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Simulation model of rubella––the effects of vaccination strategies
Applied Mathematics and Computation 153 (2004) 75–96 www.elsevier.com/locate/amc
Simulation model of rubella––the effects of vaccination strategies Anamarija Jazbec
a,*
, Marko Delimar b, Vanja Slavic Vrzic
c
a
c
Department of Mathematics and Technical Basics, Faculty of Forestry, University of Zagreb, Svetosimunska 25, Zagreb HR-10000, Croatia b Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, Zagreb HR-10000, Croatia Department of Hygiene and Epidemiology, Tresnjevka, Albaharijeva bb, Zagreb HR-10000, Croatia
Abstract The paper presents the construction of a deterministic multistage simulation model with age and sex structure. The model is based on the natural history of rubella. The dynamics of rubella is described by a system of non-linear differential equations. The model was validated by simulating the course of rubella in Tresnjevka, Zagreb, from 1961 to 1991. For the purposes of simulations, the epidemiological parameters commonly accepted to apply to the rubella were used; epidemiological classes, coefficients of transfer among the classes, and durations of stay in the classes; only the force of infection was estimated by iterative simulation until the average observed incidence in non-epidemic and epidemic years was reproduced within computational error. The force of infection was age and sex dependent. The effects of three different vaccination strategies were simulated. 2003 Elsevier Inc. All rights reserved. Keywords: Simulation; Model; Rubella; Vaccination strategies
1. Introduction The mathematical simulation model is an excellent tool for recognition and understanding the nature of the infectious diseases [1]. By the end of the 19th
*
Corresponding author. E-mail address: [email protected] (A. Jazbec).
0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00610-6
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century mechanism of epidemic spread together with bacteriological research and friendly usage of epidemiological data collection and analyses provide together great development in epidemiology. It was proven that the dynamics of infectious diseases depends on the number of susceptible and numbers of contacts between susceptible and infectious persons (contact rate). This simple mathematical assumption is the basis of all deterministic and, with some later modifications, stochastic theories [2]. At the beginning of the 20th century, first deterministic mathematical models were constructed. Kermac and McKendrich [3] proved the threshold theorem of epidemics. Mathematical models of infectious diseases are used for theoretical evaluations and comparisons of detection, education, prevention therapy, control programs and the cost-benefit analysis [4,5]. An advantage of mathematical models for infectious diseases is the economy, brightness and exactness of a mathematical formulation [6]. In model construction, it is essential to find an optimal solution based on available knowledge and model requirements. Continuous development of electronic data processing techniques has enabled a construction of more complex multistage simulation models with a large amount of parameters. Several models of rubella were developed with different approaches [7–10]. 1.1. Rubella Rubella is a mild viral infectious disease characterised by rash and lymphadenopathy. Rubella assumes great importance when infecting pregnant women, because the transmission of the disease to the foetus can have disastrous effects. The incidence of clinical illness is usually the highest in the spring, traditionally in children between 5 and 6 years old. Subclinical illnesses have a great importance in the spread of the infection (40–50% cases) [11]. In unvaccinated population, a rubella epidemic occurs approximately every 4–10 years. The susceptibility of age groups depends on the length of time since the last epidemic occurred and on immunisation coverage [12]. Incidence of congenital infection varies with percentage of available sero-negative adult women and the presence of circulating rubella virus at certain point of time. 1.2. Vaccination strategies The main goal of rubella vaccination program is to eliminate or reduce rubella infection in pregnant women and the consequential congenital rubella syndrome (CRS) in their babies. A very simple strategy is not to vaccinate at all (India, tropical Africa, Brazil) which caused most people all become infected before the adolescent age. There are two main vaccination strategies: direct, in which adolescent girls and women are protected against the effects of being exposed (UK, Israel, Australia), and indirect, in which children of one or both sexes are vaccinated in their early childhood. In indirect strategy, women are
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protected through the interruption of rubella transmission in order not to be exposed at all (USA, Canada, Scandinavian countries, Croatia) [13]. Scientists used mathematical models to find an optimal long-term vaccination strategy, which would lower the incidence of rubella infection among women in childbearing age. Knox [7], Hethcote [10] and Anderson and May [7] have similar conclusions. The UK strategy (vaccinating girls in pre-adolescent years) gives better results than the USA strategy (vaccinating all young children) if the immunisation coverage is under 80% and the other way around if the immunisation coverage is above 80%. Although Dietz [8] concluded that UK strategy is always better than USA strategy. He concluded that fixed vaccination UK strategy in long term always results with lower incidence of CRS then the USA strategy. Edmunds and his colleagues in their research suggested that in Italy and Germany the vaccination coverage in infancy has not been sufficient to interrupt rubella transmission and it seems likely that the immunisation programmes in these countries are doing more harm than good, but this may be a partly result of selective immunisation of schoolgirls. They concluded that reducing inequalities in the uptake of rubella vaccine may cause greater health benefits than increasing the mean level of coverage [14]. In Croatia, the rubella vaccination started in 1976 and the vaccination strategy against rubella changed three times since. All children were vaccinated at the age of one year and only girls at the age 14. From 1994, additional vaccination at the school entry age (6–7) was aggregated. From 1997 boys and girls were vaccinated at ages 1 and 12.
