inst/comparative_dynamics.md

In noamross/age-infects: Package for manuscript and code of age-infection interaction paper

% Comparative Dynamics % Noam Ross % 14-08-25 10:10:34

Here I examine the comparative dynamics of age classes in 2-age-class models of both $SI$, $SIV$, and macroparasite-style dynamic models.

The following parameters are identical between models:

• fecundity $(f)$
• transition rate $(g)$
• carrying capacity $(K)$

The models are also parameterized to have identical apparent mortality rates at equilibrium. That is the aggregate mortality rates of all infection classes are the same.

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Currently matching model by equilibrium mortality. Next, match the models by initial growth behavior. That is, growth rate at time zero, or R0. How do models behave comparatively if they have initial growth rate of infected individuals that is the same? What if we peg both initial rate and final equilibrium mortality rate, but free both $\alpha$ and $\lambda$ to vary independently? How will models behave differently?

Need too calculate this rate for the given initial number of infected individuals Model may need to be modified to only have initial infected individuals be of lowest disease class, not all disease classes.

At initiation, when all individuals are either susceptible or only have one infection, the growth in infected individuals is

$$\lambda_{I_1} - d - \alpha$$

Note that the 2nd derivative of $V$ is very small no matter when $I$ is small at $V=H=0$:

$$\frac{d^2V}{dt^2} = - I^{3} \lambda^{2} + I^{2} \left(2 S \lambda^{2} - 4 \alpha \lambda - 3 d \lambda\right)$$

noamross/age-infects documentation built on May 23, 2019, 9:30 p.m.