1. 1 The epidemic in a closed population Department of Mathematical Sciences The University of Liverpool U.K. Roger G. Bowers

2. 2 Plan Basic ideas The questions Tools and expected answers The main task Transmission R 0 z  i(t) and r s(  ) Conclusion

3. 3 Basic Ideas Timescales: closed population Microparasites: compartmental models Infections lead to death or immunity (removal) The main questions relate to what follows a (small) invasion by infectious hosts into an unexposed population.

4. 4 1) Can such an invasion cause an epidemic (ignoring stochastic extinctions)? 2) If so, how is this reflected in the probability of such an extinction. 3) If so, how is this reflected in the incidence (rate of appearance of new cases) particularly in the initial stages of the epidemic? 4) If so, how is this reflected in the proportion of the population which will ultimately have experienced infection? The Questions

5. 5 A) R 0 : The basic reproduction ratio - the number of secondary cases per primary in an unexposed population Answer to 1: Can cause an epidemic  R 0 >1 B) z  : The probability that the epidemic will eventually go extinct Answer to 2: Can cause an epidemic  z  <1 C) i(t): The incidence. Assume that initially Tools and expected answers Answer to 3: Can cause an epidemic  r>0 D) s(  ): The proportion of the original population remaining susceptible. Answer to 4: Can cause an epidemic  s(  )<1

6. 6 Main task This is to show how to calculate these quantities and establish that:

7. 7 Transmission Assumed to have three components: i) contact - a given individual contacts others at a rate c ii) mixing - this determines the proportion of contacts made by an infective that are with a susceptible iii) infectivity - the probability that a contact between an infective and a susceptible does result in transmission. The quantity is the probability that an individual infected for time  will infect a susceptible that it contacts

8. 8 The basic reproduction ratio R 0 Examples of infectivities We take R 0 >1 to imply a deterministic epidemic almost ‘by default’

9. 9 The probability of epidemic extinction z  q k : the probability of k secondaries per primary Generating function: Properties … particularly g´(1)=R 0 and the dynamics z n+1 =g(z n ), where z n is the probability of extinction at generation n Graphs indicating proof that R 0 1  z  <1

10. 10 i(t) and r Current new cases depend on past new cases and their present infectivity Illustration from ODE model? With the assumption Properties of f … particularly f(0)=1 Graphs indicating proof that solution for r gives R 0 >1  precisely one root r>0 Problem of actually finding r in examples.

11. 11 The fraction of susceptibles never infected s(  ) Much as for r but for a variable number of susceptibles: Graphs indicating proof that s(  ) 1

12. 12 Conclusion We have introduced key characterisations of epidemic behaviour and shown