2. Model development 2.1. Simulation model The structure of the model is based on the natural history of rubella. The dynamics of the disease is described by a non-linear system of differential equations with population dependant parameters. EulerÕs method is used for the numerical solving of the initial CauchyÕs problem. In that way, a differential system is transformed to difference system. In practice, is very difficult to determine the precise parameters of the model (an average of several authors is often used) and in this case methods error would be much lower than input data error [15,16]. 2.2. Epidemiological classes, coefficients of transfer The characteristics, modelled on the population ðN Þ, define a relation of equivalence, which results with a partition of the set. The elements of such partition are called epidemiological classes Xi ðtÞ and satisfy conditions:
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0 6 Xi ðtÞ 6 N ðtÞ for t P 0 X Xi ðtÞ ¼ N ðtÞ i
The basis of a S–I–R (susceptible–infectious–remove) model is that infectious diseases divide the population in three main epidemiological classes. The flow of the population from one epidemiological class to another goes in a predetermined order and direction. The transfer among epidemiological classes is represented by a set of coefficients Rij (probability of transfer), each representing the fraction of individuals transferred to another class. Rij is a real function: Ri;j : Rþ 0 ! ½0; 1 The coefficient of transfer from susceptible to infectious class is called rate of infection and it is a product of contact rate (probability of contact with infectious person) and the force of infection (probability that during this contact susceptible person would be infected). The input parameters of a simulation model are population data and parameters, which define the model. Population data included the population size, annual population rate (newborns, mortality) for each age and sex group separately. Parameters that define the model are: epidemiological classes, duration of population stay in epidemiological classes and coefficients of transfers among these classes. The results of simulation are the numbers of people in the epidemiological classes during a time. 2.3. Population The estimation of the input values of the model (such as the force of infection), as well as the validation of the model were based on the data on the rubella incidence in the Tresnjevka municipality in Zagreb, Croatia, between 1961 and 1991. A large number (around 80,000) of displaced persons and refugees with unknown vaccination status settled in that area in 1991 due to war activity, at which time additional measures were taken to stop possible outbreaks (vaccination of all pre-school and school children under 15 with unknown vaccination status with MRP). This is the reason why the year after 1991 were not included in parameter estimation and validation of the model. Two years, however, have been omitted, due to a complete lack of data in 1964 and a partial lack of data in 1970 (only the total number of patients was available). The data on incidence was obtained from the Tresnjevka municipal administration (Office for Hygiene and Epidemiology), while the general data on Tresnjevka population were obtained from the Zagreb Statistical Institute [17]. The epidemiological data showed no difference between the sexes before the vaccine was introduced. Since its introduction, however, the male population manifested greater rubella incidence than the female. The difference is rather significant for age groups between 15 and 19 (Fig. 1), and between 20 and 29 (Fig. 2), at which age boys lose
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INCIDENCE PER 10.000 POPULATION
300
250
200
150
100
50
1961
1965
1975 1970
1985 1980
1990
MALE FEMALE
YEARS
Fig. 1. Incidence of rubella per 10,000 population in Tresnjevka municipality for age group 15–19 years age from 1965 to 1991.
35
INCIDENCE PER 10.000 POPULATION
30
25
20
15
10
5
0 1961
1965
1975 1970
1985 1980
1990
MALE FEMALE
YEARS
Fig. 2. Incidence of rubella per 10,000 population in Tresnjevka municipality for age group 20–29 years age from 1965 to 1991.
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the immunity gained by vaccination in the first year. The difference in incidence between the sexes is a result of differing vaccination strategies for the girls and the boys. Hence the necessity to separate epidemiological classes for the rubella model by sexes. 2.4. The computational program The computer program was designed on the grounds of the mentioned formulation. The program allows changing of all parameters. Program is written in Quick Basic [15].
3. Results 3.1. Model of rubella The model is constructed as a deterministic multistage S–I–R (susceptible– infectious–remove) model with age and sex structure [4,6,11,15,18]. The flow of the population from one to another epidemiological class goes in predetermined order, direction and precise quantities resulting from one set coefficients of transfer from one epidemiological class to another in determined period of time [4]. The model is based on the natural history of rubella. Only essential epidemiological classes, which depend the dynamic of infection and which are necessary for simulation of the natural course of infection and of the effects of vaccination strategies, are retained in the model. The population is divided in 18 age groups (0, 1, 2, 3, 4, 5, 6, 7, 8–11, 12, 13, 14, 15, 16, 17–19, 20–29, 30–39, P 40 years). This partition was made to compare different vaccination strategies when it is important to observe particular age groups separately or when the force of infection is different for particular age groups. Simulations were made for 18 different age groups but results are presents in summed five age groups (0–6, 7–14, 15–19, 20–29, P 30 years) for easier understanding. Because of the difference in the force of infection, some age groups have to be separately defined in the model, but are not as important for the rubella dynamics. Partition of population by sex is necessary because of sex dependant vaccination strategies. For the purposes of simulation, epidemiological parameters related to epidemiological classes such as the duration of stay in them and coefficients of transfer were data taken from the literature. When literature data was given as intervals, the means of the interval was used. When multiple sources indicated different values, a mean of available values was used. k;s Sixteen epidemiological classes (X0k;s –X15 ) related to rubella, with duration of stay in each class are presented in the flow chart (Fig. 3).
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Maternal immunity (6 months) X2
Newborns X0 Congenital rubella (10 months) X3
Passive immunity (6 months) X11
Susceptible X1
Non -infective incubation (10 days) X5
Chronicle illness (life long) X4
Complications due to rubella (3 months) X9
Deaths due to Rubella X14
Legend:
Vaccinated (18 days) X12
Infectious incubation (8 days) X6
Active immunity (13 years) X13
Clinical illness (8 days) X8
Subclinical illness (8 days) X7
Resistant (life long) X10
Deaths due to other causes X15
Non-infectious
81
Resistant
Infectious
Fig. 3. The flow chart of the dynamics of rubella.
Where k ¼ 1; . . . ; 18 number of age group and s ¼ 1; 2 sex dependence. The flow of the population through epidemiological classes is presented on the flow chart (Fig. 3). The inter-relationship between epidemiological classes and population movement through epidemiological classes is indicated by arrows. Epidemiological classes X0 ––newborns, X2 ––maternal immunity and X3 –– congenital rubella-infectious are related only for the first age group (< 1 years).
