains having thousands and even millions of states. The ideas are illustrated by application to AIDS models in the literature which formerly had been investigated only through Monte Carlo. This introductory foray has not plumbed the depths of the computational methodology, which yet needs refinement and streamlining that comes through experience. Yet in its primitive form, it is shown herein to be adequate for a computation on the scale of a two-population partition of the San Francisco homosexual epidemic. The closing discussion compares the strengths and weaknesses of the present numerical techniques with the simulation approach to investigation of Markov epidemics.

Assessing the variability of stochastic epidemics

In predicting the course of individual realizations of an epidemic it is important to know the magnitude of the variability of such realizations about their mean. In this paper and in the context of the general stochastic epidemic, some methods of obtaining approximate estimates of this variability are investigated; one is a multivariate normal approximation based on an asymptotic Gaussian diffusion process, and another uses an approximating linear stochastic process. The extension of these methods to the more detailed models used to describe the transmission dynamics of HIV infection and AIDS is discussed.

Deterministic and stochastic epidemics with several kinds of susceptibles : Frank Ball, University of Nottingham, England

Deterministic and stochastic epidemics with several kinds of susceptibles : Frank Ball, University of Nottingham, England

Deterministic and stochastic epidemics with several kinds of susceptibles

We consider the spread of a general epidemic amongst a population consisting of invididuals with differing susceptibilities to the disease. Deterministic and stochastic versions of the basic model are described and analysed. For both versions of the model we show that assuming a uniform susceptible population, with average susceptibility, leads to an increased spread of infection. We also show how our results can be extended to the carrier-borne epidemic model of Weiss (1965).

Deterministic and stochastic epidemics with several kinds of susceptibles

We consider the spread of a general epidemic amongst a population consisting of invididuals with differing susceptibilities to the disease. Deterministic and stochastic versions of the basic model are described and analysed. For both versions of the model we show that assuming a uniform susceptible population, with average susceptibility, leads to an increased spread of infection. We also show how our results can be extended to the carrier-borne epidemic model of Weiss (1965).

