In noamross/spore: Equation-free modeling of spore-driven disease dynamics

The Parasite Exclusion Problem

(Current-time) Hamiltonian:

$$\begin{align} \mathcal{H} = &vN - ch + \ &\nu_1 \left( rN(1 - N/K) - \alpha P - d N \right) + \ &\nu_2 \left(\lambda P N - \mu P - d P - \alpha P - \alpha (P^2)/N + e^{-h} N \lambda_{ex} \right) \end{align}$$

Profit function (continuous provision of ecosystem pervice per individual host minus control cost):

$$vN - ch$$

System dynamics ($N$ = hosts, $P$ = parasites):

$$\begin{align} \frac{dN}{dt} &= r N (1 - N/K) - \alpha P - d N \ \frac{dP}{dt} &= \lambda P N - \mu P - d P - \alpha P - \alpha P^2/N + e^{-h} N \lambda_{ex} \end{align}$$

Shadow price dynamics (via adjoint principle):

$$\begin{align} \frac{d\nu_1}{dt} &= -v - \nu_1 \left(r - d - 2r \frac{N}{K}\right) - \nu_2 \left(\lambda P + \alpha \frac{P^2}{N^2} + e^{-h} \lambda_ex \right) + \delta \nu_1 \ \frac{d\nu_2}{dt} &= \nu_1 \alpha - \nu_2 \left(\lambda N - \mu - d - \alpha - \frac{2 \alpha P}{N} \right) + \delta \nu_2 \end{align}$$

Control (via maximum principle):

$$ h \frac{d\mathcal H}{dh} = 0 = h \left( -c - \nu_2 e^{-h} N * \lambda_{ex} \right)$$

$$h = \left[0, \, \log{\frac{-\nu_2 * N * \lambda_{ex}}{c}}\right]$$

noamross/spore documentation built on May 23, 2019, 9:31 p.m.