# Help understnading rates of change in differential equation

I'm confusing myself over formulating differential equations, and the units of rates of change. I can't quite find other questions that answer this for me either, but please let me know if I have missed anything already posted.

If we have the differential equation:

$$$$\frac{dP}{dt} = \beta P - \delta P$$$$

where $$P$$ is the population size, $$\beta$$ is a birth rate and $$\delta$$ is the death rate, I understand that $$\beta$$ and $$\delta$$ must have the units $$t^{-1}$$.

Let's say $$t$$ is in units of months, and everyone in the population dies at rate $$\delta$$. Can I understand $$\delta$$ to be equal to $$1/(\text{time to die})$$. So, for humans, the birth rate would be $$1/(\text{months-per-year*average years alive}) = 1/(12*80)$$, for example?

But what if we are still in time units of months, but it only takes a couple of weeks for a member of the population to die on average? For instance, if individuals only live for 2 weeks, that's around 50% of a month, and then $$\delta = 1/\text{time to die} = 1/0.5 = 2$$, which means that the outgoing rate for deaths per month ($$\delta P$$) will be greater than the number in the population ($$2*P$$), which to me doesn't make sense: deaths can't be higher than $$P$$.

Can anyone help me out with what I am missing here?

• As another example, consider the equation for recovered individuals from the SIR equation: $dR/dt = \gamma I$. The parameter $\gamma$ is often defined as $1/\text{recovery rate}$ but what if the recovery rate is < than the time units of the model, e.g. if time is in weeks, but recovery takes 2 days, then $\gamma = 1/(2/7) = 3.5$ times the number infected. Commented May 28, 2020 at 12:56
• Thanks @ChristianBlatter. I've changed to Greek letters for the constants. Commented May 28, 2020 at 15:25

The value of $$\delta P$$ corresponds to the total number of deaths per $$1$$ month. So $$\delta$$ is the number of deaths per month for one individual.

In the example from your comment, the recovery rate indeed is equal to 1/infection time. So if the infection time is less than $$1$$, then theoretically more than 100% infected get recovered. The reason it makes sense is that we measure that only on a short period of time, obtaining the derivative.

Let $$R(t)$$ and $$I(t)$$ be the number of recovered and infected people. We need to obtain the change of $$R$$ in a short period of time. We can approximate the number of people recovered during time $$\Delta t$$ by $$\gamma I(t)\cdot\Delta t$$ (and as $$\Delta t$$ is arbitrarily small, $$\gamma\Delta t$$ will be less than 1). Then $$R(t+\Delta t) = R(t) + \gamma I(t)\cdot \Delta t,$$ and we get our equation: $$\dot{R}(t)=\lim_{\Delta t\to 0}\frac{R(t+\Delta t)-R(t)}{\Delta t}=\gamma I(t)$$.

• To take an example, look at the description of the $\gamma$ parameter here: cran.r-project.org/web/packages/shinySIR/vignettes/… It's defined in terms of a recovery rate, where $1/\gamma$ corresponds to the average infectious period. But what if the average infectious period is < than the units of time in the model, then $\gamma > 1$, and the number recovered is more than 100% of the infected. Does that make sense? Commented May 28, 2020 at 15:23
• You're right, I'm sorry. I will edit my comment. I now think that to understand why it makes sense, we need to see how the equation is derived.
– M_S
Commented May 28, 2020 at 18:39
• Thanks! That makes sense to me now. Commented May 29, 2020 at 16:06
• Do you mind if I ask a follow-up question? You say that $\delta$ is the "number of deaths per month for one individual" but $\delta$ has units 1/time. If it's the number of deaths per month for one individual, isn't the units 1/time/indiividual = 1/(time * individual)? I'm getting confused when people say units are 1/time for coefficients in DE models, but then interpret them as per-capita rates. Thanks! Commented Jun 19, 2020 at 5:36
• Deaths are also measured in units of individuals (let's say $ind$). So we have $\frac{ind}{time*ind}=1/time$.
– M_S
Commented Jun 19, 2020 at 10:46