A non-fatal infectious disease divides a population into two groups: normal or ill. Assume that the average number of contacts that each ill individual has with normal individuals is $a$ multiply the size of the normal population (i.e. $aN$), and for each contact the probability of transmission is $k$. Let the average duration of the disease be $\frac{1}{c}$. Assume that the birth rate is $b$ for both normal and ill populations, and that all babies are born normal. The death rate (independent with disease) of normal group is $d_{n}$, and of ill group is $d_{i}$.
- Build differential equations describing the size of each group.
- Assuming the total population is constant, rewrite the differential equations to obtain an unique differential equation.
- Find the equilibria and determine if they are stable or not.
What I have done:
$$ \ \frac{dN}{dt}=b(I+N)+cI-d_{n}N-aNIk$$ $$\frac{dI}{dt}=aNIk-cI-d_{i}I-bI$$ $I,N$ are the size of normal and ill
Since the population is constant, we have $b(I+N)=d_{n}N+d_{i}I$, then we get $I=\frac{d_{n}-b}{b-d_{i}}N$.
Am I right? And I have no idea what "rewrite the differential equations to obtain an unique differential equation" means, and how can I get the unique equation?
Can anyone help me with this problem? Thanks very much!
There is another way to express the idea of a constant population.
$$I+N=P_{total} \implies \frac{d}{dt}(I+N)=0$$
Using this, you can deduce $\frac{dI}{dt}=-\frac{dN}{dt}$ and $I=P_{total}-N$. Thus, you can rewrite $I$ in terms of $N$ (or $N$ in terms of $I$).