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For all other age groups, the entry in the particular age group is equal to the exit from the previous age group. Exit from the last age group (P 40 years) is at the same time exit from the entire population. Exit from all epidemiological classes except these denoted on diagram have exit in the epidemiological class X15 ––deaths due to other causes. 3.2. Mathematical formulation of the model The mathematical relationship between epidemiological classes of population, shown in the flow chart (Fig. 3), is expressed by the non-linear system of differential equations: k ¼ 1; 2; . . . ; 18; s ¼ 1; 2 k;s k;s k;s k;s k;s k;s k;s k;s k;s k;s k;s s X_ 1k;s ¼ Rk;s 0;1 X þ R2;1 X2 þ R11;1 X11 þ R13;1 X13 ðR1;5 þ R1;11 þ R1;12 þ R1;15 ÞX1 0
k;s k;s k;s s X_ 2k;s ¼ Rk;s 0;2 X0 ðR2;1 þ R2;15 ÞX2 k;s k;s k;s k;s s X_ 3k;s ¼ Rk;s 0;3 X0 ðR3;4 þ R3;14 þ R3;15 ÞX3 k;s k;s k;s k;s X_ 4k;s ¼ Rk;s 3;4 X3 ðR4;14 þ R4;15 ÞX4 k;s k;s k;s k;s X_ 5k;s ¼ Rk;s 1;5 X1 ðR5;6 þ R5;15 ÞX5 k;s k;s k;s k;s k;s X_ 6k;s ¼ Rk;s 5;6 X5 ðR6;7 þ R6;8 þ R6;15 ÞX6 k;s k;s k;s k;s k;s X_ 7k;s ¼ Rk;s 6;7 X6 ðR7;9 þ R7;10 þ R7;15 ÞX7 k;s k;s k;s k;s k;s X_ 8k;s ¼ Rk;s 6;8 X6 ðR8;9 þ R8;10 þ R8;15 ÞX8 k;s k;s k;s k;s k;s k;s k;s X_ 9k;s ¼ Rk;s 7;9 X7 þ R8;9 X8 ðR9;10 þ R9;14 þ R9;15 ÞX9 k;s k;s k;s k;s k;s k;s k;s k;s X_ 10 ¼ Rk;s 7;10 X7 þ R8;10 X8 R9;10 X9 R10;15 X10 k;s k;s k;s k;s k;s X_ 11 ¼ Rk;s 1;11 X1 ðR11;1 þ R11;15 ÞX11 k;s k;s k;s k;s k;s X_ 12 ¼ Rk;s 1;12 X1 ðR12;13 þ R12;15 ÞX12 k;s k;s k;s k;s k;s X_ 13 ¼ Rk;s 12;13 X12 ðR13;1 þ R13;15 ÞX13 k;s k;s k;s k;s k;s k;s X_ 14 ¼ Rk;s 3;14 X3 þ R4;14 X4 þ R9;14 X14 k;s X_ 15 ¼
13 X
k;s Rk;s i;15 X15
i¼1
Initial conditions: X1k;s ðt0 Þ > 0 X3k;s ðt0 Þ þ X6k;s ðt0 Þ þ X7k;s ðt0 Þ þ X8k;s ðt0 Þ > 0;
k ¼ 1; 2; . . . ; 18
All equations in this system represent the age ðk ¼ 1; . . . ; 18Þ and sex ðs ¼ 1; 2Þ dependent change in time in the epidemiological classes. A change of an epidemiological class in time t ðXik;s ðtÞÞ is defined by input and output from a particular class. Input in particular class is a sum of all entries from classes
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from the same age and sex group from which a transfer in a particular class exists (see Diagram) and the identical type of class from previous age group ðk1;sÞ ðXi ðtÞÞ. Output is sum of all exits of the same age and sex group for which the transfer exists (see Diagram) and from the same class to the next age group. Input from one class to another is equal to the product of coefficients of transfer and class inputs. Output from one class to another is equal to the product of coefficients of transfer and class outputs. For k ¼ 1; 2; . . . ; 18 and s ¼ 1; 2 Ri;j ¼ Ri;j ðtÞ Xi ¼ Xi ðtÞ d X_ ik;s ¼ Xik;s ðtÞ functions of time dt 15 X Xik;s ðtÞ ¼ N k;s ðtÞ size of the kth age group of sex s i¼1 18 X
N k;s ðtÞ ¼ N s ðtÞ size of male or female population
k¼1
N 1 ðtÞ þ N 2 ðtÞ ¼ N ðtÞ size of population The non-linearity of the system of differential equations arises from the coefficient Rk;s 1;5 for k ¼ 1; . . . ; 18 and s ¼ 1; 2, Rk;s 1;5 ¼
2 X 18 f k;s X ðX i;j þ X7i;j þ X8i;j Þ N j¼1 i¼1 6
This is coefficient of transfer from class X1 (susceptible) to X5 (non-infectious incubation). It is product of two probabilities (force of infection f k;s and contact rate). On the assumption that population is homogeneous, contact rate is equal to proportion of infectious persons for all age groups and sex in the entire population: contact rate ¼
2 X 18 1 X ðX i;j þ X7i;j þ X8i;j Þ N j¼1 i¼1 6
3.3. Estimation of model parameters The force of infection is estimated by the simulation of incidence of rubella on Tresnjevka community and it is age and sex depended (Tables 1 and 2). The force of infection was estimated by iterative simulation until the average
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Table 1 Coefficients of transfer 6¼ 0 R0;1 ¼ 2:7 103 R0;2 ¼ 1:2 102 R0;3 ¼ 3:3 104 R1;5 ¼ depends on force of infection (Table 2) R1;11 ¼ depends on vaccination strategy R1;12 ¼ depends on vaccination strategy R2;1 ¼ 1 R3;4 ¼ 0:8 R3;14 ¼ 0:2 R4;14 ¼ 5 102 R5;6 ¼ 1
R6;7 ¼ 0:4 R6;8 ¼ 0:6 R7;9 ¼ 1 105 R7;10 ¼ 9:9 101 R8;9 ¼ 2:5 104 R8;10 ¼ 9:9 101 R9;10 ¼ 0:8 R9;14 ¼ 0:2 R11;1 ¼ 1 R12;13 ¼ 1 R13;1 ¼ 1
Table 2 Force of infection for non-epidemic and epidemic years by age groups and gender in Tresnjevka municipality Age groups (in years)
Females
Males
Non-epidemic
Epidemic
Non-epidemic
Epidemic
0 1 2 3 4 5 6 7 8–11 12 13 14 15 16 17–19 20–29 30–39 P 40
6.5 · 104 8.2 · 104 5.8 · 104 8.5 · 104 2.7 · 104 4.5 · 104 2.8 · 104 4.5 · 104 4.4 · 104 3.5 · 104 3.1 · 104 3.1 · 104 7.9 · 106 7.9 · 106 7.9 · 106 2.0 · 105 1.0 · 106 1.0 · 106
3.6 · 103 7.7 · 103 9.4 · 103 1.1 · 102 1.9 · 102 1.8 · 102 1.8 · 102 2.4 · 102 2.2 · 102 1.5 · 102 1.3 · 102 1.3 · 102 3.5 · 103 3.4 · 103 3.5 · 103 6.6 · 104 3.7 · 104 2.1 · 105
5.0 · 104 1.2 · 103 5.0 · 103 5.0 · 103 4.1 · 104 4.2 · 104 3.5 · 104 6.5 · 104 4.7 · 104 1.6 · 104 1.6 · 104 1.6 · 104 4.0 · 105 4.0 · 105 4.0 · 105 5.0 · 106 5.0 · 106 1.0 · 106
1.9 · 103 8.2 · 103 8.8 · 103 1.3 · 102 1.5 · 102 1.6 · 102 1.8 · 102 2.1 · 102 1.9 · 102 1.2 · 102 1.2 · 102 3.0 · 103 4.0 · 103 3.8 · 103 3.9 · 103 2.9 · 104 1.1 · 104 1.0 · 106
