[After reading this module, you will be aware of the limitations of deterministic epidemic models, like the SIR model, and understand when stochastic models are important. You will be introduced to three different methods of stochastic modelling, and have a basic understanding of the appropriate applications of each.] Contents:
- Introduction
- Stochastic epidemic simulation: stochastic differential equations
- Stochastic epidemic simulation: Markov Chain Monte Carlo
- Stochastic epidemic simulation: Agent Base Modelling
Deterministic epidemic models, like the Susceptible, Infected, Recovered (SIR) model are extraordinarily useful tools in epidemiology. Models like the SIR model can be extended to include heterogeneity of transmission (like different transmission rates for different age classes, time dependent transmission, etc), and heterogeneity of the population (like age structure, gender structure, etc). Compartments that describe isolated, quarantined, or vaccinated individuals can be added, and deaths and births can also be included. Numerical solution of deterministic models with modern computers is almost instantaneous, and these models thus lend themselves well to studies that lead to optimized disease intervention strategies, such as vaccination programs targeted at segments of the population that tend to be super-spreaders of disease. However, despite the many advantages of deterministic models, it can be difficult (but not impossible) to include realistic population networks (like family, school, and workplace networks) in such models. It can also be difficult to incorporate realistic probability distributions for the time spent in the infectious period. Deterministic models also do not allow us to assess the probability of an outbreak, and the probability distributions of the final size of an epidemic in small populations. For this, we need stochastic models.
Stochastic epidemic simulation: stochastic differential equations
There are several ways to stochastically simulate epidemics. Models based on stochastic differential equations (SDE’s) are very similar to ODE deterministic models, except that the time derivatives of the compartments include an extra stochastic term. SDE’s have the advantage that, computationally, the simulation runs almost as fast as that of the equivalent deterministic ODE model. SDE’s have the disadvantage that they are imprecise for describing the stochastic variation in small population sizes. SDE’s are also just as limited as ODE’s at describing networked populations and other heterogeneities that are important to the dynamics of disease transmission. However, tor medium to large populations SDE’s work quite well for quick assessment the probability of outbreak and/or the probability distribution of the final size of the epidemic (or other quantities of interest). For small populations, SDE’s are inappropriate because the stochastic term assumes that the change in each compartment is Normally distributed. This only occurs when the change in the compartment is fairly large (because it is only for large values of lambda that the Poisson distribution approaches the Normal distribution). Thus, for stochastic modeling with small populations, MCMC or agent based models are more appropriate.
Stochastic epidemic simulation: Markov Chain Monte Carlo
For small populations, Markov Chain Monte Carlo (MCMC) methods are useful for stochastic simulation. MCMC methods step through the simulation in very tiny time steps… so tiny that only one “event” happens on average during that step (where an “event” could be an infected person recovering, or a susceptible person getting infected). At each time step the probability of each possible “event” is assessed, and one of the events is chosen randomly (weighted by its probability of occurrence).
The disadvantage of using MCMC methods for epidemic modelling is that they can be somewhat slow for medium and large size populations because of the tiny time steps. Also, MCMC can be just as limited as SDE’s and ODE’s when it comes to incorporating networked populations and other heterogeneities.
Stochastic epidemic simulation: Agent Based Modelling
The final method of stochastic epidemic simulation that I will discuss here is Agent Based Models (ABM), also known as Individual Based Models (IBM). ABM’s have the advantage that you can put all kinds of detail into the model at the individual level (ie; you can set up your population, give them different ages, have people living in households with other people, going to school/work with other people, etc). You can also implement arbitrarily complex probability distributions for the time spent in the infectious period, etc. Because of the almost infinite complexities that can be incorporated into an ABM, they can be very, very slow to simulate epidemics in populations of even moderate size (like a town). A potential pitfall of using ABM’s is that it can be all too easy to get carried away with model complexity, resulting in lots of model parameters, especially if you have to start guessing the values of various parameters because little data exists to indicate what the true values should be. There is a saying called GIGO (“garbage in, garbage out”) that can well apply to ABM simulations that have too many arbitrarily chosen parameters. Because of the GIGO pitfall, I always ask myself if I really need to perform a simulation with all that complexity to answer a particular research question (the answer is often “no”, and I turn to a method like SDE’s or MCMC instead). The fun thing about ABM’s is that in addition to allowing a lot of freedom in complexity of simulation, they are, well… fun. Setting up your population, then introducing a disease and watching what happens is kind of like playing Sims.