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Mathematics and Computational Methods for Complex Systems summative-assessment.pdf Prof. Luc BerthouzeIntroductionThis coursework is worth 100% and is due Thursday, 6 January 2022, 16:00 (butplease always check what Canvas and Sussex Direct says!). Submission is electronic, via Canvas. Submission guidelines are identical to the formative coursework,namely: (a) Analytical work can be hand-written or typeset; (b) simulation code +analysis can be presented separately or integrated into a Jupyter / Colab notebook(whatever you prefer!). Unlike the previous coursework, there is no page limit butconciseness of exposition is always best!In this coursework, you are going to engage in some proper modelling work. You aregiven a scenario along with a couple of ordinary differential equations that describethe operation of this system (in principle I could have asked you to derive them fromthe scenario description but I don’t). The questions take you through a number ofsteps by which you will gain a detailed understanding of the system / model. Althoughall questions are related, it is possible to get results even if you find yourself unableto complete some of the steps.Statement of problemWe consider a system of individuals (the nature of which is irrelevant) that can taketwo 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 individualsin state A and by [B] the number of individuals in state B. There are N individualsin total. All individuals can be assumed to be in contact with all other individuals. Attime t = 0, there are B0 individuals in state B. All other individuals are therefore instate A. Individuals in state A turn into state B at a rate β [B]
N , i.e., proportional to thenumber of their neighbours that are in state B. Individuals in state B turn into state Aat 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 withthe coursework, I am providing the ODEs in the last page of this document. Beforeconsulting that section, you are strongly encouraged to try and come up with yourown formulation.1Mathematics and Computational Methods for Complex Systems summative-assessment.pdf Prof. Luc BerthouzeQuestionsWe start with some analytical work.Analytical work1. 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-fieldequation.(2 marks)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 writeyour results in terms of the following quantity R0 = βγ . Plot the phase portrait,i.e., [B] vs [A], identifying the equilibria and their stability (following conventiondescribed in the 2nd synchronous lecture of Unit 5).(8 marks)3. Produce the bifurcation plot for this system, that is, plot the value of the equilibriaas a function of R0, with R0 taking values from 0.1 to 5.0. For this question, thevalue of N is irrelevant (provided it’s strictly positive) so use 1000 for example.This should be done using Python.(4 marks)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 stateB in time. It is rarely the case that this can be done but with this system, it ispossible. 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]. (4 marks)• 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. Thereshouldn’t be any [B] or [A]. However, it will be helpful to use B∗ (calculatedin the 2nd question) to simplify the expression. (4 marks)2Mathematics and Computational Methods for Complex Systems summative-assessment.pdf Prof. Luc Berthouze• Integrate this equation. You should be able to do this without any help, butif help is needed, you should note that this expression looks very much likethe equation we solved during a synchronous lecture in Unit 4, replacing λand I by appropriate quantities. You can then use the result to derive anequation for y(t). Please see short document summarising the derivationfrom the lecture. (4 marks)• You can now produce a fully worked out expression for [B](t) by remembering that [B] = B0 at time t = 0. (4 marks)NB: In going through this question, do make sure to consider all scenarios possible regarding the value of R0.5. Using different values of B0 (between 1 and N – briefly discuss the case B0 = 0),plot solutions of [B](t) for various values of R0 between 0.1 and 5.0 (with γ = 0.5for example). Confirm your expression for [B](t) is correct by (a) verifying thatit converges to B∗ for large times t and (b) visually confirming agreement whenintegrating the mean-field equation using Euler (use Python). What happenswhen R0 = 1? Speculate as to what this means. We will get back to this. For agiven value of R0, what happens when the value of γ changes? Provide a briefexplanation.(11 marks)Simulation workSome more could be done analytically but let’s now turn to simulations. For thispart of the work, we will employ the Gillespie algorithm, an algorithm often used tosimulate complex systems. There is something very important to understand here.The mean-field equation provided to you is derived from considering the interactionsof a very large number of individuals. The Gillespie algorithm, instead, providesdiscrete simulations of the system with few individuals by explicitly simulating everytransitions. In other words, a single run of the Gillespie algorithm represents onesample trajectory of all possible trajectories for this system. In principle, the averageof a (sufficiently) large number of Gillespie runs should converge to the mean-field.This is what you will test here. NB: Gillespie provides