Bull. Off. int. Epiz., 1981, 93 (1-2), 1-8. II. - TRAVAUX ORIGINAUX A simple vaccination model by M.E. HUGH-JONES(*) INTRODUCTION An epidemiological and economic problem in disease control programmes is whether one should vaccinate the whole population in the shortest possible time or take a more even pace vaccinating the same proportion of the population through the year. The former needs a sudden increase in temporary staff, good logistic control and organization, and when the programme is over for the year alternative work sometimes has to be found for the permanent staff. The slower method involves a small permanent staff who can maintain and monitor the use of the vaccine, organization is simpler and there is a relative absence of bureaucratic adventures. A small conceptual model was designed to explore the epidemiological aspects of the problem, using Rabies as the disease. TIERKEL (1959) has stated that for efficient local control of Rabies mass vaccination of dogs should be carried out intensively on a schedule that aims at the vaccination of all owned dogs in the shortest possible time specifically 70% of the entire dog population of the region should be vaccinated within a 2-3 week period. At the same time there must be a very active stray dog control programme. The success of this approach is well attested to in the literature (KAPLAN, GOOR and TIERKEL, 1954; TIERKEL et al., 1950; WELLS, 1954). A number of municipal programmes have succeeded (*) Department of Epidemiology & Community Health, School of Veterinary Medicine, Louisiana State University, Baton Rouge, Louisiana 70803 (U.S.A.).
2 in Latin America only when the campaign switched from a yearround vaccination programme to crash programmes to vaccinate the same percentage of dogs in the shortest possible time (E. GONZALES LORCA, Pan American Health Organization Rabies Consultant, Brazil, and O. LARGHI, PAHO Consultant Virologist, Pan American Zoonoses Center, Argentina, personal communications). MALAGA, LOPEZ NIETO, and GAMBIRAZIO (1979) have pointed out the need to carefully time the vaccination programme just before the population of 3-6 month old dogs peaks but this has been noted subsequent to TIERKEL'S work and therefore cannot but play a minor, though important part in controlling Rabies. The question remains as to why intensive programmes work while on-going year-round campaigns are less efficient and less cost effective. THE MODEL The model shown in Figure 1 was built on the following assumptions : (1) When a population is vaccinated, the proportion becoming immune is the product of the proportion vaccinated and the efficacy of the vaccine. (2) The currently immune population is made up of animals which have not died and whose immunity has persisted. Replacement animals are fully susceptible and were spontaneously generated by having a constant total population size. (3) For simplicity it was assumed that a constant proportion of immune animals become susceptible in each unit of time. In reality the proportion is initially very small but increases with time to effectively reach 100%. The effects of this misassumption will be discussed. The model works on a time unit of one month. Firstly it calculates the number of immune animals that will survive to the end of the month, and of these how many will still be immune. By having a fixed population size, immune animals that die are replaced by fully susceptible animals. Vaccination can be carried out over a number of months but the model assumes that these are always contiguous months. Non-contiguity can be modelled by adjusting the cycle time and the proportion vaccinated in each cycle, but this is clumsy for short cycles. Thus, for example, the model can examine vaccination once per year or for three contiguous months in the year. If it is a
3 FIG. 1. System diagram of conceptual vaccination model.
4 vaccination month a proportion of the susceptible population is transferred to the immune group. This proportion is determined by the efficacy of the vaccine and the fraction of the annual percentage of the population to be vaccinated in one month. The model calculates the proportion of the population which is currently immune for each month in turn. If N = population size, S =.P of immunes becoming susceptible each month, V =.P of population to be vaccinated during any campaign, C = number of contiguous months each vaccination campaign lasts, M =.P of population dying annually, E = vaccination efficacy (0 to 1.0), and I =.P of population currently immune, then for each month or unit of time, t : (1) Population alive and immune (2) Number vaccinated and becoming immune each month = I1N = It-l.(l-S). [1-(M/12)].N = [N.(l -I t )].(E.V/C) This model will «work» for most diseases but is more efficient with some than others. Logically this efficiency is a function of the loss of immunity in relation to the normal vaccination cycle. Thus if the fall-off of immunity rapidly increases after a certain period of time so that the timing of revaccination is critical, the model cannot be said to be accurate because it assumes a constant loss of immunity. In this trial of an urban Rabies control programme for vaccinating dogs, we avoided this problem by taking a Rabies vaccine with a notionally persistent immunity (50% of the immune population becomes susceptible by 36 months, or a 1.91% loss of immunity per month), and a vaccination cycle of twelve months. We also assumed that it was 100% effective in the 80% of the population vaccinated each year which suffered a 35% annual mortality (MALAGA, 1973). A copy of the model is available on request, suitable for an HP 65 programmable hand calculator. RESULTS (a) Monthly versus annual, biannual or triannual vaccination. This trial's results are shown in Figure 2. It can be readily seen that the optimum strategy is to annually vaccinate the whole population during one month. If the population is vaccinated throughout
5 FIG. 2. Levels of immunity following monthly (. ), annual ( - ), biannual (....) or triannual (. -) vaccination campaigns. FIG. 3. Levels of immunity following one ( ), two (- -) or three (. ) month vaccination campaigns during each year.
