Consider a population of constant size $N + 1$ that is suffering from an infectious disease. We can model that spread of the disease as Markov process. Let $X(t)$ be the number of healthy individuals at time t and suppose that $X(0) = N$. We assume that if $X(t)$ $$\lim_{h\rightarrow 0}\frac{1}{h}\mathbb P [X (t + h) = n - 1|X (t) = n] = \lambda n (N + 1 - n)$$ For $0 \le s \le 1, 0 \le t$, define $$G(s,t):=\mathbb E(s^{X(t)})$$ Please find a non-trivial partial differential equation for $G(s,t)$, which involves $\partial_t G$.
It seem that it is difficult to find the distribution function of $X$ (otherwise the implicit form of $G(s,t)$ can be write out), so I think a good way is to find a partial differential equation involving distribution function, but the expression $\lim_{h\rightarrow 0}\frac{1}{h}\mathbb P [X (t + h) = n - 1|X (t) = n] = \lambda n (N + 1 - n)$ is hard to me. Does anyone has a hint or a thought for this expression?