Forced Bionomics with Lags

Computing Mosquito Survival and Dispersal through a Time-Varying EIP

In delay differential equations describing mosquito infection dynamics, we are faced with the challenge of dealing with time-varying mosquito bionomic parameters, including a time-varying EIP. In particular, we must compute \(\Upsilon(t),\) the fraction of mosquitoes that survived to become infectious at time \(t.\)

In the following, we explain the mathematics and the SimBA algorithm for computing mosquito survival and dispersal through a time-varying EIP.

The EIP

To describe the dynamics of infection in mosquitoes, we must take some care in defining a time varying parasite latent period, traditionally called the extrinsic incubation period (EIP) If the lag 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. Here, we must define the EIP with respect to both:

• Let \(\tau(t)\) denote the delay counting 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 \[\tau'(t) = \tau(t-\tau'(t)).\]

The Demographic Matrix

The demographic matrix, \(\Omega\) includes all the parameters affecting mosquito survival and dispersal:

• \(g(t)\) is the patch mortality rate of adult mosquitoes;

• \(\sigma(t)\) the patch emigration rate;

• \(\mu(t)\) the loss associated with emigration (including emigration related mortality and emigration out of the system); and

• \(K(t)\) is the dispersal matrix.

After leaving a patch, the proportion surviving dispersal (\(1-\mu(t)\)) that end up in every other patch is given by a kernel, \(K(t).\)

\[\Omega(t) = g(t) + \sigma (1-\mu(t)) \cdot K(t).\]

In a delay differential equation, we must compute survival and dispersal through the EIP, \(\Upsilon(t)\). If all the parameters were constant, and \(\tau(t)=n,\) then we would need to compute:

\[ \Upsilon = e^{-\gamma} \]

where

\[\gamma = \int_{t-\tau'(t)}^t \Omega(s) ds\]

Upsilon

To motivate the algorithm used to compute it, we introduce a new variable, \(G\) that integrates \(\Omega\) over time:

\[\begin{equation} \frac{d{G}}{dt} = \Omega(t) dt. \end{equation}\]

Letting \(s\) denote the moment a mosquito becomes infectious, cumulative mortality from time \(t\) to time \(t+s\) is:

\[\begin{equation} \gamma(t,s) = { G}(t+s) - { G}(t). \end{equation}\]

The probability a mosquito becoming infectious at time \(t\) survived the interval to become infectious is:

\[\begin{equation} \Upsilon(t) = e^{-\gamma\left(t-\tau'(t), t\right)} = e^{-\left({ G}\left(t\right) - { G}\left(t-\tau'(t) \right)\right)}. \end{equation}\]

In solving the equations, we compute \(\Upsilon\) as a set of accessory variables with dynamics:

\[\begin{equation} \frac{d\Upsilon}{dt} = \left(\Omega\left(t-\tau'(t)\right)\left(1-\frac{d\tau'(t)}{dt}\right)-\Omega(t)\right)\Upsilon(t) \label{dUpsilon} \end{equation}\]

The initial condition, \(\Upsilon(0) = e^{-\Omega(0)},\) can be computed using the expm package in R.