observed incidence in non-epidemic and epidemic years was reproduced within a computational error. The maximum relative aberration for the size of age group is 0.03%. The force of infection is age and sex dependent. Another values were estimated as mean or weighted mean value taken from literature [12,19–22].
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3.4. Validation of the model The model needs to be validated to show if simulations adequately fit the real life data, and possibly, to modify the model if needed. The model was validated by simulating the course of rubella in Tresnjevka municipality in Zagreb from 1961 to 1991 (Fig. 4). The data received from the Municipal Health Centre Tresnjevka does not precisely represent the real situation for more reasons. One of them lies in the method of registering the rubella cases. This model assumed that 60% of clinical illnesses were registered. While in some cases, if the epidemic occurred in some children institution like school, kindergarten or scholarÕs house number of registered cases is very high (near 100%), sometimes in such cases some susceptive to rubella are registered too (Fig. 4). The simulations were made with an average natural birth-rate of 1.5% and mortality for particular age groups. The natural flow of the disease was simulated from 1961 to 1973, and after that vaccination strategies applied in Croatia were used: from 1974 vaccination of girls at age 14, and from 1977 vaccination of both girls and boys at age 1 and girls at 14. The simulations were made for the fixed effectiveness of vaccine of 95%, but for different immunisation coverage (60%, 70%, 80% and 90%) comparing the best fit with real data. The best fit was obtained for simulation
INCIDENCE PER 100.000 POPULATION
2000
1500
1000
500
0 1960
1965
1970
1975
1980
1985
1990
OBSERVED SIMULATED
YEARS
Fig. 4. Observed and simulated rubella incidence per 100,000 population in Tresnjevka municipality from 1961.
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with immunisation coverage of 70% for age group from 1 to 2 years old girls and boys, and for 80% for age group from 14 to 15 years old girls (Fig. 4). It was also simulated that until 1973, an epidemic occurred every 3–4 years, which is obvious from the observed data (Fig. 4). Fig. 7 shows that the epidemics that occurred in 1972 and 1973 with low intensity was in fact one epidemic which occurred at the end of 1972 and at the beginning of the 1973. 3.5. Simulations The model was used to simulate the changes in the dynamics of the disease that occurred with three different vaccination strategies. The effects of three different vaccination strategies were simulated assuming vaccination during 60 years with immunisation coverage of 75% and vaccine effectiveness of 95% [23]. The natural flow of rubella was simulated for the first 5 years, and different vaccination strategies were simulated in years 6–60: • G1,14B1 Vaccination of girls at ages 1 and 14, boys at age 1. • G1,7,14B1,7 Vaccination of girls at ages 1, 7 and 14, boys at ages 1 and 7. • G1,12B1,12 Vaccination of both and girls at ages 1 and 12. The strategy G1,14B1 is the strategy applied in Croatia till 1994. The strategy G1,7,14B1,7 is applied in Croatia from 1994. The strategy G1,12B1,12 is strategy applied in Croatia from 1997. All simulations are graphically represented according to age groups for both sexes and for the whole population. Simulation for age groups are represent as incidence per 10,000 population while simulation for the whole population are represent as incidence per 100,000 population. The figures do not contain the first 5 years of natural flow of rubella because it is the same for all vaccination strategies and it is not very important for comparisons of the strategies (Figs. 5–17). The incidence of rubella decreased in all age groups after vaccination, regardless of the vaccination strategy as shown in Figs. 5–17. While interpreting Figs. 5–17, one should take into account that local variations of less than 5 per 10,000 (i.e. 0.05%) in certain graphs are of no epidemiological importance whatsoever being the result of erroneous approximations and inaccurate database entry. The simulations indicate that all three strategies would have similar effects and would barely differ in the incidence for the next 23 years. The vaccination strategy here marked as G1,14B1 would result in comparable incidence decrease for the first 8–10 years, until the first epidemic that would occur sooner and at a greater scale than with the vaccination strategies G1,7,14B1,7 and G1,12B1,12 (Fig. 5). A close look at the simulation results shows that such difference in incidence between the vaccination strategy G1,14B1 and the others could primarily be attributed to the incidence in the
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87
140
INCIDENCE PER 100.000 POPULATION
120
100
80
60
G1,12 B1,12 40
G1,7,14 B1,7
20 0
10
20
30
40
50
G1,14 B1
YEARS
Fig. 5. Simulation of three vaccination strategies against rubella on whole population in Tresnjevka municipality for period of 55 years.
