scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (04) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

scholarly journals An epidemic model with exposure-dependent severities

2005 ◽  
Vol 42 (4) ◽  
pp. 932-949 ◽  
Author(s):  
Frank Ball ◽  
Tom Britton
Keyword(s):  
Branching Process ◽  
Large Population ◽  
Final Outcome ◽  
Model Parameters ◽  
Final Size ◽  
Strong Law ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Infectious State ◽  
Process Approximation

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

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.

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.

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

1999 ◽  
Vol 31 (01) ◽  
pp. 220-234 ◽  
Author(s):  
Mikael Andersson
Keyword(s):  
Branching Process ◽  
Counting Process ◽  
Contact Structure ◽  
Finite Population ◽  
Convergence Result ◽  
System Of Equations ◽  
Final Size ◽  
Linear System Of Equations ◽  
Non Linear System ◽  
Process Approximation

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.

scholarly journals The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

2015 ◽  
Vol 52 (04) ◽  
pp. 1195-1201 ◽  
Author(s):  
Peter Windridge
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Large Population ◽  
Extinction Time ◽  
Exponential Tail ◽  
Sir Epidemic Model ◽  
Multitype Branching Process ◽  
Type Distribution ◽  
Sir Epidemic

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

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

1999 ◽  
Vol 31 (1) ◽  
pp. 220-234 ◽  
Author(s):  
Mikael Andersson
Keyword(s):  
Branching Process ◽  
Counting Process ◽  
Contact Structure ◽  
Finite Population ◽  
Convergence Result ◽  
System Of Equations ◽  
Final Size ◽  
Linear System Of Equations ◽  
Non Linear System ◽  
Process Approximation

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.

scholarly journals An SIR epidemic on a weighted network

2019 ◽  
Vol 7 (4) ◽  
pp. 556-580
Author(s):  
Kristoffer Spricer ◽  
Tom Britton
Keyword(s):  
Weighted Graph ◽  
Configuration Model ◽  
Final Size ◽  
Sir Epidemic ◽  
Wide Range ◽  
Major Outbreak ◽  
Standard Normal ◽  
Model Graph ◽  
Special Case ◽  
Unweighted Graph

AbstractWe introduce a weighted configuration model graph, where edge weights correspond to the probability of infection in an epidemic on the graph. On these graphs, we study the development of a Susceptible–Infectious–Recovered epidemic using both Reed–Frost and Markovian settings. For the special case of having two different edge types, we determine the basic reproduction numberR0, the probability of a major outbreak, and the relative final size of a major outbreak. Results are compared with those for a calibrated unweighted graph. The degree distributions are based on both theoretical constructs and empirical network data. In addition, bivariate standard normal copulas are used to model the dependence between the degrees of the two edge types, allowing for modeling the correlation between edge types over a wide range. Among the results are that the weighted graph produces much richer results than the unweighted graph. Also, while R0 always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of a major outbreak nor for the relative final size of a major outbreak. When using copulas we see that these can produce results that are similar to those of the empirical degree distributions, indicating that in some cases a copula is a viable alternative to using the full empirical data.

scholarly journals The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

2015 ◽  
Vol 52 (4) ◽  
pp. 1195-1201 ◽  
Author(s):  
Peter Windridge
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Large Population ◽  
Extinction Time ◽  
Exponential Tail ◽  
Sir Epidemic Model ◽  
Multitype Branching Process ◽  
Type Distribution ◽  
Sir Epidemic

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

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

2001 ◽  
Vol 33 (1) ◽  
pp. 99-123 ◽  
Author(s):  
Frank Ball ◽  
Owen D. Lyne
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Stochastic Model ◽  
Central Limit ◽  
Epidemic Model ◽  
Branching Process ◽  
Finite Population ◽  
Exact Distribution ◽  
Final Outcome ◽  
Process Approximation

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.

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