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Vaccination and Herd Immunity to Infectious Diseases Roy M _NA'-"TU-'---'-'R-"E'--V-'---O'-'-L.-'3_1.c...8.c.2.c...8c...N--'.O_V.=EM'-'-"-BECCR'-'----'-l'---'98'-'--5-------REVIEW ARTICLE-------------------3~23 Vaccination and herd immunity to infectious diseases Roy M. Anderson & Robert M. May· Department of Pure and Applied Biology, Imperial College, London University, London SW7 2BB, UK * Biology Department, Princeton University, Princeton, New Jersey 08540, USA An understanding of the relationship between the transmission dynamics of infectious agents and herd immunity provides a template for the design of effective control programmes based on mass immuniz­ ation. Mathematical models of the spread and persistence of infection provide important insights into the problem of how best to protect the community against disease. MUCH attention is now focused on the development and com­ of acquired immunity and maternally derived protection, age­ munity-wide use of vaccines for the control of infectious dis­ related changes in the degree and intimacy of contacts among eases. This interest is a consequence of many factors; these people, and the prevailing levels of genetic, spatial and include the worldwide eradication of smallpox by vaccination behavioural heterogeneity in susceptibility/resistance to infec­ in the late 1970s1, the current success of child immunization tion. Intuition built on many years of practical epidemiological programmes for the control of measles, mumps, pertussis and experience often fails to predict the outcome of a particular rubella in the United States2·3, the failure of frequent appeals vaccination programme. Mathematical models which clearly by public health authorities in the United Kingdom to raise and accurately define the details of the association between vaccination rates that are at present low by developed-country disease agent and host (at the individual and population levels) 4 standards ·', the Expanded Programme on Immunization (EPI) can help to overcome such difficulties. This article reviews recent of the World Health Organisation (WHO) for the control of a theoretical and empirical work on the transmission dynamics variety of serious childhood viral and bacterial diseases in the of infectious disease. Emphasis is placed on the relevance of 8 12 developing world • , and the rapid advances in molecular bio­ such research to the design of disease control programmes based logy and biotechnology which promise new vaccines for the on vaccination. 13 16 future - • The main thrust of research today is at the molecular level, where it is stimulated by the urgent need to develop General theory vaccines for the major killing diseases of the Third World such Models for the transmission dynamics of directly transmitted 11 1 as malaria • 9, and to combat new infections such as the current viral and bacterial infections are typically compartmental in 23 35 epidemic of AIDS (acquired immune deficiency syndrome), structure · • They represent changes, with respect to age, a, caused by the recently isolated human T-lymphotropic virus and time, I, in the populations of infants protected by maternally 20 21 (HTL V-III) • • derived antibodies (M(a, t)), susceptible individuals (X(a, 1)), The development of a safe, effective and cheap vaccine, infected individuals who are not yet infectious (H(a, t)), infec­ however, is only a first step (albeit an essential one) towards tious people ( Y(a, t)), and people who have recovered and are community-wide control. Epidemiological, economic and moti­ immune from subsequent infection (Z,(a, t)). The total popula­ vational issues are at least as important as technological ones. tion is N(a, 1) = M(a, t) + X(a, t) + H(a, t) + Y(a, t) + Z(a, t). 12 A vaccine for measles (a very cost-effective immunization ) has The deterministic equations describing the flows from one class 24 25 27 been available since the late 1960s, yet the infection remains to another are of the general form • • 8 22 one of the world's major causes of child mortality • • Concomi­ aM(a, 1) aM(a, t) tant with research on vaccine development, there is a need for ---+--- -[µ(a)+d]M(a, t) an improved understanding ofhow best to use vaccines to protect at aa the community as well as the individual. The persistence of aX(a, t) aX(a, t) infectious disease within a population requires the density of ---+--- dM(a, t) susceptible individuals to exceed a critical value such that, on at aa average, each primary case of infection generates at least one -[v(a, t)+A(a, t)+ µ(a)]X(a, t) secondary case. It is therefore not necessary to vaccinate every­ aH(a, 1) aH(a, t) one within a community to eliminate infection; the level of herd ---+--- A(a, t) X(a, t) immunity must simply be sufficient to reduce the susceptible at aa fraction below the critical point. The central epidemiological -[µ(a)+a]H(a, 1) (1) questions are thus: what proportion of the population should be vaccinated to achieve elimination (in a local programme), a Y(a, t) a Y(a, t) ---+--- aH(a, t)-[µ(a)+y]Y(a, 1) eradication (in a global programme) or a defined level of con­ a1 aa trol? How is this affected by demographic factors (for example, az(a, 1) aZ(a, 1) birth rate)? What is the best age at which to immunize? How ---+---= yY(a, t)+ v(a, t) X(a, t) does mass immunization affect the age distribution of susceptible at aa individuals, particularly in those classes most at risk from serious -µ(a) Z(a, t) disease, and how important is genetic and spatial heterogeneity in suceptibility to infection (or response to immunization) to aN(a, t) aN(a, t) 23 26 ---+--- -µ(a) N(a, 1) the creation of effective herd immunity • ? a, aa To answer these questions, we need to understand the interac­ tion between the transmission dynamics of the disease agent Here µ(a) denotes age-related human mortality, 1/d is the and the level of naturally acquired (or artificially created) average duration of protection provided by maternal antibodies, immunity to infection. This relationship is complex and depends 1/ a and 1/ y are the average incubation and infectious periods on such factors as the precise course of infection within an respectively, and v( a, t) records the age- and time-specific individual, the demography of the host population, the duration vaccination schedule. Mortality explicitly associated with the © 1985 Nature Publishing Group 3_24_____________________ REVIEWARTICLE---------~N~A.c..cTUc...c..c.RE=-V-O'-'L'--.-'-3_18_2C-'8-'-N'--O'--V_E=MB~E=R'--1=9~85 Fig. 1 a, Time series analysis of the case so,ooo a resports for measles in England and Wales over the pre-vaccination period 1948-68. The top ~ 4 O, 0 0 0 e graph denotes the recorded cases and the bottom ; c 1.4 ~--------------, st 8 b graph the serial corrclogram • Note the sig- ~ 30,000 ., 1.2 nificant yearly (seasonal) and 2-yearly peaks in a. 0 incidence (the horizontal axis in the bottom "' 20,000 .; graph denotes weeks). b, Model predictions of 5 e the equilibrium (the state to which the system u 10,000 .! 0.8 .; settles in the long term) ratio of the number of 0.6 cases of mumps in the age range 14-50 yr (the ., 1953 1958 1963 1968 "' age range in which males are at risk to serious = 0.4 Year u disease resulting from infection) under the C 1.0 0.2 impact of a vaccination programme in which p .!2 o iii of each yearly cohort arc vaccinated at age 2 yr, 0.5 ~ 0.2 0.4 0.6 0.8 divided by the number of cases in the same age 0 0.0 range before vaccination. If the ratio is above u Proportion vaccinated unity, the control programme has a deterimental ~ -0.5 <( effect and vice versa. The average age at infection -1.0 Time lag <weeks) before control was assumed to be 6.5 yr e d (appropriate for the United Kingdom) and the c calculations were based on a model defined in g 2 ,----------------, 0 ref. 24. The straight line denotes the ratio of ~ 1.8 C " unity. c, Age-specific incidence of reported ~ ., ,_ 6 measles encephalitis cases ( defined per 1,000 a. ., 1 4 cases of measles) in the United States in 1973- :i ~ · 116 1 2 79 • d, Similar to b but representing the ratio ,. ~ · of cases after vaccination divided by cases before u 1. 0 O vaccination for rubella in the United Kingdom ~ ., 0. 8 in the age range 16-40 yr. The graph denotes .; ~ o.6 model predictions (equilibrium values) for a -a ~ o.• two-stage vaccination policy involving vaccina· _ g 0 2 tion of only girls at age 12 yr (the UK policy) w and vaccination of boys and girls at age 2 yr (the 0 2 10 12 14 16 18 20 25 67 US policy ) • • Note that the addition of a Age <yr) second-stage US policy over the current UK policy is only of substantial benefit if moderate to high levels of vaccination coverage are achieved at 2 yr of age. The average age at infection before control was 25 67 assumed to be 9-10 yr of age • . infection has been ignored. The term A(a, t) denotes the 'force' unity. Within communities in which the infection is endemic, 23 37 or rate of infection, which under the 'mass action' assumption ~ is inversely correlated with A, the average age at infection - 24 30 adopts the form - A(a, t) = J~f3(a, a') Y(a', t) da'. Here (~= B/(A- D)). This relation holds both for developed coun­ {3(a, a') represents the transmission coefficient for infected tries (growth rates essentially zero) and for developing countries individuals in age class a' who come into contact with suscep­ (with their growing populations)52 .
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