10 KIYOSHI IGUSA0.2. Kermack-McKendrick. This is from the book Epidemic Mod-elling, An Introduction, D.J. Daley & J.Gani, Cambridge UniversityPress. Kermack-McKendrick is the most common model for the gen-eral epidemic. It is usually more realistic with many subpopulationswith different characteristics. But we are only interested in the concept,not in an accurate model. So, I made two simplifying assumptions:• The population is homogeneous• No births or deaths by other meansSince there are no births, the size of the population N is constant.This model is similar to the Markov processes which we will studystarting next week: There are “states” and people move from one stateto another according to certain rules. In a Markov process, the move-ment is random. Here it is deterministic.In this model there are three states:S: = susceptibleI: = infectedR: = removed (immune)Let x = #S, y = #I, z = #R. SoN = x + y + z.I assume that z0= 0 (If there are any “removed” people at t = 0 weignore them.)As time passes, susceptible people b ecome infected and infected “re-cover” and become immune. So the size of S decreases and the size ofR increases. People move as shown by the arrows:S −→ I −→ R0.2.1. recovery rate. The infected recover at an exponential rate. Weassume that the infection has a half-life, say one week, and half of theinfected will recover in that time and 3/4 will recover in two weeks, etc.So, the number of infected tends to decrease at the rate proportional toits size. However, there are also newly infected which keep appearing.So, this recovery process only describes the flow I → R. The equationis:dzdt= γy, γ > 0.The rate of change of y is equal to the rate of infection minus the rateof recovery.MATH 56A SPRING 2008 STOCHASTIC PROCESSES 110.2.2. infection rate. The infection rate is given by the Law of massaction which says:The rate of interaction between two different subsets of the populationis proportional to the product of the number of elements in each subset.So,dxdt= −βxy, β > 0.To solve these equations, we divide them:dxdz=dx/dtdz/dt=−βxyγy=−βxγThis is a linear differential equation with solutionx = x0exp!−βγz"= x0e−z/ρwhere ρ := γ/β is called the threshold population size. This is anexponential decay equation. It says that the size of the susceptiblepopulation is decreasing at an alarming rate. Bad news!However, something happens before we all die: the number of in-fected y goes to zero and the infection stops!Since N = x + y + z is fixed we can find y as a function of z:y = N − x − z = N − x0e−z/ρ− zDifferentiating gives:dydz=x0ρe−z/ρ− 1d2ydz2= −x0ρ2e−z/ρ< 0So, the function is concave down with initial slopedydz=x0ρ− 1.I graphed these functions for different values of the parameters to showyou what this means.0.2.3. Case 1: x0> ρ. When the initial susceptible population size isgreater than the threshold ρ, the infected population increases at thebeginning. This is becausex0> ρ ⇒x0ρ> 1 ⇒dydz=x0ρ− 1 > 0.12 KIYOSHI IGUSAHowever, it eventually comes back down, although this may not beobvious. Here is the plot in the case whenN = 10, 000x0= 9, 900y0= 100ρ = 5, 000.The plot shows x, y, ρ as a function of z. (ρ is constant.)Note that there are approximately 2,000 uninfected at the end ofthe epidemic. (The infected line crosses the z axis at z = 8, 000 andx = N − z = 2, 000 is the final value.)In fact, this model predicts that there will always be survivors of anyepidemic. I.e., there will always be people who never get infected.0.2.4. case 2: x0< ρ. If the initial susceptible population size is lessthan the threshold ρ, the infected population is decreasing at the be-ginning. Since the infected curve is concave down, it decreases evenfaster as z increases. Here is the plot in the caseN = 10, 000x0= 8, 000y0= 2, 000MATH 56A SPRING 2008 STOCHASTIC PROCESSES 13ρ = 9, 000.Here, almost half of the population survives the epidemic.Another way to look at it is that this is the tail end of the epidemic.The worst is over. In case 1 we saw the beginning of the epidemic.Exercise 0.7. Prove that the highest point in the infected curve occurswhen the susceptible curve crosses the
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