6 the year, that is the vaccinating teams are out all year round, the eventual maximum level of protection is 60%. But with one month a year vaccination the population has a higher level of protection for nine months and the level never falls to 50% population immunity. With biannual and triannual vaccination the average levels of population immunity are lower than for the year-round programme. (b) Yearly vaccination taking one, two or three months. If the yearly vaccination programme is spread out over one, two or three months, the results can be seen in Figure 3. There is a marked difference in the protection afforded the population with a one month programme compared to a two month programme, but much less difference between a two month and a three month programme; in fact a seven or eight month programme of vaccination every year provides the same average protection as a twelve month programme. DISCUSSION It seems clear therefore that for this set of parameters which describe a slow fall in immunity but a rapid population turn-over, a single, short, sharp vaccination programme, once a year, is the optimum strategy. It has the advantage of a great temporal width of immunity. Not only does one get a higher level of immunity but it appears to last for at least three incubation periods to be effective. From Figure 2, after the first year, the following applies : 80% herd immunity for 3 months; 70% herd immunity for 6 months; 60% herd immunity for 9 months. It would also appear to be the simplest strategy because any other must go for higher population coverages to attempt to do as well. Eighty percent coverage is probably the effective maximum. It must be pointed out that this is a conceptual model awaiting field verification. It is because of this that the results are shown in graphical form only as it was believed that tabular results would suggest a premature accuracy in precision. However, it is adequate to demonstrate how the crash programmes succeed. Obviously any advantages would be maximized if the vaccination programme were scheduled before the 3-6 month old puppy population peaked. * * *
SUMMARY A simple conceptual model of a vaccination programme is presented to demonstrate why a crash programme, such as is used in emergency Rabies control, succeeds while on-going programmes may not, even though the same proportion of animals are vaccinated. The crash programme provides an overall higher average level of herd immunity. It also provides a higher barrier of immune animals between programmes which would have to be passed by incubating cases for the epidemic to persist. * * * RÉSUMÉ Un modèle conceptuel simple de vaccination est présenté afin de démontrer pourquoi un programme intensif, comme ce qui est fait pour les campagnes d'urgence contre la rage, donne de bons résultats alors que les programmes étalés dans le temps peuvent échouer, bien que la proportion des animaux vaccinés soit identique. Le programme intensif confère à l'ensemble de la population un niveau moyen d'immunité plus élevé. Les animaux immuns opposent à la maladie, entre deux campagnes, une barrière plus solide; il faudrait la brèche constituée par les animaux en incubation pour que l'épidémie puisse persister. * * * REFERENCES KAPLAN (M.M.), GOOR (H.) and TIERKEL (E.S.). A Field Demonstration of Rabies Control Using Chicken-Embryo Vaccine in Dogs. Bull. Wld. Health Org., 1954, 80, 743-752. MALAGA (H.). Características de la Población Canina y Felina de Lima Metropolitana. Ministerio de Salud, Lima, Peru, 1973, pp. 36. MALAGA (H.), LOPEZ NIETO (E.) and GAMBIRAZIO (C). Canine Rabies Seasonality. Int. J. Epidem., 1979, 8 (3), 243-246.
8 TIERKEL (E.S.), GRAVES (L.M.), TUGGLE (H.G.) and WADLEY (S.L.). Effective Control of an Outbreak of Rabies in Memphis & Selby County, Tennessee. Am. J. Publ. Health, 1950, 40 (9), 1084-88. TIERKEL (E.S.). Rabies. Advances in Veterinary Sciences, 1959, 5, 183-223. WELLS (C.W.). The Control of Rabies in Malaya through Compulsory Mass Vaccination of Dogs. Bull. Wld. Health Org., 1954, 10, 731-742.