160
INCIDENCE PER 100.000 POPULATION
140
120
100
80
60
40
G1,12 G1,7,14 20 0
10
20
30
40
50
G1,14
YEARS
Fig. 6. Simulation of three vaccination strategies against rubella for female population in Tresnjevka municipality for period of 55 years.
88
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INCIDENCE PER 100.000 POULATION
160
140
120
100
80
60
40
B1,12 B1,7
20 0
10
20
30
40
50
B1
YEARS
Fig. 7. Simulation of three vaccination strategies against rubella for male population in Tresnjevka municipality for period of 55 years.
INCIDENCE PER 10.000 POPULATION
13
12
11
10
9
G1,12 G1,7,14 8 0
10
20
30
40
50
G1,7
YEARS
Fig. 8. Simulation of three vaccination strategies against rubella for age group 0–6 years female population in Tresnjevka municipality for period of 55 years. All vaccination strategies have the same effect.
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89
16
INCIDENCE PER 10.000 POPULATION
14 12 10 8 6 4 2 0
G1,12 G1,7,14 0
10
20
30
40
50
G1,7
YEARS
Fig. 9. Simulation of three vaccination strategies against rubella for age group 7–14 years female population in Tresnjevka municipality for period of 55 years.
3.5
INCIDENCE PER 10.000 POPULATION
3
2.5
2
1.5
1
0.5
0
G1,12 G1,7,14 0
10
20
30
40
50
G1,14
YEARS
Fig. 10. Simulation of three vaccination strategies against rubella for age group 15–19 years female population in Tresnjevka municipality for period of 55 years.
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INCIDENCE PER 10.000 POPULATION
8
6
4
2
0
G1,12 G1,7,14 0
10
20
30
40
50
G1,14
YEARS
Fig. 11. Simulation of three vaccination strategies against rubella for age group 20–29 years female population in Tresnjevka municipality for period of 55 years.
3.5
INCIDENCE PER 10.000 POPULATION
3
2.5
2
1.5
1
0.5
0
G1,12 G1,7,14 0
10
20
30
40
50
G1,14
YEARS
Fig. 12. Simulation of three vaccination strategies against rubella for female older than 30 years in Tresnjevka municipality for period of 55 years.
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91
20
INCIDENCE PER 10.000 POPULATION
18
16
14
12
10
8
B1,12 B1,7 6 0
10
20
30
40
50
B1
YEARS
Fig. 13. Simulation of three vaccination strategies against rubella for age group 0–6 years male population in Tresnjevka municipality for period of 55 years. All vaccination strategies have the same effect.
20
INCIDENCE PER 10.000 POPULATION
18
16
14
12
10
8
B1,12 B1,7 6 0
10
20
30
40
50
B1
YEARS
Fig. 14. Simulation of three vaccination strategies against rubella for age group 7–14 years male population in Tresnjevka municipality for period of 55 years.
92
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INCIDENCE PER 10.000 POPULATION
140 120 100 80 60 40 20 0
B1,12 B1,7 0
10
20
30
40
50
B1
YEARS
Fig. 15. Simulation of three vaccination strategies against rubella for age group 15–19 years male population in Tresnjevka municipality for period of 55 years.
7
INCIDENCE PER 10.000 POPULATION
6
5
4
3
2
1
0
B1,12 B1,7 0
10
20
30
40
50
B1
YEARS
Fig. 16. Simulation of three vaccination strategies against rubella for age group 20–29 years male population in Tresnjevka municipality for period of 55 years.
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1.4
INCIDENCE PER 10.000 POPULATION
1.2
1
0.8
0.6
0.4
0.2
0
B1,12 B1,7 0
10
20
30
40
50
B1
YEARS
Fig. 17. Simulation of three vaccination strategies against rubella for men older than 30 years Tresnjevka municipality for period of 55 years.
male population (Fig. 7). Namely, strategy G1,14B1 ceases to protect the boys after reaching the average age of 13. The epidemic has a greater effect when the vaccination is first introduced, while after continued vaccination treatment for 10–15 years, this effect tends to weaken notably as the general population becomes well protected. As for the girls, the simulations show comparably good protection with all three strategies (Figs. 6,8–12). The rubella incidence in women aged between 20 and 29 (part of the reproductive age), however, does not depend on the vaccination strategy (Fig. 11).
4. Discussion This paper presents the simulation model for rubella only. There is a possibility to make simulation models for specific age groups such as women in reproductive age. Such specific models require highly reliable/precise data input that includes both the female population belonging to the age group and congenital syndrome incidence [6,8,9,13]. It is worth mentioning that all parameters and epidemiological classes in this model may be modified according to intentions and knowledge on the development course of rubella. This model can provide useful information in the choice of appropriate protection strategy, and in the evaluation of actual results.
