Figure 1.
Schematics of a branching process.
Two generations of infections are shown. Every horizontal bar segment represents the time interval that a specified individual remains infectious; these time intervals follow a negative exponential distribution with average μ−1. The time intervals between infections for any given individual follow a negative exponential distribution with an average of β−1.
Figure 2.
Schematics of our individual-level model (ILM).
The model tracks individuals through growing a transmission network by using infection and removal rules [10]. Individuals are represented as the nodes of the network; two individuals a and b are connected by a directed link from b to a if b has infected a. In the Figure, a is a newly infected individual added to the growing transmission network. As an example of an infection rule, a node b is uniformly randomly chosen to be the infectious individual who has infected a. If a removal occurs, an individual c is randomly chosen from the infected group, and is removed. Under the assumption that the number of the secondary infections caused by c is a proxy for the progression of the disease, we choose that the probability that c is removed to be proportional to the number of secondary infections caused by c plus one; i.e., to be proportional to the total number of connections of c. The node c remains connected to the network, but can not cause any new infections; i.e., c becomes the same as node d who previously infected c. The rates of infections and removals per infectious capita are β and μ, respectively. The branching process presented in Fig. 1 yields the same expected incidence as our ILM which is given by the SI ODE model.
Figure 3.
Comparison plots between our example ILM and branching process.
A The average prevalence versus time. The open circles represent results for the branching process while the dots represent results for our ILM. We used β = 0.015, μ = 0.01, and we averaged over 1,000 stochastic realizations. On a logarithmic scale, the results fit very well with a straight line, with slope (β−μ) and intercept 0, that corresponds to the ODE solution I(t) = I(0)exp[(β−μ)t], where I(0) = 1. B The average number of secondary infections stratified versus the date of infection. The open circles represent results for the branching process while the dots represent results for our ILM. We used β = 0.015, μ = 0.01, and we averaged over 15,000 stochastic realizations. The R0 of the branching process is 1.5, while the R0 of our ILM is approximately 1.4.