Branching Process Approximation Research Articles

Stochastic models for the spread of HIV amongst intravenous drug users

In this paper we examine the spread of HIV when this disease is transmitted through the random sharing of contaminated drug injection equipment. We first model the spread of disease using a standard set of behavioral assumptions discussed by Kaplan [1]. We demonstrate that deterministic and stochastic models based on these assumptions behave very similarly and use a branching process approximation to show that if the basic reproductive number, R 0, is less than or equal to unity then the disease will always become extinct. If R 0>1 then, although the disease might take off, it is still possible for it to die out, and we calculate the probability of extinction. This is not of the simple form R 0 −a , where a is the initial number of infectious addicts, which might have been expected from Whittle's stochastic threshold theorem [2]. We next discuss an extended model that incorporates a three-stage AIDS incubation period and again examine a branching process approximation. We finally explore the extent to which control strategies such as needle exchange and improved needle cleaning can reduce the risk of a HIV epidemic before concluding with a brief discussion.

Persistence patterns of scrapie in a sheep flock.

The epidemiology and transmission dynamics of sheep scrapie is as yet poorly understood. Here we present a theoretical analysis of the transmission dynamics within a sheep flock, concentrating on how persistence properties depend on transmission scenario and flock size. Patterns of disease persistence and extinction are studied analytically using branching-process approximations and numerically using stochastic model simulations. For a given basic reproduction number, disease extinction is most likely when late-stage infected animals are responsible for most of the transmission. This effect can be understood in terms of aggregation in the distribution of the number of secondary infections arising from a single primary infection. The presence of an environmental reservoir reduces the probability of extinction.

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 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.

The asymptotic final size distribution of multitype chain-binomial epidemic processes

A multitype chain-binomial epidemic process is defined for a closed finite population by sampling a simple multidimensional counting process at certain points. The final size of the epidemic is then characterized, given the counting process, as the smallest root of a non-linear system of equations. By letting the population grow, this characterization is used, in combination with a branching process approximation and a weak convergence result for the counting process, to derive the asymptotic distribution of the final size. This is done for processes with an irreducible contact structure both when the initial infection increases at the same rate as the population and when it stays fixed.

The asymptotic final size distribution of multitype chain-binomial epidemic processes

A multitype chain-binomial epidemic process is defined for a closed finite population by sampling a simple multidimensional counting process at certain points. The final size of the epidemic is then characterized, given the counting process, as the smallest root of a non-linear system of equations. By letting the population grow, this characterization is used, in combination with a branching process approximation and a weak convergence result for the counting process, to derive the asymptotic distribution of the final size. This is done for processes with an irreducible contact structure both when the initial infection increases at the same rate as the population and when it stays fixed.

A computer exploration of some properties of non-linear stochastic partnership models for sexually transmitted diseases with stages

In this paper, branching process approximations to non-linear stochastic partnership models for sexually transmitted diseases in heterosexual populations were used to find points in the parameter space such that an epidemic would occur. At selected points in the parameter space, samples of Monte Carlo realizations of the process were computed and analyzed statistically to gain insights into the stochastic evolution of epidemics seeded by one infective single female and male. Non-linear difference equations were embedded in the stochastic processes, making it possible to compare trajectories computed according to the deterministic model with those computed from samples of Monte Carlo realizations. From these trajectories it was shown that stochastic fluctuations may have a profound effect on the long-term evolution of an epidemic, and examples demonstrate that an investigator may be misled if a deterministic model alone were used to project an epidemic, particularly when there is a significant probability of extinction.

Limit theorems for a random graph epidemic model

We consider a simple stochastic discrete-time epidemic model in a large closed homogeneous population that is not necessarily homogeneously mixing. Rather, each individual has a fixed circle of acquaintances and the epidemic spreads along this social network. In case the number of initially infective individuals stays small, a branching process approximation for the number of infectives is in force. Moreover, we provide a deterministic approximation of the bivariate process of susceptible and infective individuals, valid when the number of initially infective individuals is large. These results are used in order to derive the basic reproduction number and the asymptotic final epidemic size of the process. The model is described in the framework of random graphs.

