Dynamics of simultaneous epidemics on complex graphs
Janes, Denys Zachary Alexander
The subject of this thesis is the study of a system of multiple simultaneously spreading diseases, or strains of diseases, in a structured host population. The disease spread is modelled using the well-studied SEIR compartmental model; host population structure is imposed through the use of random graphs, in which each host individual is explicitly connected to a predetermined set of other individuals. Two different graph structures are used: Zipf power-law distributed graphs, in which individuals vary greatly in their number of contacts; and Poisson distributed graphs, in which there is very little variation in the number of contacts. Three separate explorations are undertaken. In the first, the extent to which two SEIR processes will overlap due to chance is examined in the case where they do not affect each other's ability to spread. The overlap is found to increase with increased heterogeneity in the number of contacts, all things equal. Introducing differences in infection probability or a delay between introducing the two strains produces more complex dynamics. I then extend the model to allow strains to modify each other's transmissibility. This is found to lead to modest changes in the size of the outbreak of affected strains, and larger effects on the size of the overlap. The extent of the effect is found to depend strongly on the order in which the strains are introduced to the population. Zipf graphs experience somewhat larger reductions in outbreak size and less reduction of overlap size, but overall the two graphs experience similar effects. This is due to the reduced effect of modification in key high-degree vertices in the Zipf graph being offset by higher local clustering. Finally, I introduce recombination and competition by replacement into the model from the first project. The number of recombinant strains that arise is found to be either very low or very high, with chance governing which occurs. Recombinant strains in Zipf distributed graphs have a significant chance of failing to spread, but not in Poisson distributed graphs. Replacement competition in the presence of a growing number of strains is found to both increase the chance of a strain failing to spread, and to reduce the overall size of outbreaks. This effect is equal in both graph types.