Statement of problem
We consider a system of individuals (the nature of which is irrelevant) that can take two states A and B. At any point in time, an individual can only be in one state. There are no other states possible. We will denote by [A] the number of individuals in state A and by [B] the number of individuals in state B. There are N individuals in total. All individuals can be assumed to be in contact with all other individuals. At time t = 0, there are B0 individuals in state B. All other individuals are therefore in state A. Individuals in state A turn into state B at a rate β [B] N , i.e., proportional to the number of their neighbours that are in state B. Individuals in state B turn into state A at a fixed rate γ. Now, in an ideal world, you should be able to write the two ordinary differential equations (ODE) that describe this system. To make sure that everyone can proceed with the coursework, I am providing the ODEs in the last page of this document. Before consulting that section, you are strongly encouraged to try and come up with your own formulation.
Questions
We start with some analytical work.
Analytical work
1. Using the fact that [A] + [B] = N at all times, write ˙ [B] as a function of [B], i.e., the expression should no longer involve [A]. This is your (so-called) mean-field equation.
2. Find the equilibria of the system and determine their stability. From now on, we will refer to the non-zero equilibrium as B∗. You may find it useful to write your results in terms of the following quantity R0 = β γ . Plot the phase portrait, i.e., [B] vs [A], identifying the equilibria and their stability (following convention described in the 2nd synchronous lecture of Unit 5).
3. Produce the bifurcation plot for this system, that is, plot the value of the equilibria as a function of R0, with R0 taking values from 0.1 to 5.0. For this question, the value of N is irrelevant (provided it’s strictly positive) so use 1000 for example. This should be done using Python.
4. In this question, you are going to integrate ˙ [B] analytically to obtain an expression for [B](t), i.e., an expression that gives the number of individuals in state B in time. It is rarely the case that this can be done but with this system, it is possible. You will do this in four steps:
• Starting from the mean-field equation, factorise the right hand side by [B] 2, then write an expression for 1 [B]2 ˙ [B].
• Consider the following variable substitution: y = 1 [B] . Using the chain rule, express y˙ in terms of [B], then derive a simple expression for y˙, i.e., this expression should only involve y terms and parameters of the system. There shouldn’t be any [B] or [A]. However, it will be helpful to use B∗ (calculated in the 2nd question) to simplify the expression.
• Integrate this equation. You should be able to do this without any help, but if help is needed, you should note that this expression looks very much like the equation we solved during a synchronous lecture in Unit 4, replacing λ and I by appropriate quantities. You can then use the result to derive an equation for y(t). Please see short document summarising the derivation from the lecture.
• You can now produce a fully worked out expression for [B](t) by remembering that [B] = B0 at time t = 0.
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