1. How does mass immunisation affect disease incidence? Niels G Becker (with help from Peter Caley ) National Centre for Epidemiology and Population Health Australian National University A valuable feature of mathematical models that describe the transmission of an infectious disease is their ability to anticipate the likely consequences of interventions, such as introducing mass vaccination. This feature is illustrated in these tutorial-style lectures, by address some specific questions in simplified settings.

2. Specific Questions 1.How well does immunisation control epidemics? 2.How well does immunisation control endemic transmission? 3.Is it always a good thing to promote vaccination? 4.What is a good strategy to protect a vulnerable group?

3. Question 1 How well does immunisation control epidemics? More specifically: Suppose the infection is absent and everyone is susceptible. (a) What happens when the infection is imported? (b) How is this changed when part of the community has been immunised prior to the importation?

4. Assume that the community size (n) is constant over the duration of the epidemic, and that n is large. Suppose the infection is transmitted primarily by person-to- person contacts. For example, measles or a respiratory disease. Suppose that all n members of community are initially susceptible to this infection. At time t = 0, one recently infected individual arrives. 1. What happens? 2. How is ‘what happens’ altered if a fraction of community members is totally immune? The setting

5. infection Susceptible S t Infectious I t Removed (immune) The solution (found numerically) depends on the initial values I 0 and S 0, and on the values of the parameters , the transmission rate, and , the recovery rate. Assume individuals are homogeneous and mix uniformly Rate of change in susceptibles = Rate of change in infectives = The deterministic SIR epidemic model for this process is

6. Setting 1 Suppose I 0 = 1 n =S 0 = 1000  = 0.3  = 0.1 What happens ? The model predicts that the outbreak takes off and the epidemic is described by Total number infected = area under curve / 10 ≈ 941 Question: Could something else happen?

7. Setting 2 Suppose I 0 = 1 n =S 0 = 1000  = 0.3/4  = 0.1 What happens ? The model predicts that the outbreak peters out and is described by Total number infected = area under curve / 10 ≈ 4 Question: What really happens?

8. This poses the questions: (a) What determines whether an outbreak takes off? (b) How large will the outbreak be? We have seen these two types of outcomes:

9. Setting 1: (the outbreak takes off) Setting 2: (it does NOT take off) I t increases initially when I t always decreases when Setting 1: (the outbreak takes off) Setting 2: (it does NOT take off) I t increases initially when I t always decreases when determines whether the outbreak takes off. (actually, there’s an element of chance) (a) What determines whether an outcome takes off?

10. If R 0 < 1 there can not be an epidemic. No intervention is required. If R 0 > 1an epidemic occurs. (Can occur) is the basic reproduction number. It is (rate at which the infective transmits) × (mean duration of the infectious period), R 0 = mean number of individuals a person infects during their infectious period when everyone they meet is susceptible, and there is no intervention. The word basic is used when everyone else is susceptible and no intervention is in place. ASIDE:

11. If R 0 > 1 an epidemic is prevented when R 0 S 0 /n <1. That is, when the susceptible fraction has been reduced to less than 1/R 0, by immunisation. (b)How large will the outbreak be ? Let s 0 = S 0 /n = fraction initially susceptible C ∞ = eventual number of cases c ∞ = C ∞ /S 0 = fraction of initial susceptibles eventually infected Then

12. Heuristic derivation

13. What happens if some community members are immunised? The initial reproduction number is Illustrate this for Setting 1 I 0 = 1, n = 1000,  = 0.3,  = 0.1 Proportion of infections among susceptible individuals )1(ln 0 0     cR c s Question: What really happens?

14. Question 2 How well does immunisation control endemic transmission? More specifically: Suppose transmission is endemic in the community. (a) What does this mean? (b) How is endemic transmission changed when the community is partially immunised? (c) What happens to endemic transmission in response to a pulse of mass vaccination?

15. immunisation infection Susceptible S t Infectious I t Recovered (immune) Death Birth An infection is endemic in the community when transmission persists. It requires replenishment of susceptibles. This happens by births, so we add births and deaths.

16. Assume no immunisation The solution depends substantially on I 0 and S 0, but eventually settles down to steady state endemic transmission. We determine this state by solving the equations This gives

17. Numerical illustration n = 1,000,000 R 0 = 15 (e.g. measles) = 1/(70*365) (life expectancy of 70 years)  = 1/7 (mean infectious period of 1 week) s E = 1/15, that is 1,000,000/15 = 66,667 susceptibles i E ≈ [7/(70*365)]*(1  1/15), that is 256 infectives In practice the numbers fluctuate around those values, because of chance fluctuations and seasonal waves driven by seasonal changes in the transmission rate. Question: Would imported infections change this?

18. What if we immunise a fraction of the newly born infants? Eventually Transmission can not be sustained when (1 – v)R 0 ≤ 1 The infection is eliminated when the immunity coverage exceeds 1 – 1/R 0. [Or s ≤ 1/R 0.] Question: Why is s E not affected by the immunisation, (as long as v ≤ 1 – 1/R 0 )? Question: What happens when i E is small?

