Extrinsic Incubation Period
The latent period for malaria infection in mosquitoes has traditionally been called the extrinsic incubation period (EIP). It is defined as the time elapsed from the blood meal that infects a mosquito to the time when sporozoites appear in the salivary glands, and a mosquito is infective. The time lag presents a technical challenge. Here, we give an overview of methods and for handling the EIP, and some closely related issues.
Forcing |
Fixed Lag, Time-Varying
If the EIP is a fixed lag that varies with time (e.g., if it is a function of temperature), then we need notation for the lag computed with respect to the moment when a mosquito becomes infected and when it becomes infectious.
Let \(\tau(t)\) as it would be computed in a degree-day model, from the moment a mosquito becomes infected: a mosquito becoming infected at time \(t\) would become infectious at time \(t+\tau(t).\)
Let \(\tau'(t)\) denote the lag for a mosquito at the point in time when it becomes infectious: a mosquito becoming infectious at time \(t\) would have become infected at time \(t-\tau'(t).\)
The two are related by the identities \[\tau(t) = \tau'(t+\tau(t))\] and \[\tau'(t) = \tau(t-\tau'(t)).\]
In Differential Equations
Taking a step back from models for parasite / pathogen transmission by mosquitoes, a big difference is how the models handle the EIP. There have been five main approaches for:
- In Ross’s original model [1], there was no delay. The SI compartment model implementation in SimBA ignores the delay but incorporates the mortality:
\[\begin{array}{rl} \frac{dM}{dt} &= \Lambda - g M \\ \frac{dY}{dt} &= fq\kappa(M-Y) - Y/\tau - g Y \\ \end{array}\] and
\[Z = e^{-g\tau} Y\]
- Sharpe & Lotka introduced a fixed delay in a delay differential equation (DDE) [2]. If the model. Macdonald’s model for the sporozoite rate used the fixed delay [3,4]
\[\begin{array}{rl} \frac{dM}{dt} &= \Lambda - g M \\ \frac{dY}{dt} &= fq\kappa(M-Y) - g Y \\ \frac{dZ}{dt} &= e^{-g\tau} fq\kappa_\tau(M_\tau-Y_\tau) - gZ \\ \end{array}\]
- It is common practice to use an SEI compartment model.
\[\begin{array}{rl} \frac{dM}{dt} &= \Lambda - g M \\ \frac{dY}{dt} &= fq\kappa(M-Y) - (\psi + g) Y \\ \frac{dZ}{dt} &= \psi Y - gZ \\ \end{array}\]
- Argues for a sigmoidal distribution [5]. One way to achieve this is to implement a two-stage latent period. There is a fixed delay before mosquitoes start to emerge, and then there is a delay of \(\tau + \psi^{-1}\)
\[\begin{array}{rl} \frac{dM}{dt} &= \Lambda - g M \\ \frac{dE}{dt} &= fq\kappa(M-E-Y-Z) - g E \\ \frac{dY}{dt} &= e^{-g\tau} fq\kappa_\tau(M_\tau-E_\tau-Y_\tau-Z_\tau) - \psi Y - g Y \\ \frac{dY}{dt} &= \psi Y - g Z\\ \end{array}\] + The boxcar gives a \(\Gamma-\) distributed EIP:
\[\begin{array}{rl} \frac{dM}{dt} &= \Lambda - g M \\ \frac{dY_1}{dt} &= fq\kappa(M-\sum Y-Z) - (g + \psi/n) Y_1 \\ \frac{dY_i}{dt} &= \psi/n Y_{i-1} - \psi/n Y_i\\ \frac{dZ}{dt} &= \pi/n Y_n - g Z\\ \end{array}\]