The following direct quotations give an index of the concepts addressed in the paper:

* The ODE epidemic models serve as a framework for formulating analogous stochastic models and as a source of comparison with the stochastic models.

* This primer is restricted to two types of stochastic settings, continuous-time Markov chains (CTMCs) and stochastic differential equations (SDEs).

* Some well-known examples are used for illustration such as an SIR epidemic model and a host-vector malaria model.

* Stochastic modeling of epidemics is important when the number of infectious individuals is small or when the variability in transmission, recovery, births, deaths, or the environment impacts the epidemic outcome. The variability associated with individual dynamics such as transmission, recovery, births or deaths is often referred to as demographic variability. The variability associated with the environment such as conditions related to terrestrial or aquatic settings is referred to as environmental variability.

* In the SIR deterministic model, S(t), I(t), and R(t) are the number of susceptible, infectious, and recovered individuals, respectively. In the simplest model, there are no births and deaths, only infection and recovery

* The discrete random variables for the SIR CTMC model satisfy: \\(S(t), I(t) \in \lbrace 0,1,2,\ldots N \rbrace\\)

* 3.3 Branching process approximation. In this brief introduction, we study the stochastic behavior near the disease-free equilibrium to determine whether an epidemic (major outbreak) occurs when a few infectious individuals are introduced into the population.

* 4. SIR stochastic differential equations. Stochastic differential equations for the SIR epidemic model follow from a diffusion process. The random variables are continuous: \\(S(t), I(t) \in [0, N] \\).

* 4.2 Numerical simulation. The Euler-Maruyama method is a simple numerical method that can be used to simulate sample paths of SDEs.

* Malaria continuous time Markov chain

* Malaria stochastic differential equations

* Environmental variability. For the SIR epidemic model or the malaria host-vector model, changes in the environment may impact the parameters for birth, death, recovery, or transmission. For example, if birth, death, or transmission rates fluctuate with changes in the environmental conditions, then a stochastic differential equation for the model parameter can be formulated as a mean-reverting process (fluctuation about some average value).