The spread of the infection has a logistic growth: at the beginning, it grows exponentially until a large fraction of the population is infected. From that, the growth rate starts to decrease until herd immunity takes over and the infection vanishes. See Figure 1, where the plot of Rₜ – the number of individuals who have been infected by time t – is a logistic curve.
This model is a key element of our recently published work, where we studied two questions:
- Will the disease die off before any significant number of individuals get infected, or will it take root and spread until the population reaches herd immunity?
- Is travelling always making the spread of the virus worse?
To answer the first question, we model the individuals as agents and assign a certain probability to every state transition, as illustrated in Figure 2. In expectation, one step of this discrete-time model is described by the above equations. However, any given infected agent may infect more or fewer other agents than expected, or it may remain infected for a shorter or longer time than expected. When we take that into account, we capture the scenarios when the disease is extinguished early in the process. We calculate the exact probability with which this happens. Modelling this behaviour leads to a more precise prediction of the expected spread of the disease than what we obtain from the differential equations.
For the second question, we assume there are multiple countries, which have their own, potentially different, infection rates β₁,β₂,…, but the same recovery rate γ. Figure 3 shows the transition diagram for the case of two countries, where the arrows with the label p represent travelling and a larger p means that travelling is more frequent.