19. Response to enhanced vaccination Suppose we have endemic transmission (without immunisation) and have a mass vaccination day. That is, we immunise a fraction v of susceptibles at t = 0. So transmission declines immediately. How much? And what happens then? Consider the earlier example: n = 1,000,000 R 0 = 15 = 1/(70*365)  = 1/7

20. Here’s what happens if we immunise 1%, namely 667, of the susceptibles: Question: What really happens?

21. Here’s what happens if we immunise 5%, namely 3333, of the susceptibles: Question: What really happens?

22. Two of our Specific Questions remain, namely 3.Is it always a good thing to promote vaccination? 4.What is a good strategy to protect a vulnerable group? We will look at these questions with regard to one simple model, which we now introduce. We choose a situation with two types of individual. One type is more vulnerable to illness, while the other type contributes more to the transmission. Practical examples include (a) rubella, and (b) influenza.

23. First the demography Partition age into ‘young’ and ‘old’. People are ‘young’ when they are aged less than c years. The mortality rate is negligible for the ‘young’, and  for the‘old’. The total community size is specified by  and N 0 are estimated from demographic data

24. Suppose c = 50*365 = 18250 days and  = 1 / (10*365). Then the life expectancy is 50+10 = 60 years. The age distribution is Age With N 0 = 100 the community size is 6000.

25. Next the transmission model Consider an SIR model in which the transmission rate is age- dependent We consider only the steady state of transmission. The steady state force of infection acting on the ‘young’ is, and that acting on the ‘old’ is ’. Estimate and ’ from age-specific surveillance data (perhaps using incidence data for a period before immunisation). , the recovery rate, is estimated from disease-specific data Suppose that = 0.0001, ’ = 0.00002,  = 0.1

26. The steady state transmission equations are For a in [0, c) For a in [c, ∞) The solution can be found analytically or numerically.

27. The solution is as follows: For a in [0, c) For a in [c, ∞)

28. Proportion infectious at different ages – no vaccination Age (years) Proportion infected 01020304050607080 0.0000 0.0002 0.0004 0.0006 0.0008 0.001

29. Age (years) Proportion susceptible 01020304050607080 0.0 0.2 0.4 0.6 0.8 1.0 Proportion susceptible at different ages – no vaccination

30. and ’, the forces of infection, change when we change the vaccination coverage. In contrast, the rates of making close contacts do not change, so it is useful to determine the corresponding transmission rates. The forces of infection and transmission rates are related by Can not determine 4 parameters from 2 equations

31. Assume proportionate mixing; i.e. the WAIFW matrix is We find and

32. Question 3 Is it always a good thing to promote vaccination? More specifically: Consider a disease with more serious consequences for older people, but young people transmit more infection. Practical examples include (a) rubella, and (b) influenza. Any type of immunisation reduces the overall incidence, but some strategies may actually increase the incidence among older people, and so increase their risk.

33. Our parameter values are c = 50*365 days,  = 1 / (10*365),  = 0.1,  = 0.51,  ’ = 0.10 Now suppose that a fraction v of individuals are vaccinated, essentially at birth. Then S 0 is reduced from N 0 to (1-v)N 0. We first need to find the new expressions for S a and I a from the steady state transmission equations. Then substitute these in and solve for the new and ’.

34. Vaccination coverage Ratio of cases for age>C 0.00.20.40.60.81.0 0.0 0.5 1.0 1.5 Vaccination at birth Graph of cases aged over 50 at v relative to v = 0.

35. Question 4 What is a good strategy to protect a vulnerable group? More specifically: As above, consider a disease which is more serious for older people and young people transmit more infection. To protect the old people, is it better (i) to vaccinate the young, or (ii) to vaccinate the older people? Practical examples again include (a) rubella, and (b) influenza.

36. Our parameter values are again c = 50*365 days,  = 1 / (10*365),  = 0.1,  = 0.51,  ’ = 0.10 The above strategy vaccinated a fraction v of individuals at birth. For comparison, consider a strategy which, instead, vaccinates a fraction v of individuals as they reach the age of c years. Then S c+ is reduced from S c  to (1-v) S c . With this change we find the new expressions for S a and I a from the steady state transmission equations and substitute these in to solve for the new and ’.

37. Vaccination coverage Ratio of cases for age>C 0.00.20.40.60.81.0 0.0 0.5 1.0 1.5 Vaccination at birth Vaccination at C=50 years Graph of cases aged over 50 at v relative to v = 0.

38. Limitations of these deterministic SIR model I t and S t are taken as continuous when they are really integers. (Of concern when I t or S t are small) They suggest that an outbreak always takes off when R 0 s 0 > 1. (Not always the case.) They ignore the chance element in transmission. (Of particular concern when I t or S t are small, e.g. during early stages) The End