Inference and inverse problem:
The inverse problem consists of identifying parameters of process based model or a statistical model that might have produced the observed data.
Binomial/Beta distribution
Here we will begin with a simple example of estimating the bias of a coin based on one experiment after N=400 tosses resulting in 200 heads. The probability P(N,r) that will inform the bias (or lack of it ) of the coin is given the binomial/beta distribution
This probability function peaks at p=0.5 if N=400 and r=200. Hence the maximum likelihood estimator predicts and unbiased coin given the data.
However we repeat the coin toss experiment sufficiently large number of times, then we will get a normal distribution for the parameter p, whose expectation value will be true estimate of the biasness of the coin. For a single coin toss experiment then we can construct confidence interval, that will quantify the uncertainity in the estimate of the true value of p based on single experiment.
Stochastic Simulation: Extinction
Introducing an infectious individual into a population implies that the infection will likely spread to a final size, denoted as I_size. This is possible if the basic reproduction number is greater than one. However, this deterministic prediction may not hold true when considering the stochastic nature of the spreading process. The probability of disease transmission diminishes with an increase in the population size. In larger populations, the likelihood of successful infection decreases due to the stochastic variability inherent in the spreading dynamics. In the simulations showed in Figure 2, the initial conditions were set to be S=0.998, E=0.001, I=0.001 fraction of the total population.
Figure 2: Probability of extinction as a function of population size. As the population increases the probability of extinction decreases and eventually becomes zero.