I am trying to predict the spread of a type of infectious disease demonstrated by the following problem. Suppose we have 1000 people and each infected person infects 10% of uninfected people per hour. Initially, 100 people are infected. Suppose further that a health agency "disinfects" 120 of the infected people per hour and I need to know how long it will take to eradicate the disease. Note that there is no mention of people being immunised and so people can get re-infected again in a very short space of time. That is, this is not a classic SIR problem with a system of two differential equations.
If I had to use a model that reflects this situation, I would say:
$\frac{dx}{dt} = kxy – P$
… where x(t) is the ammount of infected people as a function of time, and y(t) is the ammount of uninfected people as a function of time, k is the infection rate and P is the fixed ammount of people that are disinfected (not immunized) per hour.
If I had to simplify and substitute the known's into the model, I would get:
$\frac{dx}{dt} = kx(N – x) – P$
$\frac{dx}{dt} = 0.1x(1000 – x) – 120$
Now, the problem I am encountering is that when I try and solve this system and predict how long itll take to eradicate the disease I get a negative number. I have quadripple checked my solutions and still end up with a negative number.
My question fundamentally is: Is the negative number derived due to my very first steps, that is, the models inherent inability to deal with such situation (i.e. I used the wrong model), or, the answers provided by the model should be reasonable and should yield a positive number, in which case, there must be an error in my calculations. Intuitively speaking, I'm expecting a positive number with the lowest bound of slightly under an hour.
Best Answer
The problem is that the disease will never be eradicated. You will approach a stable equilibrium solution at approximately $x = 998.8$ (one of the two solutions of the quadratic $0.1 x (1000 - x) - 120 = 0$). The other solution of the quadratic is an unstable equilibrium, approximately $x = 1.2$. In order for the disease to be eradicated you must start with $x(0)$ less than that value.
To help see why this is reasonable (given your model), consider what is happening when $x = 100$. People are being infected at a rate of $0.1 \cdot 100 \cdot 900 = 9000$ per hour and disinfected at a rate of $120$ per hour, so $x$ is increasing at $8880$ per hour.