a mathematically rigorous andcomputationally efficient alternative to agent-based modelling. You are not asked tocompare the two approaches even if that would be an interesting thing to do (NB: butcertainly not within this assessment!).To make sure everybody can work, I provide a Python implementation of the Gillespiealgorithm for this problem. Use the code although feel free to write your own if you feel3Mathematics and Computational Methods for Complex Systems summative-assessment.pdf Prof. Luc Berthouzeso inclined – you would need to read up on the Gillespie algorithm obviously. Functiongillespie_ABA takes 5 arguments: N, B0, β, γ and Tmax (max simulation time) andproduces three outputs: T, A and B which contain the time of each transition (whenone, and only one, individual changed state, whatever that state was), the numbersof individuals in state A and in state B after that transition, respectively. NB: TheGillespie algorithm uses exponentially-distributed times to next event so each run willhave a different timeline T of events. This won’t be apparent to you if plotting withlines but will matter in what comes below.1. Explore the behaviour of the system when considering suitably chosen scenarios, i.e., focus on the limit cases (e.g., small R0, large R0 and R0 = 1; smallN, large N; small B0, large B0). For each scenario, use the code provided togenerate many realisations of the stochastic process. Plot all realisations on asingle plot. Make relevant qualitative observations.(10 marks)2. For each scenario, calculate the average (and standard deviation) of the realisations. Here, you are going to face a problem linked with the nota bene fromthe introductory paragraph. You will need to think of a solution and implementit. Superimpose the average (and error bars) to the realisations. Use a largerline width for visibility.(11 marks)3. Finally, superimpose the mean-field solution. Again, use a larger line width anddifferent colour for visibility. Describe and interpret agreement between averageof stochastic realisations and mean-field in relation to the choice of parameters.In this question, using B0 = 1 (i.e., only one individual in state B at t = 0) canhelp exacerbate the differences and help you think about what is happening.You may want to refer to your bifurcation plot.(8 marks)4. [Slightly challenging question]: Consider 100 replications for N = 1000, β =0.51, γ = 0.5 and 100 replications for N = 1000, β = 0.95, γ = 0.5. You shouldnotice a substantial difference in agreement between the mean-field and the average of the stochastic realisations depending on which scenario is considered.How could you improve agreement for the scenario with the poorest agreement.Please note: The difference in B∗ is not the quantity of interest here. Ratheryou should think about why agreement is so poor. This does not actually involveanalytical work. An excellent answer would see you implement your proposedsolution and provide evidence of improved agreement.(12 marks)4Mathematics and Computational Methods for Complex Systems summative-assessment.pdf Prof. Luc BerthouzeA bit of critical thinkingIn this last part of the coursework, you are invited to think a bit more about what youhave just done.1. So far the brief has provided no information whatsoever regarding what thestates A and B are and what the individuals are. Thinking about what is happening in this system, provide at least one example of real-world scenario towhich this model could apply. Bonus points will be given for any answer thatprovides two examples, one in which the equilibria are of interest and the otherin which the critical regime (when R0 = 1) is of interest. Either way, what is thebenefit of being able to model the system?(8 marks)2. [Very tough question]: The model provided implicitly assumes that all individuals are potentially in contact with each other. What would be a more likelyscenario? What changes would have to be made to the code of the Gillespiealgorithm in order to include such a scenario? If you are able to do this, doit. Then, speculate as to what could affect the results observed in the previousquestions. If you feel so inclined, demonstrate it experimentally. NB: Only 10marks have been given to this question. However, anyone managing it successfully would receive an extra 10 marks for the assignment (with the totalmark capped to 100 obviously).(10 marks)Total marks: 100.5Mathematics and Computational Methods for Complex Systems summative-assessment.pdf Prof. Luc BerthouzeThe ODEsThe system is fully described by the following two ordinary differential equations:˙ [B] = β [B]
N [A] − γ[B]
˙ [A] = −β [B]
N [A] + γ[B]
where β and γ are strictly positive real numbers. The first equation provides the rateof change of the number of individuals in state B and says that it is the sum of thenumber of individuals in state A that become B with a rate β [B]
N (proportional to thenumber of other individuals in state B) minus the number of individuals in state B thatturn to state A with rate γ. I leave it to you to interpret the second equation. Summingboth equations lead to 0 which is consistent with the fact that the total population[A] + [B] = N at all times. To complete the description we need the initial conditions.We will call B0 the number of individuals in state B at t = 0. The number of individualsin state A is therefore N − B0.6
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