Figure 1: Compartmental SI model of disease spread, in which S = susceptible group, I = infectious group, and λ = risk (or force) of infection which determines the rate of transition of the population from S to I.
We use differential equations to illustrate the process of disease transmission as a continual process, although difference equations using a time-step process could also be used (Vynnycky & White, 2010). The risk of infection, known as λ(t) , is the rate at which susceptible individuals become infected and is dependent on both the number or proportion of infected individuals in the population,I(t) , and the effective contact rate (ecr ), or ß , more formally defined as the per capita rate at which two specific individuals come into effective contact per unit time (Equations 1 and 2).
\(\lambda_{\left(t\right)}=\ \beta.I_{(t)}\) Equation 1
\(\beta\ =\ ecr/N\) Equation 2
The disease model can then be defined in terms of 2 equations in which the S compartment loses individuals and the I compartment gains individuals as disease spreads (Equations 3 and 4; Figure 2).
\(\frac{\text{dS}_{\left(t\right)}}{\text{dt}}=\ -\ \beta.I_{\left(t\right)}\text{. }S_{\left(t\right)}\)Equation 3
\(\frac{\text{dI}_{\left(t\right)}}{\text{dt}}=\ \beta.I_{\left(t\right)}\text{.}S_{\left(t\right)}\)Equation 4
We can further define effective contact by considering the basic reproductive rate, R0 . This is the number of infected individuals that arise from one typically infectious individual during their entire infectious period when introduced to a totally susceptible population. Therefore, R0 is equivalent to the effective contact rate (ecr ) multiplied by the duration of infectiousness, D (Equation 5), and the per capita rate at which two specific individuals come into effective contact per unit time is described in Equation 6.
\(R_{0}=ecr\ .\ D\) Equation 5
\(\beta=\ R_0/(N\ .\ D)\ \) Equation 6
Further modification of equations to describe effective contact depend on whether we assume contacts are dependent on the density of the population, or are limited to a finite number of contacts between two individuals (Begon et al., 2002). This is determined by our understanding of the disease transmission process. The former is called density dependent transmission, the latter frequency dependent transmission. The density dependent assumption is generally considered appropriate for animal diseases in which the population is constrained within a given space. This might be the case for livestock, and perhaps also many wild animal species. The frequency dependent assumption might be more appropriate in the case of sexually-transmitted diseases, or human or companion animal diseases in which contact is determined by social networks and constraints, rather than population size.
To produce a realistic output, data to parameterise the effective contact rate is essential (Kirkeby et al., 2020). Without knowing details about the way in which the disease is transmitted, a modeller can use previous or current outbreak data to determine the likely duration of infectiousness of individuals and R0 . However, relevant outbreak data are often unavailable, and although it might be possible to generalise R0 and the duration of infectiousness from other situations,R0 depends on local context (for example, theR0 of measles is estimated to range from 12 to 18; Guerra et al., 2017). Instead, the frequency of contact at disease transmission interfaces and the probability of transmission given the potential routes of transmission can be used to infer an effective contact rate. This approach is often used when building models in contexts in which outbreaks are yet to occur and the model is being used to predict the pattern of disease spread and the efficacy of control measures. A range of experimental and observational data might be used, for example, laboratory transmission experiments to define transmission probability, and field telemetry data to define contact rates. In the next section we describe a range of methods that have been used in rabies spread models in free-roaming domestic and wild dogs in northern Australia to parameterise the effective contact rate using the probability of contact (bite) and the probability of rabies virus transmission, given a contact.
Models become more complex and data requirements increase for diseases with multiple routes of transmission because each route has a probability of effective contact (more than one ß) for which species-specific data about the probability of contact needs to be collected, as well as disease-specific data such as the probability of infection associated with the route. For disease transmission between populations, such as at a wildlife interface, population density, distribution, and the behaviour of the populations such as pack size, home range and seasonality, or spatial variation of movements might be required, to reflect the nature of interaction between the populations. This contact heterogeneity (rather than homogenous mixing) can be important for valid predictions of disease spread patterns and assessment of control options.
Population dynamics are also needed to derive birth and death rates (which could also be seasonal), and knowledge of the progression of disease in individuals is required to parameterise the rate of transition between states other than susceptible to infected (for example duration of infectiousness to determine rate of recovery in an SIR model). Finally, uncertainty can be reflected by introducing stochasticity of events so that outputs represent a range of possibilities (for example, the number of individuals infected, or the duration of an outbreak), and by using distributions of parameters to represent known natural variability (for example, in disease parameters such as latent period or in population parameters such as group size) or limited knowledge (the influence of such parameters can be assessed using sensitivity analysis).