Epidemic Modelling: Simulations Using Stochastic Methods
Stochastic Models| Epidemics| Angeline Xiao
Branching Processes are a method of modelling the processes of populations that evolve independently with chance. These stochastic models can be used to simulate epidemics. Say we have an infected individual, they will inflict more cases with a certain probability, and the new infected individuals will independently continue to infect at the same rate. [1] The offspring distribution, and mean daily offspring are very useful in showing how epidemics may spread stochastically, given different parameters and models. Over summer, as part of the UoA summer research programme, we conducted epidemic simulations for a variety of models and infection rates.
We can determine if a branching process will die or not based on its mean daily offspring, popularly known as an Reff value. If each individual has on average less than 1 offspring per generation then it will almost surely die. We call this process the subcritical process. If the mean offspring is 1, this is the critical process, and the population will still, almost surely die. If Reff is greater than 1 then we call this a supercritical process, and it is not certain that the process will die out, although it is still possible. [2]
We can have branching processes in both discrete and continuous time. Both have their uses, with discrete time being a simpler model to use and understand. Continuous time branching processes are still useful as viruses do not adhere to human set intervals such as days, and a more accurate picture can be painted.
For a discrete time branching process model, the offspring of a population can be expressed by the sum of all the offspring that each member of the population will have, independent of each other. For a population in generation n, Zn, the population in the next time generation, Zn₊1, can be expressed by the sum of independent identically distributed random variables which is same offspring distribution, Xn,i, where Xn,i represents the offspring per individual per time generation.
That is:
To simulate total active cases, we define that when an individual has n infected offspring in a new day, [latex6] of those offspring are new infections, and 1 offspring is the individual surviving. If an individual has 0 offspring then they have 'recovered'.
Perhaps the most intuitive offspring distribution is the Poisson distribution. Every day each infected individual infects more individuals at a rate λ, where λ-1 is the mean number of individuals they infect (as they are not infecting themselves again). 100 iterations of this simulation were performed with λ rates of 0.8, 1 & 1.2 to demonstrate a subcritical, critical & supercritical process. We can observe the trend towards death of the subcritical process population while the supercritical process population explodes (Fig. 1).
Figure 1
An extremely important but less intuitive offspring distribution we can use is the Geometric distribution. We define a Geometric distribution to represent the probability of the number of failures before the first success in a sequence of Bernoulli trials, where we can set the parameter p to be the probability of success. In context, this would signify the number of individuals infected before an individual stops infecting for the time unit.
For X ~ Geom(p),
The Geometric distribution has special properties that allow us to explicitly calculate the branching process with a Geometric offspring distribution using generating functions. This allows us to conduct sanity checks on our simulations before we dive into more complex, noncalculable simulations. To compare subcritical, critical, and supercritical processes, p values of 5/9, 1/2, and 5/11 were used so that we have a mean daily infection rate of 0.8, 1, and 1.2 respectively. We can see that the simulation run for our 3 mean daily offspring rates looks similar to the Poisson model (Fig. 2). However what we are really interested in is the extinction proportion. On an arbitrary day (take day 10) we can see the distribution of extinction for each other mean daily offspring rates. (Fig. 3, Fig. 4, and Fig. 5).
Figure 2
By comparing the histogram on day 10 to the explicit survival probability on day 10, we can confirm the accuracy of our simulations.
Using the generating function in the geometric case, we can find the extinction probability on day n,
for an initial population of 1 explicitly. [3] Given the offspring distribution:
Let us assume:
Notice:
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Alex Chapple - MSc, Physics
Alex is a Master's student in the Department of Physics. He likes pretzels and churros.