Skip to contents

The GeM module for the MY component

Source: R/adult-MY-GeM.R
GeM.Rd

GeM is a system of delay differential equation model for mosquito ecology and infection dynamics. The core dynamics trace back to Macdonald's mosquito model (1952). Macdonald's model was extended by Aron & May (1982) to include a variable for mosquito population density and a term for emergence (\(\Lambda\)). Aron & May's model was extended by Wu et al. (2023) to include spatial dynamics. In this version, a new method is introduced to non-autonomous dynamics: an accessory variable computes time-varying survival and dispersal through a time varying EIP.

State Variables

M

density of adult mosquitoes

P

density of parous adult mosquitoes

Y

density of infected adult mosquitoes

Z

density of infectious adult mosquitoes

Parameters

f

blood feeding rate

q

human blood fraction

nu

egg batch laying rate

eip

extrinsic incubation period (\(\tau\))

g

mosquito mortality rate

sigma

emigration rate (\(\sigma\))

mu

emigration loss rate (\(\mu\))

K

mosquito dispersal matrix

eggs_per_patch

# eggs per batch

Mosquito Demography

The demographic matrix \(\Omega\) is: $$ \Omega(t) = \mbox{diag} \left( g(t) + \sigma(t) \mu(t) \right) - K \cdot \mbox{diag} \left( \sigma(t) \left(1-\mu(t)\right) \right) $$

EIP

Let \(\tau(t)\) denote the EIP for a mosquito, if infected at time \(t\): the mosquito would become infectious at time \(t+\tau(t).\) Similarly, let \(\tau'(t)\) denote the lag for a mosquito that became infectious at time \(t\). It was 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 the implementation, a function F_eip returns \(\tau'(t)\). Implementation of dynamically changing EIP also requires a function to compute the derivative dF_eip (below).

The Accessory Variables, \(\Upsilon\)

To compute time-varying survival and dispersal through a time-varying EIP, we use a set of accessory variables, called \(\Upsilon\). To motivate the algorithm used to compute it, we introduce a new variable, \(U\) that integrates \(\Omega\) over time: $$dU/dt = \Omega(t).$$

Cumulative mortality from time \(t\) to time \(t+s\) is: $$\gamma(t,s) = U(t+s) - U(t).$$ Survival and dispersal for the mosquitoes that became infectious at time \(t\) is: $$\Upsilon(t) = e^{-\gamma\left(t_\tau, t\right)} = e^{-\left(U\left(t\right) - U\left(t_\tau \right)\right)}.$$ where \(t_\tau = t-\tau'(t)\).

In GeM, \(\Upsilon\) is computed as an accessory variable, with derivatives: $$d\Upsilon/dt = \left(\Omega\left(t_\tau\right)\left(1-\frac{d\tau'(t)}{dt}\right)-\Omega(t)\right) \cdot \Upsilon(t)$$ Initial conditions are set to: $$\Upsilon(t_0) = e^{-\Omega(t_0)}$$

Inputs

Emergence – Lambda or \(\Lambda\), is computed by the ML-Interface using outputs of the L Component

Net Infectiousness – kappa or \(\kappa\), is computed by the XY-Interface using outputs of the XH Component

Dynamics

In the following we use the subscript \(\tau\) to denote the value of a parameter, term, or variable at time \(t-\tau'(t)\): $$X_\tau = X(t-\tau'(t))$$.

The derivatives of the state variables are: $$ \begin{array}{rl} dM/dt &= \Lambda - \Omega \cdot M \\ dP/dt &= f(M-P) - \Omega \cdot P \\ dY/dt &= fq\kappa(M-Y) - \Omega \cdot Y \\ dZ/dt &= \Upsilon \cdot (fq\kappa)_\tau(M_\tau-Y_\tau) - \Omega \cdot Z \\ \end{array}$$

The function dMYdt.GeM also computes the variables: \(\Upsilon\); and an accessory variable that internalizes computation of \((fq\kappa)_\tau\).

Egg Laying

The number of egg batches laid, per patch, per day is \(\nu M\). The total of eggs laid is \(\nu M \times\) eggs_per_batch

Infectious Biting

The number of bites on humans, per patch, per day is \(fqM\), and the number of infectious bites, per patch, per day is \(fqZ\).

References

  • Macdonald G (1952) The analysis of the sporozoite rate. Tropical Diseases Bulletin 49:569-586.

  • Aron JL, May RM (1982) The population dynamics of malaria. Chapter 5 in The Population Dynamics of Infectious Diseases: Theory and Applications, Springer, Boston, MA.

  • Wu SL, et al. (2023) Spatial dynamics of malaria transmission. PLoS Computational Biology 19(6): e1010684. https://doi.org/10.1371/journal.pcbi.1010684