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).