The self-study problem from the last lecture showed that the basic SIR model can be simplified to: du dt dv . = -Rouv (Rou-1)v = V where u is the proportion

of hosts that are susceptible • v is the proportion of hosts that are infected (and infectious) w is the proportion of hosts that are recovered • Ro=BN is the basic reproduction number 3 is the ("pairwise") transmission rate is the recovery rate Why do we not need the third equation? Basic reproduction number, Ro The basic reproduction number ¹, Ro, is the most important quantity in mathematical epidemiology. The basic reproduction number tells you the average number of secondary infections produced by one infected individual in an otherwise susceptible population. Simply put, it is the ratio of the rate at which new infections are produced by each infected host when the entire population is susceptible (3N), to the rate at which infections are lost (7). ¹Note that Ro is not a rate. It is dimensionless. Sometimes it is referred to as the basic reproduction ratio, but should never be referred to as the basic reproduction rate!/nreproduction ratio, but should never be referred to as the basic reproduction rate! Algebraic analysis Find and determine the stability of any equilibria. What do you notice? Algebraic analysis Let's tackle this another way. Suppose initially that almost everyone is susceptible (u(0) ≈ 1). When can an epidemic take off (the epidemic threshold)? 5/nGeometric analysis What would a phase plane look like for this system? Note that u + v ≤ 1, so we have to solve the system on a triangle. Q: What are the nullclines and direction field? Geometric analysis Let's sketch two phase planes to understand the dynamics of the model. 7/nNote that So the host population size (density) must be greater than N, for an epidemic to occur. Q: How do the transmission rate, population size, and recovery rate affect whether an epidemic will occur or not? Final size of an epidemic How many people will get infected during an epidemic? Will everyone become infected? For this SIR model, any point on v = 0 is a disease-free equilibrium (DFE) and the epidemic eventually burns out, leaving the population in one of these states. The final size of an epidemic is the total number of individuals who become infected over the course of the epidemic. If all individuals are initially susceptible, it is given by the number of individuals eventually in the recovered class. 10/nFinal size of an epidemic Recall that Then Idea: If we solve this equation for w as a function of u with initial condition w = wo =0 when u = Uo = 1 then we find a relationship between w and u that is valid everywhere on the solution trajectory that starts arbitrarily close to the DFE uo = (1,0,0). Self-study problem 1) Integrated from (uo, Wo) = (1,0) to a general point (u₁, W₁) and rearrange to find equations for w₁ in terms of u₁ and for u₁ in terms of W₁. 2) Using the fact that the epidemic eventually burns out ( v→0 as t→∞), and your equation for u above to shown that the final size of the epidemic, w₁, satisfies 1- Wy=e-Rowy 3) Sketch 1- w and e-Row for Ro <1 and Ro> 1 to show that at most there is a single root (solution) to the equation above, giving the final size of an epidemic. 11