The threshold behaviour of epidemic models

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

The threshold behaviour of epidemic models

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

Interaction Models and Spatial Diffusion Processes

Geographical AnalysisVolume 14, Issue 2 p. 95-108 Free Access Interaction Models and Spatial Diffusion Processes Robert Haining, Robert Haining Robert Haining is lecturer in geography, University of Sheffield. The author wishes to express his thanks to the referees for providing valuable advice in the presentation of the arguments in this paper.Search for more papers by this author Robert Haining, Robert Haining Robert Haining is lecturer in geography, University of Sheffield. The author wishes to express his thanks to the referees for providing valuable advice in the presentation of the arguments in this paper.Search for more papers by this author First published: April 1982 https://doi.org/10.1111/j.1538-4632.1982.tb00059.xCitations: 14 Robert Haining is lecturer in geography, University of Sheffield. The author wishes to express his thanks to the referees for providing valuable advice in the presentation of the arguments in this paper. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED Athreya, K. R., and S. Karlin (1971). “On Branching Processes in Random Environments.” Annals of Mathematical Statistics, 42, 5: 1499– 520. Bailey, N. T. J. (1953). “The Total Size of a General Stochastic Epidemic.” Biometrika, 403: 177– 85. Bailey, N. T. J. (1967). “The Simulation of Stochastic Epidemics in Two Dimensions.” Proc. 5th Berkeley Symposium Maths., Stats, and Prob., 4: 237– 58. Bailey, N. T. J. (1968). “Stochastic Birth, Death and Migration Processes for Spatially Distributed Populations.” Biometrika, 55: 189– 98. Bartlett, M. S. (1957). “Measels Periodicity and Community Size.” Journal of the Royal Statistical Society, A, 120: 48– 70. Bartlett, M. S.. (1975). The Statistical Analysis of Spatial Pattern. London: Chapman Hall. Bartoszynski, R. (1967). “Branching Processes and the Theory of Epidemics,” Proc. 5th Berkeley Symposium Maths., Stats, and Prob., 4: 259– 70. Besag, J. E. (1974). “Spatial Interaction and the Statistical Analysis of Lattice Systems.” Journal of the Royal Statistical Society, B. 36: 192– 236. Besag, J. E.. (1975). “The Statistical Analysis of Non-Lattice Data.” The Statistician, 24, 3, 179– 95. Biggs, N. L. (1976). Interaction Models. Cambridge: C.U.P. Cox, D. R.(1970). Analysis of Binary Data. London: Methuen. Dacey, M. F. (1969). “Some Properties of a Cluster Point Process.” Canadian Geographer, 13, 2: 128– 40. Hagerstrand, T. (1967). Innovation Diffusion as a Spatial Process. Chicago: University of Chicago Press. Haining, R. P. (1979). “Statistical Tests and Process Generators for Random Field Models.” Geographical Analysis, 11: 45– 64. Hammersley, I. M., and D. J. Welsh (1965). “First Passage Percolation, Sub-Additive Processes, Stochastic Networks and Generalized Renewal Theory.” Bemoulli-Bayes-Laplace Anniversary Vol., 61-110, Berlin: Springer. Hay, A. M., and R. J. Johnston (1980). “Spatial Variations in Grocery Prices: Further Attempts at Modelling.” Urban Geography, 1, 2, 189– 201. Hudson, T. C. (1972). Geographical Diffusion Theory. Northwestern University Studies in Geography 19. Kendall, D. G. (1956). “Deterministic and Stochastic Epidemics in Closed Populations.” Proc. 3rd Berkeley Symposium Maths., Stats, and Prob., 4: 149– 65. Kingman, J. F. C. (1975). “Markov Models for Spatial Variation.” The Statistician, 24: 167– 74. Morrill, R. L., and F. R. Pitts (1967). “Marriage, Migration and the Mean Information Field.” Annals, Association of American Geographers, 57: 401– 42. Murray, G. D., and A. D. Cliff (1977). “A Stochastic Model for Measels Epidemics in a Multi-Region Setting.” Trans. Inst. British Geog., 2, 2: 158– 74. Newell, G. F., and E. W. Montroll (1953). “On the Theory of the Ising Model of Ferromagnetism.” Rev. Mod. Physics, 25: 353– 89. Neyman, J., and E. Scott (1964). “A Stochastic Model of Epidemics.” In Stochastic Models in Medicine and Biology, (edited by J. Gurland, pp. 45– 85. Madison, Wis.: University of Wisconsin Press. Preston, D. J. (1974). Gibbs States on Countable Sets. Cambridge: Cambridge University Press. Renshaw, E. (1972). “Birth Death and Migration Processes.” Biometrika, 59, 1. Samuelson, P. A. (1952). “Spatial Price Equilibrium and Linear Programming.” American Economic Review, 42: 283– 303. Smith, W. L., and W. E. Wilkinson (1971). “Branching Processes in Markovan Environments.” Duke Mathematical Journal, 38, 4, 749– 63. Whittle, P. (1954). “On Stationary Processes in the Plane.” Biometrika, 41: 434– 49. Whittle, P.. (1955). “The Outcome of Stochastic Epidemics—a Note on Bailey's Paper.” Biometrika, 42: 116– 22. Wrigley, N. (1977). “Probability Surface Mapping: a New Approach to Trend Surface Mapping.” Trans. Inst. Brit. Geog., 2, 2: 129– 40. Citing Literature Volume14, Issue2April 1982Pages 95-108 ReferencesRelatedInformation

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Qualitative behavior of stochastic epidemics

The qualitative behavior of the distribution of final sizes of an epidemic is considered. The main features of this distribution depend upon two parameters. The first parameter is the extinction probability for a branching process, which approximates the early stages of the epidemic. The second parameter is the expected final size of a “quasi-deterministic” approximation, which describes the later stages of the epidemic. By combining the branching process and quasi deterministic approximation, an approximation to the final size distribution of the epidemic is obtained.

Optimal isolation policies for deterministic and stochastic epidemics

Optimal policies for the isolation of infectives in the Kermack-McKendrick model of an epidemic in a closed population are derived. The bounded control variable appears linearly in the dynamic equations and integral cost functional, and influences the rate at which infectives are isolated from the rest of the population. In a class of deterministic problems the cost of any admissible policy is shown to depend only on the resulting final number of susceptibles. This fact is used to show that optimal policies expend either all or none of the available control effort and that there is no switching between these extremes. Some less complete results for analogous stochastic epidemics are combined with a description of policies computed by dynamic programming to provide a reasonable guide to the form of optimal policies in the stochastic case.

On the probability distributions of some stochastic epidemic models

The probability distributions of certain Markov stochastic epidemic models in a finite homogeneous population are derived. It is assumed that the epidemic is generated by contagion as well as by mutation (or random occurrences) and no recovery and removal of infectives are allowed for. This model includes the simple stochastic epidemic defined by Bailey (1957) as a special case, in which no mutation is assumed. Let P j ( t) be the probability that there are j infectives up to time t. The usual approach of investigating this type of models is to examine the system of differential-difference equations or the generating function of the probabilities P j ( t). If this approach leads to a closed form solution, the computation required is often extremely laborious. One such example is the solution for simple stochastic epidemics obtained by Bailey (1957, 1963). A much simpler method is presented in this paper in which the information on the interarrival times of the infectives is utilized to calculate the exact probability P j ( t) and the distribution of the duration of the epidemic. The procedure involves essentially the convolution of exponential densities which avoids the necessity of solving the differential equations associated with P j ( t).