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Rubella follow-up shows that the epidemics occur every 4 years, and that about 1500 of 100,000 people are affected by the disease. After the vaccination, the incidence decreased in all age groups regardless of the vaccination strategy (Figs. 5–17). All simulated strategies are indirect, that is, women are protected by halting the expansion of the virus within the population. All simulations start from the same premises: 75% immunisation coverage, 95% vaccine efficiency, and sex/age dependent data for infection force. The simulations for the general population show that all three strategies would have similar effects and would barely differ in the incidence for the next 23 years. The vaccination strategy here marked as G1,14B1 would result in comparable incidence decrease for the first 8–10 years until the first epidemic that would occur sooner and at a greater scale than with the vaccination strategies G1,7,14B1,7 and G1,12B1,12 (Fig. 5). A close look at the simulation results shows that such difference in incidence between the vaccination strategy F1,14B1 and the others could primarily be attributed to the incidence in the male population (Fig. 7). Namely, strategy G1,14B1 ceases to protect the boys after reaching the average age of 13. The epidemic has a greater effect when the vaccination is first introduced, while after continued vaccination treatment for 10–15 years, this effect tends to weaken notably as the general population becomes well protected. As for the girls, the simulations show comparably good protection with all three strategies (Figs. 6,8–12). The rubella incidence in women aged between 20 and 29 (part of the reproductive age), however, does not depend on the vaccination strategy (Fig. 11). Strategies G1,7,14B1,7 and G1,12B1,12 show comparable effects for all age groups and both sexes (Figs. 5–17). Their differences in the incidence move around 10 per 100,000, and are even smaller within individual age groups (Fig. 5). The difficulties related to the rubella vaccination cost-benefit analysis are quite complex when it comes to different strategies. Part of the difficulties relates to unreliable, inaccessible, or incomplete input data. Besides, an exhaustive cost-benefit analysis would require data on post-vaccination and post-rubella complications, as well as on the number of children born with congenital malformations due to foetal infection with the rubella virus. It is sad, however, that no such information is recorded. The existing records refer to connate malformations, yet without reference to their cause. The data on post-vaccination or post-rubella complications have been taken from the published papers and further elaborated in our study. Due to unavailable or partial data, the cost-benefit analysis is based on the immunisation cost exclusively. It is rather important to stress that rubella is associated with high expenses due to CRS [24]. The US researchers have estimated that the cost of healthcare services amount to US$ 200,000.00 for a single child with CRS during her/his lifetime [25]. The opinion prevailing in some authors [24,26] is that selective immunisation of adolescent girls is more cost-effective than the general
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immunisation since, in the prevention of congenital syndrome, the invested funds return far sooner with the selective immunisation [24,26]. Secondly, however arbitrary the cost estimations of the early rubella vaccination, they can hardly indicate at the financial gain, and thirdly, if the immunity of adolescent girls results lower than 100%. The theories of other authors (Knox, Dietz) rely exclusively on CRS incidence model that is estimated to correlate with the rubella incidence in women between 25.0 and 25.1 years of age. These models show that the vaccination limited to 14-year-old girls results more yields higher results and financial benefit than the vaccination of all 2-year-olds if the immunisation coverage is lower or equal to 80%. As we mentioned earlier, a complete cost-benefit analysis, that is, an exhaustive estimation was not possible due to missing data for the Tresnjevka municipality. The purpose of this study relying on comparison of estimated cost and rubella incidence in the general population was to see or let the reader decide which of the three simulated strategies would yield more benefit. The meaning of cost-benefit, if one reviews the literature, seems to shift from country to country. A good example is Island where the cost-benefit analysis resulted to favour previous sera testing of 14-year-old girls and vaccination of those who had never had rubella than undertaking general vaccination of 2-year-olds [26]. The question is which of the strategies protecting the boys to chose. Strategy G1,7,14B1,7 achieves better immunisation coverage as the boys who were not covered at year 1 have a fair chance to be covered at the age of 7. Strategy G1,12B1,12 achieves almost identical effect while the immunisation cost is lower.
References [1] N.T.J. Bailey, The Mathematical Theory of Epidemics, Griffin, London, 1957. [2] L. Billard, A stochastic general epidemic in m sub-populations, J. Appl. Prob. 13 (1976) 567– 572. [3] W.O. Kermac, A.G. McKendrich, Contributions to the mathematical theory of epidemics, Parts I–V, Proc. Roy. Soc. A 115 (1927) 700–721; Proc. Roy. Soc. A 138 (1932) 55–83; Proc. Roy. Soc. A 141 (1933) 94–122; J. Hyg. Camb. 37 (1937) 172–187; J. Hyg. Camb. 39 (1939) 271–288. [4] B. Cvjetanovic, B. Grab, H. Dixon, Computerized epidemiological model of typhoid fever with age structure and its use in the planning and evaluation of Antityphoid Immunization and Sanitation Programmes, Math. Modelling 7 (1986) 719–744. [5] B. Cvjetanovic, N. Delimar, M. Kosicek, Modelling infectious diseases, Supplement Medicinskih Razgledov, Ljubljana (1990) 54–59 (in Croatian). [6] H.W. Hethcote, Current issues in epidemiological modelling, Int. Conf. on Diff. Equat. and Appl. to Biol. and Popul. Dyn., Clearmont, CA, 10–13 January 1990, pp. 1–31. [7] E.G. Knox, Strategy for rubella vaccination, Int. J. Epidemiol. 9 (1980) 13–23.
