ANALYSING DATA ON MEASLES IN ITALY WITH THE HELP OF DETERMINISTIC MODELS

1. ANALYSING DATA ON MEASLES IN ITALY WITH THE HELP OF DETERMINISTIC MODELS • A. Lunelli(a), A. Pugliese(a), P. Manfredi(b), E. M. Cleur(b) • Department of Mathematics, University of Trento • Department of Statistics and Mathematics Applied to Economics, University of Pisa Workshop on Computational Life Sciences, October 13-15 2005, Innsbruck

2. MEASLES DYNAMICS • childhood disease that can lead to complications and death • caused by a highly infective virus belonging to the morbillivirus group • incubation period of 6-9 days and infective period of 6-7 days • long-life immunity or death of the host In developed countries death occurs very rarely, but measles still causes about 1 million deaths worldwide each year, especially in developing countries because of attracted secondary infections.

3. Relatively violent epidemics generally at biennial or triennial frequency decrease in number of infected rapid depletion of susceptible slow replenishment of susceptibles MEASLES DYNAMICS - CONTINUED Especially good data sets exist for UK where measles showed a clear biennial pattern.

5. THE ITALIAN DATA We base our analysis on pre-vaccination measles notification and demographic data from Italy, considering the provincial, regional and national scale. Lombardia Trentino Alto Adige All available studies of the pre-vaccination epidemiology of measles in Italy have put forth the absence of clear patterns of measles oscillations

6. THE ITALIAN DATA: SPECTRAL ANALYSIS Provincial scale Regional scale National scale • For essentially all Italian regions: a sharp peak at the annual frequency and a second peak at a lower frequency • In the province of Trento: nice symptoms of better defined oscillation patterns and a much better defined long term oscillation around three years.

7. Mass-action law THE SEIR MODEL The simplest way to model measles dynamics is to use the SEIR model Bt new born at time t ßtransmission rate µdeath rate νrate at which individuals become infective γrecovery rate Ssusceptible E exposed I infective R recovered The choice of using a discrete-time model to investigate measles dynamics is motivated by the fact that the distributions of latent and infectious periods are closer to a constant than to an exponential distribution and from the discrete nature of data available.

8. CONSTANT BIRTH AND TRANSMISSION RATES Analytical results We assume equal birth and death rates: Bt = µNt where Nt = St + Et + It + Rt Equilibria: disease free equilibrium: endemic equilibrium (if R0>1): Basic reproduction rate: The disease free equilibrium is stable if R0<1 and unstable otherwise. For reasonable values of the parameters, the model exhibits damped oscillationswith inter-epidemic period

9. CONSTANT BIRTH AND TRANSMISSION RATES Numerical results The model exhibits damped oscillationswith inter-epidemic period Large R0 or high birth rate favour annual or biennial dynamics, whereas smaller R0 and smaller birth rates lead to long-period oscillations.

10. ßs during school terms ßh during holidays ßt = ß1=0.1 annual dynamics ß1=0.35 triennial dynamics ß1=0.45 annual dynamics SEASONALLY FORCED MODEL WITH CONSTANT BIRTH RATE Numerical results A crucially important process in the dynamics of measles is the seasonal variation in the transmission rate, induced by the annual pattern of aggregation of children in schools

11. SEASONALLY FORCED MODEL WITH CONSTANT BIRTH RATE Numerical results Small differences between terms and holidays lead to annual dynamics; as the amplitude of the seasonal variation increases, the solution may pass to a multi-annual period cycle, tending to a transition to chaotic fluctuations. The constant birth rate also act as a bifurcation parameter: for values typical of the developed countries we can have either annual or multi-annual dynamics.

12. VARIABLE BIRTH RATES During the pre-vaccination era, birth rates in the areas considered are subject to a time variation (the “baby-boom” of the 60s. We simulate measles dynamics using the forced SEIR model previously analyzed together with a periodic birth rate with a realistic shape. Province of Trento Lombardia

13. ßs during school terms ßh during holidays ßt = SEASONALLY FORCED MODEL WITH TIME-VARYING PERIODIC BIRTH RATE Numerical results Bt = f(t) Depending on the function considered, different dynamics arise: annual dynamics with little variation in the amplitude of the peaks of the infectives (scenarios on the left), or transition from annual to multi-annual cycles (above).

14. ß1=0.1 ß1=0.35 ß1=0.45 SEASONALLY FORCED MODEL WITH TIME-VARYING PERIODIC BIRTH RATE - Numerical results

15. THE ITALIAN DATA We base our analysis on pre-vaccination measles notification and demographic data from Italy, considering the provincial, regional and national scale. Lombardia Trentino Alto Adige All available studies of the pre-vaccination epidemiology of measles in Italy have put forth the absence of clear patterns of measles oscillations

16. A DISCRETE TIME MODEL WITH A SEIR STRUCTURE Passive immunity to measles is passed from recovered mothers to infants and lasts between 3 and 9 months • Time scale: 1 week • Latent period: 10 days • Infectious period: 1 week • Alldeaths are assumed to occur among immune individuals • Unknown parameters: • transmission rate • initial conditions • reporting rate Expected number of cases in the next time step: we can assume that half of the individuals in the latent period will become infectious after one week and half of them after two weeks

17. MODEL FIT To estimate the unknown parameters and initial conditions we minimize the error between the predicted (Ct) and the observed (Ot) time series of cases, where αis the under-reporting rate. Correspondingly, we use, as a measure of goodness of fit: We have decided to divide the period considered into shorter subperiods, because the method yields worst results if applied to long time series influenced by dynamic stochasticity.

18. Notifications Model predictions RESULTS – PROVINCIAL SCALE Province of Trento Province of Milano

19. RESULTS – NATIONAL SCALE Notifications Model predictions

20. THE ITALIAN DATA: SPECTRAL ANALYSIS Provincial scale Regional scale National scale • For essentially all Italian regions: a sharp peak at the annual frequency and a second peak at a lower frequency • In the province of Trento: nice symptoms of better defined oscillation patterns and a much better defined long term oscillation around three years.

21. PREDICTION FROM SEASONAL SEIR MODEL WITH VARYING BIRTH RATE During the pre-vaccination era, birth rates in the areas considered are subject to a time variation (the “baby-boom” of the 60s); choosing a function that represents this variation and the parameters that we have estimated we can simulate measles dynamics using the forced SEIR model previously analyzed to see what kind of dynamics are to be expected. Province of Trento Lombardia

22. PREDICTION FROM SEASONAL SEIR MODEL WITH VARYING BIRTH RATE - CONTINUED Province of Trento Province of Milano

23. PREDICTION FROM SEASONAL SEIR MODEL WITH VARYING BIRTH RATE - CONTINUED Lombardia Trentino Alto Adige

24. PREDICTION FROM SEASONAL SEIR MODEL WITH VARYING BIRTH RATE - CONTINUED Italy

25. CONCLUSIONS • the fit of the model to the observed data is reasonably good, considering the simplicity of the model and the very low reporting rate • the estimates of the basic reproductive rate are generally within an interval of reasonable values for measles • the estimates of the reporting rate are similar to other independent estimates • the observed patterns seem to agree with the predictions of the seasonal SEIR model with varying birth rates • The SEIR model seems to be useful in understanding pre-vaccination measles dynamics in Italy. Hence, it could constitute a good basis on which further analysis can be developed; for example, it could be helpful in planning current interventions to investigate post-vaccination dynamics.