The final outcome of an epidemic model with several different types of infective in a large population

1995 ◽  
Vol 32 (3) ◽  
pp. 579-590 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Epidemic Model ◽  
Infected Individual ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Initial Population ◽  
Group Model ◽  
Final Size ◽  
Different Types

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.

The final outcome of an epidemic model with several different types of infective in a large population

1995 ◽  
Vol 32 (03) ◽  
pp. 579-590 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Size Distribution ◽  
Epidemic Model ◽  
Infected Individual ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Initial Population ◽  
Group Model ◽  
Final Size ◽  
Different Types

We consider a stochastic model for the spread of an epidemic amongst a closed homogeneously mixing population, in which there are several different types of infective, each newly infected individual choosing its type at random from those available. The model is based on the carrier-borne model of Downton (1968), as extended by Picard and Lefèvre (1990). The asymptotic distributions of final size and area under the trajectory of infectives are derived as the initial population becomes large, using arguments based on those of Scalia-Tomba (1985), (1990). We then use our limiting results to compare the asymptotic final size distribution of our model with that of a related multi-group model, in which the type of each infective is assigned deterministically.

Asymptotic final size distribution of the multitype Reed–Frost process

1986 ◽  
Vol 23 (03) ◽  
pp. 563-584
Author(s):  
Gianpaolo Scalia-Tomba
Keyword(s):  
Size Distribution ◽  
Total Population ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Reproduction Rate ◽  
Final Size ◽  
Basic Reproduction Rate ◽  
Infection Pattern ◽  
Large Numbers ◽  
A Chain

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

scholarly journals An epidemic model with infector-dependent severity

2007 ◽  
Vol 39 (4) ◽  
pp. 949-972 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Vaccination Coverage ◽  
Epidemic Model ◽  
Branching Process ◽  
Infected Individual ◽  
Large Population ◽  
Final Size ◽  
Stochastic Epidemic ◽  
Major Outbreak ◽  
Stochastic Epidemic Model ◽  
The Individual

A stochastic epidemic model is defined in which infected individuals have different severities of disease (e.g. mildly and severely infected) and the severity of an infected individual depends on the severity of the individual he or she was infected by; typically, severe or mild infectives have an increased tendency to infect others severely or, respectively, mildly. Large-population properties of the model are derived, using branching process approximations for the initial stages of an outbreak and density-dependent population processes when a major outbreak occurs. The effects of vaccination are considered, using two distinct models for vaccine action. The consequences of launching a vaccination program are studied in terms of the effect it has on reducing the final size in the event of a major outbreak as a function of the vaccination coverage, and also by determining the critical vaccination coverage above which only small outbreaks can occur.

Asymptotic final size distribution of the multitype Reed–Frost process

1986 ◽  
Vol 23 (3) ◽  
pp. 563-584 ◽  
Author(s):  
Gianpaolo Scalia-Tomba
Keyword(s):  
Size Distribution ◽  
Total Population ◽  
Large Population ◽  
Asymptotic Distributions ◽  
Reproduction Rate ◽  
Final Size ◽  
Basic Reproduction Rate ◽  
Infection Pattern ◽  
Large Numbers ◽  
A Chain

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

scholarly journals An epidemic model with infector-dependent severity

2007 ◽  
Vol 39 (04) ◽  
pp. 949-972 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Vaccination Coverage ◽  
Epidemic Model ◽  
Branching Process ◽  
Infected Individual ◽  
Large Population ◽  
Final Size ◽  
Stochastic Epidemic ◽  
Major Outbreak ◽  
Stochastic Epidemic Model ◽  
The Individual

A stochastic epidemic model is defined in which infected individuals have different severities of disease (e.g. mildly and severely infected) and the severity of an infected individual depends on the severity of the individual he or she was infected by; typically, severe or mild infectives have an increased tendency to infect others severely or, respectively, mildly. Large-population properties of the model are derived, using branching process approximations for the initial stages of an outbreak and density-dependent population processes when a major outbreak occurs. The effects of vaccination are considered, using two distinct models for vaccine action. The consequences of launching a vaccination program are studied in terms of the effect it has on reducing the final size in the event of a major outbreak as a function of the vaccination coverage, and also by determining the critical vaccination coverage above which only small outbreaks can occur.

A STOCHASTIC MODEL FOR EPIDEMICS BASED ON THE RENEWAL EQUATION

2000 ◽  
Vol 08 (01) ◽  
pp. 1-20 ◽  
Author(s):  
J. L. W. GIELEN
Keyword(s):  
Stochastic Model ◽  
Deterministic Model ◽  
Formal Solution ◽  
Infected Individual ◽  
External Source ◽  
Mean Value ◽  
Renewal Equation ◽  
Infection Model ◽  
Final Size ◽  
Infinite Population

A stochastic continuous-infection model is developed that describes the evolution of an infectious disease introduced into an infinite population of susceptibles. The proposed model is the natural stochastic counterpart of the deterministic model for epidemics, based on the renewal equation. As in the deterministic model, the infectivity of an infected individual is a function of his age-of-infection, that is the time elapsed since his own infection. A time-dependent external source of infection is included. The model provides analytical expressions that describe the stochastic infective-age structure of the population at any moment of time. It is shown that the mean value of the number of infectives predicted by the stochastic model satisfies the renewal equation, which furnishes a formal solution of this equation. The model also yields simple expressions for the expected arrival times of infectives, that can be useful for the inverse problem. An explicit expression for the final size distribution is obtained. This leads to a precise quantitative threshold theorem that distinguishes between the possibilities of a minor outbreak or a major build-up of the epidemic.

The final outcome and temporal solution of a carrier-borne epidemic model

1995 ◽  
Vol 32 (02) ◽  
pp. 304-315 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Explicit Expression ◽  
Exponential Distribution ◽  
Epidemic Model ◽  
Joint Distribution ◽  
Time Dependent ◽  
Infectious Period ◽  
Final Size ◽  
Probabilistic Structure ◽  
Dependent State

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

scholarly journals Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

2007 ◽  
Vol 44 (01) ◽  
pp. 41-57 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Final Size ◽  
Sir Epidemic ◽  
Novel Approach ◽  
Stochastic Epidemic ◽  
Sir Epidemic Models ◽  
Variable Susceptibility

The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

The final size and severity of a generalised stochastic multitype epidemic model

1993 ◽  
Vol 25 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Frank Ball ◽  
Damian Clancy
Keyword(s):  
Stochastic Model ◽  
Epidemic Model ◽  
Exact Results ◽  
Final Size ◽  
Asymptotic Results ◽  
Current Group ◽  
Total Size ◽  
Original Group

We consider a stochastic model for the spread of an epidemic amongst a population split into m groups, in which infectives move among the groups and contact susceptibles at a rate which depends upon the infective's original group, its current group, and the group of the susceptible. The distributions of total size and total area under the trajectory of infectives for such epidemics are analysed. We derive exact results in terms of multivariate Gontcharoff polynomials by treating our model as a multitype collective Reed–Frost process and slightly adapting the results of Picard and Lefèvre (1990). We also derive asymptotic results, as each of the group sizes becomes large, by generalising the method of Scalia-Tomba (1985), (1990).

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