1. A drug is available in doses of Xmg, which raise the blood plasma concentration of an average adult by 5mg per liter. Once in the blood plasma, the drug concentration decays according to the equation c=4c where time is taken in hours. (i) Health regulations require that the concentration never exceed 20mg per liter. What is the minimum concentration for t>0 a person experiences if doses are given at certain time interval t0 ? (ii) Forgetting specific constants, in general, what will be the minimum concentration of a drug that decays according to the equation (1), and is given in doses that raise plasma concentration to c0 at regular intervals to ensure that its concentration never exceeds cM? [10] 2. Suppose a population of cells grows logistically according to equation p=0.1pap2 time is taken in hours. (i) What does the rate 0.1h1 mean? How fast is this rate? Find a suitable factor an initial concentration increases by every specific interval of time. (ii) If the initial population is 10% of carrying capacity, how long will it take for the population to reach 90% of the carrying capacity? (iii) What are the major unrealistic assumptions of this model? (iv) Find equilibria of the system (2) and analyse their stability. Draw phase diagram. (v) Find the linearization of equation (2) near each equilibrium point. [10] 5. This question requires Matlab/Octave codes for both parts. (i) Consider the following classical epidemics model: S=TinfRfISE=TinfRtISTinc1EI=Tinc1ETinf1IR=Tinf1I whereas S is the number of susceptibles, E is the number of exposed, R is the number of removed, I is the number of infected. Rt is the reproduction number, Tinc is the length of incubation period in days, Tinf is the duration a patient is infectious. Starting with Rt=0.77,Tinc=5.2,Tinf=7 simulate this model in MATLAB/OCTAVE with initial number of infected people being one. Make sure the number of infected doesn't go above 10% of the population at all times by changing the parameters. (ii) Generate or replicate a model for phage and bacteria interaction and present your code together with the figure showing phages fighting bacteria. List the assumptions, variables and parameters of the model. 6. Consider a model of the glycolitic oscillations: {x=1xy,y=y(x1+ry1+r) Find the steady state, find the characteristic equation of the linearised system near this steady state and state which parameter value(s) there is only one real eigenvalue when =1. What kind of stability of the steady state does this eigenvalue show