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[8] K. Dietz, The evaluation of rubella vaccination strategies, in: The Mathematical Theory of the Dynamic of Biological Populations, Academic Press, New York, 1981. [9] R.M. Anderson, R.M. May, Vaccination against rubella and measles: Quantitative investigations of different policies, J. Hyg. Camb. 90 (1983) 259–325. [10] H.W. Hethcote, in: Rubella, Applied Mathematical Ecology, Springer Verlag, Berlin– Heidelberg, 1989. [11] G.L. Mendell, R.G. Douglas, J.E. Benett, Priciples and Practice of Infectious Disease, John Wiley, 1985. [12] S.A. Plotkin, Rubella Vaccine, in: Vaccines, Saunders, Philadelphia, PA, 1988. [13] E.G. Knox, Theoretical aspects of rubella vaccination strategies, Rev. Infect. Disease 7 (1985) 194–197. [14] W.J. Edmunds, O.G. Van de Heijden, M. Eerola, N.J. Gay, Modelling rubella in Europe, Epidemiol. Infect. 125 (3) (2000) 617–634. [15] M. Kosicek, Baza podataka i opci simulacijski model dinamike zaraznih bolesti (in Croatian), Ph.D. Thesis, Institute for medical research and occupational health, University of Zagreb, 1992. [16] N.V. Kopchenova, I.A. Maron, Computational Mathematics, Mir Publishers, Moscow, 1975. [17] Statistical yearbooks of Zagreb, 1961, 1971, 1981, 1991. [18] N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London, 1975. [19] H. Friedman, J.E. Prier, Rubella, C.C. Thomas Publisher, Springfield, IL, 1973. [20] D. Mardesic, Pediatrics, Skolska knjiga, Zagreb, 1976 (in Croatian). [21] HarrisonÕs Principles of Internal Medicine, McGraw-Hill, 12th edition, 1991. [22] B. Borcic, M. Ferdebar, M. Rajniger-Miholic, Imuni odgovor na revakcinaciju protiv rubeole 13 godina nakon primovakcinacije, Arhiv Zast. Majke & Djeteta 35 (1991) 167–172. [23] B. Borcic, S. Smrdel, J.A. Eldan, J. Kolic, M. Ferdebar, O efikasnosti domace kombinirane vakcine protiv ospica, zausnjaka i crljenke, Arhiv. Zast. Majke & Djeteta, 32 (1988) 85–92 (in Croatian). [24] S.C. Schoenbaum, Benefit-cost aspects of rubella immunization, Rev. Infect. Disease 7 (1985) 210–211. [25] M.L. Lindegren, L.J. Fehrs, S.C. Hadler, A.R. Hinman, Update: Rubella and congenital rubella syndrome, 1980–1990, Epidemiol. Rev. 13 (1991) 341–348. [26] M. Gudnadottir, Cost-effectiveness of different strategies for prevention of congenital rubella infection: a practical example from iceland, Rev. Infect. Diseases 7 (1985) 200–208.
Simulation model of rubella––the effects of vaccination strategies Anamarija Jazbec
a,*
, Marko Delimar b, Vanja Slavic Vrzic
c
a
c
Department of Mathematics and Technical Basics, Faculty of Forestry, University of Zagreb, Svetosimunska 25, Zagreb HR-10000, Croatia b Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, Zagreb HR-10000, Croatia Department of Hygiene and Epidemiology, Tresnjevka, Albaharijeva bb, Zagreb HR-10000, Croatia
Abstract The paper presents the construction of a deterministic multistage simulation model with age and sex structure. The model is based on the natural history of rubella. The dynamics of rubella is described by a system of non-linear differential equations. The model was validated by simulating the course of rubella in Tresnjevka, Zagreb, from 1961 to 1991. For the purposes of simulations, the epidemiological parameters commonly accepted to apply to the rubella were used; epidemiological classes, coefficients of transfer among the classes, and durations of stay in the classes; only the force of infection was estimated by iterative simulation until the average observed incidence in non-epidemic and epidemic years was reproduced within computational error. The force of infection was age and sex dependent. The effects of three different vaccination strategies were simulated. 2003 Elsevier Inc. All rights reserved. Keywords: Simulation; Model; Rubella; Vaccination strategies
1. Introduction The mathematical simulation model is an excellent tool for recognition and understanding the nature of the infectious diseases [1]. By the end of the 19th
*
Corresponding author. E-mail address: [email protected] (A. Jazbec).
0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00610-6
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century mechanism of epidemic spread together with bacteriological research and friendly usage of epidemiological data collection and analyses provide together great development in epidemiology. It was proven that the dynamics of infectious diseases depends on the number of susceptible and numbers of contacts between susceptible and infectious persons (contact rate). This simple mathematical assumption is the basis of all deterministic and, with some later modifications, stochastic theories [2]. At the beginning of the 20th century, first deterministic mathematical models were constructed. Kermac and McKendrich [3] proved the threshold theorem of epidemics. Mathematical models of infectious diseases are used for theoretical evaluations and comparisons of detection, education, prevention therapy, control programs and the cost-benefit analysis [4,5]. An advantage of mathematical models for infectious diseases is the economy, brightness and exactness of a mathematical formulation [6]. In model construction, it is essential to find an optimal solution based on available knowledge and model requirements. Continuous development of electronic data processing techniques has enabled a construction of more complex multistage simulation models with a large amount of parameters. Several models of rubella were developed with different approaches [7–10]. 1.1. Rubella Rubella is a mild viral infectious disease characterised by rash and lymphadenopathy. Rubella assumes great importance when infecting pregnant women, because the transmission of the disease to the foetus can have disastrous effects. The incidence of clinical illness is usually the highest in the spring, traditionally in children between 5 and 6 years old. Subclinical illnesses have a great importance in the spread of the infection (40–50% cases) [11]. In unvaccinated population, a rubella epidemic occurs approximately every 4–10 years. The susceptibility of age groups depends on the length of time since the last epidemic occurred and on immunisation coverage [12]. Incidence of congenital infection varies with percentage of available sero-negative adult women and the presence of circulating rubella virus at certain point of time. 1.2. Vaccination strategies The main goal of rubella vaccination program is to eliminate or reduce rubella infection in pregnant women and the consequential congenital rubella syndrome (CRS) in their babies. A very simple strategy is not to vaccinate at all (India, tropical Africa, Brazil) which caused most people all become infected before the adolescent age. There are two main vaccination strategies: direct, in which adolescent girls and women are protected against the effects of being exposed (UK, Israel, Australia), and indirect, in which children of one or both sexes are vaccinated in their early childhood. In indirect strategy, women are
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protected through the interruption of rubella transmission in order not to be exposed at all (USA, Canada, Scandinavian countries, Croatia) [13]. Scientists used mathematical models to find an optimal long-term vaccination strategy, which would lower the incidence of rubella infection among women in childbearing age. Knox [7], Hethcote [10] and Anderson and May [7] have similar conclusions. The UK strategy (vaccinating girls in pre-adolescent years) gives better results than the USA strategy (vaccinating all young children) if the immunisation coverage is under 80% and the other way around if the immunisation coverage is above 80%. Although Dietz [8] concluded that UK strategy is always better than USA strategy. He concluded that fixed vaccination UK strategy in long term always results with lower incidence of CRS then the USA strategy. Edmunds and his colleagues in their research suggested that in Italy and Germany the vaccination coverage in infancy has not been sufficient to interrupt rubella transmission and it seems likely that the immunisation programmes in these countries are doing more harm than good, but this may be a partly result of selective immunisation of schoolgirls. They concluded that reducing inequalities in the uptake of rubella vaccine may cause greater health benefits than increasing the mean level of coverage [14]. In Croatia, the rubella vaccination started in 1976 and the vaccination strategy against rubella changed three times since. All children were vaccinated at the age of one year and only girls at the age 14. From 1994, additional vaccination at the school entry age (6–7) was aggregated. From 1997 boys and girls were vaccinated at ages 1 and 12.
