Abstract
We study the spread of sexually transmitted infections (STIs) and other infectious diseases on a dynamic network by using a branching process approach. The nodes in the network represent the sexually active individuals, while connections represent sexual partnerships. This network is dynamic as partnerships are formed and broken over time and individuals enter and leave the sexually active population due to demography. We assume that individuals enter the sexually active network with a random number of partners, chosen according to a suitable distribution and that the maximal number of partners that an individual can have at a time is finite. We discuss two different branching process approximations for the initial stages of an outbreak of the STI. In the first approximation we ignore some dependencies between infected individuals. We compute the offspring mean of this approximating branching process and discuss its relation to the basic reproduction number R_0. The second branching process approximation is asymptotically exact, but only defined if individuals can have at most one partner at a time. For this model we compute the probability of a minor outbreak of the epidemic starting with one or few initial cases. We illustrate complications caused by dependencies in the epidemic model by showing that if individuals have at most one partner at a time, the probabilities of extinction of the two approximating branching processes are different. This implies that ignoring dependencies in the epidemic model leads to a wrong prediction of the probability of a large outbreak. Finally, we analyse the first branching process approximation if the number of partners an individual can have at a given time is unbounded. In this model we show that the branching process approximation is asymptomatically exact as the population size goes to infinity.
Highlights
Transmitted infections (STIs) are among the world’s most common diseases remaining as a major global threat
We implicitly assume that the number of susceptible individuals that are not connected to infectious individuals is very large and their properties, such as the distribution of the number of other susceptible partners etc., does not change as long as the branching process approximation is valid, i.e. we study the initial phase of the epidemic
3 Branching process approaches to the spread of epidemics 3.1 A first naive approach In this subsection, we study the spread of an sexually transmitted infections (STIs) on the partnership network in the beginning of an epidemic by employing an appealing but wrong branching process approximation approach
Summary
Transmitted infections (STIs) are among the world’s most common diseases remaining as a major global threat. Leung and co-authors use deterministic models to study different epidemic models on the dynamic graphs introduced in their work (that we briefly discuss in the following paragraph) In this deterministic approach, one implicit assumption is that the initial fraction of the population which is infectious might be very small, but always positive, which implies that the number of initially infectious individuals is large, because it is effectively assumed that the total population size is infinite. 4, the first approximation of the model is used to study the epidemic on the dynamic network when the partnership capacity is infinite, i.e. when n = ∞ In this particular case, dependencies fall away and we may use branching processes to analyse the early phase of an S I epidemic spreading through a dynamic sexual network.
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