Waiting with and without Recombination: The Time to Production of a Double Mutant

R. A. Fisher and H. J. Muller argued in the 1930s that a major evolutionary advantage of recombination is that it allows favorable mutations to be combined within an individual even when they first appear in different individuals. This effect is evaluated in a two-locus, two-allele model by calculating the average waiting time until a new genotypic combination first appears in a haploid population. Three approximations are developed and compared with Monte Carlo simulations of the Wright–Fisher process of random genetic drift in a finite population. First, a crude method, based on the deterministic accumulation of single mutants, produces a waiting time of 1/Nμ2with no recombination and 1/13RNμ2with recombination between the two loci, whereμis the mutation rate,Nis the haploid population size, andRis the recombination rate. Second, the waiting time is calculated as the expected value of a heterogeneous geometric distribution obtained from a branching process approximation. This gives accurate estimates forNμlarge. The estimates for small values ofNμare considerably lower than the simulated values. Finally, diffusion analysis of the Wright–Fisher process provides accurate estimates forNμsmall, and the time scales of the diffusion process show a difference betweenR=0 and forR⪢0 of the same order of magnitude as seen in the deterministic analysis. In the absence of recombination, accurate approximations to the waiting time are obtained by using the branching process for highNμand the diffusion approximation for lowNμ. For lowNμthe waiting time is well approximated by 1/8N2μ3. WithR⪢0, the following dependence onNμis observed: ForNμ>1 the waiting time is virtually independent of recombination and is well described by the branching process approximation. ForNμ≈ the waiting time is well described by a simplified diffusion approximation that assumes symmetry in the frequencies of single mutants. ForNμ⪡1 the waiting time is well described by the diffusion approximation allowing asymmetry in the frequencies of single mutants. Recombination lowers the waiting time until a new genotypic combination first appears, but the effect is small compared to that of the mutation rate and population size. For largeNμ, recombination has a negligible effect, and its effect is strongest for smallNμ, in which case the waiting time approaches a fixed fraction of the waiting time forR=0. Free recombination lowers the waiting time to about 45% of the waiting time for absolute linkage for smallNμ. Selection has little effect on the importance of recombination in general.

Epidemics with two levels of mixing

We consider epidemics with removal (SIR epidemics) in populations that mix at two levels: global and local. We develop a general modelling framework for such processes, which allows us to analyze the conditions under which a large outbreak is possible, the size of such outbreaks when they can occur and the implications for vaccination strategies, in each case comparing our results with the simpler homogeneous mixing case. More precisely, we consider models in which each infectious individual i has a global probability $p_G$ for infecting each other individual in the population and a local probability $p_L$, typically much larger, of infecting each other individual among a set of neighbors $\mathscr{N}(i)$. Our main concern is the case where the population is partitioned into local groups or households, but our approach also applies to cases where neighborhoods do not form a partition, for instance, to spatial models with a mixture of local (e.g., nearest-neighbor) and global contacts. We use a variety of theoretical approaches: a random graph framework for the initial exposition of the simple case where an individual's contacts are independent; branching process approximations for the general threshold result; and an embedding representation for rigorous results on the final size of outbreaks. From the applied viewpoint the key result is that, compared with the homogeneous mixing model in which individuals make contacts simply with probability $p_G$, the local infectious contacts have an "amplification" effect. The basic reproductive ratio of the epidemic is increased from its individual-to-individual value $R_G$ in the absence of local infections to a group-to-group value $R_* = \mu R_G$, where $\mu$ is the mean size of an outbreak, started by a randomly chosen individual, in which only local infections count. Where the groups are large and the within-group epidemics are above threshold, this amplification can permit an outbreak in the whole population at very low levels of $p_G$, for instance, for $p_G = O(1/Nn)$ in a population of N divided into groups of size n. The implication of these results for control strategies is that vaccination should be directed preferentially toward reducing $\mu$; we discuss the conditions under which the equalizing strategy, aimed at leaving unvaccinated sets of neighbors of equal sizes, is optimal. We also discuss the estimation of our threshold parameter $R_*$ from data on epidemics among households.

The effect of preferential mixing on the growth of an epidemic

The spread of infection in a community stratified into classes, where individuals have a preference for either within- or between-class contact, is discussed. The effect of such classification is assessed through the comparison of heterogeneous epidemic models with corresponding homogeneous models. Stratification of the community is modeled via a multitype branching process approximation to the epidemic. This allows us to conveniently study its effect through such quantities as the epidemic threshold parameter and the probability of a major outbreak. Attention is focused on the model of May and Anderson; however, a number of other heterogeneous structures reflecting preferential mixing are also studied.

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.