A model of an infectious disease in a plant population is given by

$$\begin{aligned} \dot{S} &=(S+I)-(S+I) S-\beta I S \\ \dot{I} &=-(S+I) I+\beta I S \end{aligned}$$

where $S(t)$ is the density of healthy plants and $I(t)$ is the density of diseased plants at time $t$ and $\beta$ is a positive constant.

(a) Give an interpretation of what each of the terms in equations (1) and (2) represents in terms of the dynamics of the plants. What does the coefficient $\beta$ represent? What can you deduce from the equations about the effect of the disease on the plants?

(b) By finding all fixed points for $S \geqslant 0$ and $I \geqslant 0$ and analysing their stability, explain what will happen to a healthy plant population if the disease is introduced. Sketch the phase diagram, treating the cases $\beta<1$ and $\beta>1$ separately.

(c) Define new variables $N(t)$ for the total plant population density and $\theta(t)$ for the proportion of the population that is diseased. Starting from equations (1) and (2) above, derive equations for $\dot{N}$ and $\dot{\theta}$ purely in terms of $N, \theta$ and $\beta$. Without carrying out a full fixed point analysis, explain how this system can be used directly to show the same results you had in part (b). [Hint: start by considering the dynamics of $N(t)$ alone.]

(d) Suppose now that in an attempt to control disease, plants are culled at a rate $k$ per capita, independently of whether the plants are healthy or diseased. Write down the modified versions of equations (1) and (2). Use these to build updated equations for $\dot{N}$ and $\dot{\theta}$. Without carrying out a detailed fixed point analysis, what can you deduce about the effect of culling? Give the range of $k$ for which culling can effectively control the disease.