Stochastic epidemic models in structured populations

Lead Research Organisation: University of Nottingham
Department Name: Sch of Mathematical Sciences

Abstract

The aim of the proposed research is to develop and analyse stochastic epidemic models that incorporate realistic population structures and are mathematically tractable. Random network models that explicitly include household structure will be investigated, as will models that allow different severities of infection and models incorporating more complex structures, such as hierarchical mixing (e.g. a population of cities, each of which is partitioned into households, with different infection rates for within-household, within-city and between-city contacts), overlapping subgroups (in which the population is partitioned in several ways, e.g according to both the household and workplace of individuals) and spatial local mixing. For each model, a threshold parameter that governs whether or not an outbreak can become established will be determined, together with other properties, such as the probability that an outbreak does become established and the final outcome if it does. Implications for vaccination strategies will be explored, using a varitey of models for how vaccination affects a vaccinee's susceptibility to the disease in question and his/her ability to transmit the disease if he/she becomes infected. The theory developed will be tested by using simulations on realistic contact networks that have been proposed for diseases such as SARs and pandemic influenza. The planned research will lead to novel methods for analysing epidemic models and to increased understanding of the mechanisms underlying the spread of infection and the effect of such mechanisms on the performance of vaccination schemes.
 
Description A stochastic model was developed and analysed for the spread of an epidemic amongst a population with a random network of social contacts, that is partitioned into households. A threshold parameter that determines if an epidemic with few initial infectives can become established and lead to a major outbreak was determined, as was the probability and average size of a major outbreak. Household size distribution, the distribution of the number of neighbours a typical individual has in the network and the level of clustering present in the overall population structure were each shown to impact significantly on disease dynamics. Vaccination strategies were explored using a range of models for vaccine action. A variety of households-based vaccine allocation schemes was investigated and the households-based scheme for preventing major outbreaks with minimum vaccination coverage was determined. A new model for acquaintance vaccination, a method for targeting vaccination at individuals with many neighbours, which is amenable to analysis for imperfect vaccines, was developed and acquaintance vaccination was shown to appreciably outperform households-based allocations in many situations. The households-network model and its analysis were extended to include several types of individuals (e.g. adults and children). The above results were proved rigorously by obtaining limit theorems as the number of households tends to infinity and validated using simulations on moderately-sized finite contact networks.



Various models for infectious diseases with two severities of infection, mild and severe, were investigated. For each model, a threshold parameter was determined, together with other properties, such as the probability and average size of a major epidemic. A homogeneously mixing model, in which an individual can become severely infectious directly upon infection (with probability depending on the type of its infector) or if additionally exposed to infection, was developed and analysed. Implications for vaccination strategies were explored, using a variety of models for vaccine action. Two different models with a household structure were analysed. In the first model, the infection status (mild or severe) of an individual is predetermined, perhaps due to prior immunity, and in the second the infection status of an individual depends on that of its infector and on whether the individual was infected by a within- or between-household infection. Large population properties of the model were derived and numerical studies were used to show that, given final size household outbreak data (containing mild and severe cases) on sufficiently many households, it is generally possible to determine which of the two hypothesised explanations is causing the varying response.



A model for the spread of an epidemic on a network of individuals described by a random intersection graph (i.e. a population in which each individual belongs to a random number of groups and infection can be transmitted only between individuals that share a group) was analysed, using infinite-type branching processes, and qualitatively similar results as for the households-network model were obtained. A network epidemic model in which individuals can also make casual contacts, i.e. with people chosen randomly from the population, was also studied. In addition to threshold results, the size of a major outbreak in a large population was shown to be approximately normally distributed, and numerical studies demonstrated that the inclusion of casual contacts can have a major impact on the outcome of an epidemic.



In summary, a range of new stochastic epidemic models that incorporate realistic population structures was developed, together with techniques for their rigorous mathematical analysis, and the novel features in these models were shown to have significant impact on disease dynamics and the performance of vaccination strategies.
Exploitation Route Not applicable
Sectors Healthcare
 
Description This was a theoretical project and there is no direct non-academic impact.