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I am working on a differential equation model, and I am struggling to set up the appropriate flows.

Imagine a population of young individuals, $N$, subject to births, $B(N)$ and deaths, $D(N)$, and a rate of maturing to adults, $M(N)$:

\begin{equation} \begin{split} \frac{dN}{dt} & = births - deaths - maturing\\ & = B(N) - D(N) - M(N) \end{split} \end{equation}

Imagine that we are modelling the population on a weekly time step but our knowledge of the demographics comes from annual statistics.

I know that individuals give birth, on average, at an annual rate of $b_{year} = 0.1$. The annual death rate is $d_{year} = 0.15$. It also takes 5 years to reach maturity. If I was modelling the population at a yearly time step, I could just use $b_{year}$ and $d_{year}$ as is, and define $M(N) = mN = \frac{1}{5}N$.

First question: am I right in thinking that to model this population at a weekly time step, I should scale the parameters by 52 weeks? That is, if 10% give birth per year, then 10/52 $\approx 0.19\%$ give birth each week. In other words, the equation should be:

\begin{equation} \begin{split} \frac{dN}{dt} & = births - deaths - maturing\\ & = B(N) - D(N) - M(N)\\ & = \frac{b}{52}N - \frac{d}{52}N - \frac{1}{5(52)}N \end{split} \end{equation}

Second question If the death rate is $d$, should the the number of individuals maturing be weighted by the probability of surviving, i.e. should $M(N) = (1 - d)mN$, leading to something like:

\begin{equation} \begin{split} \frac{dN}{dt} & = births - deaths - maturing\\ & = B(N) - D(N) - M(N)\\ & = bN - dN - (1-d)mN\\ \end{split} \end{equation}

Or, because this model is cast in continuous-time, is this complementary probability not required?

Third question Individuals being born will not die immediately, but will die at some average point of youth. In that case, should there be some additional rate included for $D(N) = deM$, where $e$ is $1/\text{average time to death}$. For instance, individuals might die 2 years after they are born on average, meaning $D(N) = d 0.5 N$.

Thanks a lot for any help you might have clear up these issues!

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You could model it is a homogeneous system of ODEs with two equations for mature, $M$, and immature, $I$. The deaths will diverge as the system runs long enough. $$ \dot{M} = g*I-d*.05*M $$ $$ \dot{I} = b*M - g*I $$ where $b$ is the birth rate, only mature people reproduce, $g$ is the rate at which the young transition to mature, only mature people die, and $d*.05$ is the rate at which the mature die. You can easily make it more complicated if you want. These are pretty easy to solve, and there's lots of computational tools for plotting phase diagrams.

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  • $\begingroup$ Thanks. I see that there are other ways of formulating the problem, but imagine that I want to keep the same structure as my example, i.e. one differential equation with two outgoing rates. $\endgroup$
    – user_15
    Apr 2, 2020 at 15:06

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