Who is the infector? Epidemic models with symptomatic and asymptomatic cases

What role do asymptomatically infected individuals play in the transmission dynamics? There are many diseases, such as norovirus and influenza, where some infected hosts show symptoms of the disease while others are asymptomatically infected, i.e. do not show any symptoms. The current paper considers a class of epidemic models following an SEIR (Susceptible  →  Exposed  →  Infectious  →  Recovered) structure that allows for both symptomatic and asymptomatic cases. The following question is addressed: what fraction ρ of those individuals getting infected are infected by symptomatic (asymptomatic) cases? This is a more complicated question than the related question for the beginning of the epidemic: what fraction of the expected number of secondary cases of a typical newly infected individual, i.e. what fraction of the basic reproduction number R₀, is caused by symptomatic individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction ρ also depends on timing and hence on the probabilistic distributions of latent and infectious periods of the two types (not only their means). Bounds on ρ are derived for the situation where these distributions (and even their means) are unknown. Special attention is given to the class of Markov models and the class of continuous-time Reed-Frost models as two classes of distribution functions for latent and infectious periods. We show how these two classes of models can exhibit very different behaviour.

Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

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.

Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

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.

An epidemic model with infector-dependent severity

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.

Non-exponential tolerance to infection in epidemic systems--modeling, inference, and assessment

The transmission dynamics of infectious diseases have been traditionally described through a time-inhomogeneous Poisson process, thus assuming exponentially distributed levels of disease tolerance following the Sellke construction. Here we focus on a generalization using Weibull individual tolerance thresholds under the susceptible-exposed-infectious-removed class of models which is widely employed in epidemics. Applications with experimental foot-and-mouth disease and historical smallpox data are discussed, and simulation results are presented. Inference is carried out using Markov chain Monte Carlo methods following a Bayesian approach. Model evaluation is performed, where the adequacy of the models is assessed using methodology based on the properties of Bayesian latent residuals, and comparison between 2 candidate models is also considered using a latent likelihood ratio-type test that avoids problems encountered with relevant methods based on Bayes factors.

An epidemic model with infector and exposure dependent severity

A stochastic epidemic model allowing for both mildly and severely infectious individuals is defined, where an individual can become severely infectious directly upon infection or if additionally exposed to infection. It is shown that, assuming a large community, the initial phase of the epidemic may be approximated by a suitable branching process and that the main part of an epidemic that becomes established admits a law of large numbers and a central limit theorem, leading to a normal approximation for the final outcome of such an epidemic. Effects of vaccination prior to an outbreak are studied and the critical vaccination coverage, above which only small outbreaks can occur, is derived. The results are illustrated by simulations that demonstrate that the branching process and normal approximations work well for finite communities, and by numerical examples showing that the final outcome may be close to discontinuous in certain model parameters and that the fraction mildly infected may actually increase as an effect of vaccination.

Reproduction numbers for epidemics on networks using pair approximation

One way to describe the spread of an infection on a network is by using the method of pair approximation. This method is a deterministic pair-based variant of the usual methods used to describe the progress of an epidemic in randomly mixing populations. However, although the ideas of pair approximation are intuitively clear, it is not straightforward to make all assumptions used explicit. Furthermore, in literature problems arise in defining basic quantities like the basic reproduction number R(0) and the real-time epidemic growth rate parameter r. We formulate the pair approximations and the needed assumptions explicitly. We discuss problems inherent to this method. Furthermore, we define a new reproduction number, similar to R(0) and a new real-time growth rate parameter similar to r. We illustrate the methods of the paper by an example for which we can compare the approximation of the reproduction number with exact results.

Control of transmission with two types of infection

The effectiveness of a vaccination strategy to control transmission of an infectious disease depends on the way vaccine doses are distributed to individuals in a community of households. Here we show that this dependence is more complicated when acquisition and severity of illness are determined by the size of the infecting dose, as is thought to be the case for measles and varicella. Two alternative formulations for the way vaccination changes an individual's susceptibility and infectivity show that vaccination coverage, the nature of the vaccine response and the distribution of household size also have a big impact on which strategy is more effective. These judgements are made by comparing the post-vaccination reproduction numbers corresponding to different vaccination strategies.