Rumor Spread Differential Equation

Question

Given

• A closed population with an external news source
• There is a chance someone who has heard the rumor transmits it to someone who has not
• There is a chance someone who believes the rumor begins to doubt it and will stop spreading it

Let

• $P(t)$ - the number of people who believe the rumor
• $T$ - total population
• $r$ - Chance each day of any one person hearing the rumor from an external source
• $c$ - Probability of a person who knows the rumor telling a person they meet who does not know the rumor
• $s$ - The probability of any two people meeting
• $d$ - The proportion of rumor believers who come to doubt the rumor each day

Attempt

I have tried to model this with:

$$\frac{dp}{dt} = r\left[T - P(t) - \frac{P(t)}{d}\right] + csP(t)\left[T - P(t) - \frac{P(t)}{d}\right] - \frac{P(t)}{d}$$

with the idea being that $\frac{P(t)}{d}$ represents the number of people who have stopped believing the rumor so $T - P(t) - \frac{P(t)}{d}$ represents the number of people who have not heard the rumor.

Although this model seems appropriate when $t=0$, when $t$ is very large I would expect $\frac{dp}{dt}$ to be zero, as everyone has heard the rumor, but no longer believes it.

Question:

How can I improve the model so that $\frac{dp}{dt}$ is $0$ when $t$ is very large?

Answer 1

I'd rather consider a simpler model: $$\dot{P}=r(T-P)-dP+\beta(T-P)P.$$ Note that

1. The number of people who haven't heard the rumor is $T-P$. No need to subtract $P/d$ as this is already a part of $P$.
2. If $d$ is a proportion of rumor believers that change their views, than you should write $dP$, not $P/d$
3. The last term in the DE can vary depending on the assumptions about the rumor propagation. In particular, it could be $\beta\frac{(T-P)P}{T}$.

Regarding your question: did you mean $P(t)=0$ as $t\rightarrow \infty$? Otherwise the question is not clear.