Universiteit Utrecht

Department of Mathematics

Generalised random graphs as a model for epidemic spread
in an inhomogeneous population

Piter Trapman

8 May 2007

We consider an SIR (Susceptible -> Infective -> Removed) epidemic model on a (possibly complete) network. Every individual has a 2-dimensional random vector  (wwj)  assigned to it, denoting its susceptibility and its infectivity if it becomes infected. An infective individual  i  contacts a neighbour  j  at rate  wi  and if  j  is still susceptible the probability that it becomes infectious at a contact is  wj.

We study properties of the above model by relating it to generalised random graphs as described by Britton et al. (J. Statist. Phys. 2006). In this model the individuals have i.i.d. weights assigned to them and two individuals are connected with a probability depending on the weights of the two individuals.

We compare different epidemics with given expected “infectivity” and “susceptibility” and show that if  w  and  w  are independent, then fixed  w  and  w  is a worst case scenario, in the sense that the probability of a large outbreak as well as the expected number of ultimately removed individuals is maximal. To obtain this result we use an idea of Kuulasmaa (J. Appl. Probab. 1982).

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Yuri Yakubovich