2. Model development 2.1. Simulation model The structure of the model is based on the natural history of rubella. The dynamics of the disease is described by a non-linear system of differential equations with population dependant parameters. EulerÕs method is used for the numerical solving of the initial CauchyÕs problem. In that way, a differential system is transformed to difference system. In practice, is very difficult to determine the precise parameters of the model (an average of several authors is often used) and in this case methods error would be much lower than input data error [15,16]. 2.2. Epidemiological classes, coefficients of transfer The characteristics, modelled on the population ðN Þ, define a relation of equivalence, which results with a partition of the set. The elements of such partition are called epidemiological classes Xi ðtÞ and satisfy conditions:
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0 6 Xi ðtÞ 6 N ðtÞ for t P 0 X Xi ðtÞ ¼ N ðtÞ i
The basis of a S–I–R (susceptible–infectious–remove) model is that infectious diseases divide the population in three main epidemiological classes. The flow of the population from one epidemiological class to another goes in a predetermined order and direction. The transfer among epidemiological classes is represented by a set of coefficients Rij (probability of transfer), each representing the fraction of individuals transferred to another class. Rij is a real function: Ri;j : Rþ 0 ! ½0; 1 The coefficient of transfer from susceptible to infectious class is called rate of infection and it is a product of contact rate (probability of contact with infectious person) and the force of infection (probability that during this contact susceptible person would be infected). The input parameters of a simulation model are population data and parameters, which define the model. Population data included the population size, annual population rate (newborns, mortality) for each age and sex group separately. Parameters that define the model are: epidemiological classes, duration of population stay in epidemiological classes and coefficients of transfers among these classes. The results of simulation are the numbers of people in the epidemiological classes during a time. 2.3. Population The estimation of the input values of the model (such as the force of infection), as well as the validation of the model were based on the data on the rubella incidence in the Tresnjevka municipality in Zagreb, Croatia, between 1961 and 1991. A large number (around 80,000) of displaced persons and refugees with unknown vaccination status settled in that area in 1991 due to war activity, at which time additional measures were taken to stop possible outbreaks (vaccination of all pre-school and school children under 15 with unknown vaccination status with MRP). This is the reason why the year after 1991 were not included in parameter estimation and validation of the model. Two years, however, have been omitted, due to a complete lack of data in 1964 and a partial lack of data in 1970 (only the total number of patients was available). The data on incidence was obtained from the Tresnjevka municipal administration (Office for Hygiene and Epidemiology), while the general data on Tresnjevka population were obtained from the Zagreb Statistical Institute [17]. The epidemiological data showed no difference between the sexes before the vaccine was introduced. Since its introduction, however, the male population manifested greater rubella incidence than the female. The difference is rather significant for age groups between 15 and 19 (Fig. 1), and between 20 and 29 (Fig. 2), at which age boys lose
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INCIDENCE PER 10.000 POPULATION
300
250
200
150
100
50
1961
1965
1975 1970
1985 1980
1990
MALE FEMALE
YEARS
Fig. 1. Incidence of rubella per 10,000 population in Tresnjevka municipality for age group 15–19 years age from 1965 to 1991.
35
INCIDENCE PER 10.000 POPULATION
30
25
20
15
10
5
0 1961
1965
1975 1970
1985 1980
1990
MALE FEMALE
YEARS
Fig. 2. Incidence of rubella per 10,000 population in Tresnjevka municipality for age group 20–29 years age from 1965 to 1991.
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the immunity gained by vaccination in the first year. The difference in incidence between the sexes is a result of differing vaccination strategies for the girls and the boys. Hence the necessity to separate epidemiological classes for the rubella model by sexes. 2.4. The computational program The computer program was designed on the grounds of the mentioned formulation. The program allows changing of all parameters. Program is written in Quick Basic [15].
3. Results 3.1. Model of rubella The model is constructed as a deterministic multistage S–I–R (susceptible– infectious–remove) model with age and sex structure [4,6,11,15,18]. The flow of the population from one to another epidemiological class goes in predetermined order, direction and precise quantities resulting from one set coefficients of transfer from one epidemiological class to another in determined period of time [4]. The model is based on the natural history of rubella. Only essential epidemiological classes, which depend the dynamic of infection and which are necessary for simulation of the natural course of infection and of the effects of vaccination strategies, are retained in the model. The population is divided in 18 age groups (0, 1, 2, 3, 4, 5, 6, 7, 8–11, 12, 13, 14, 15, 16, 17–19, 20–29, 30–39, P 40 years). This partition was made to compare different vaccination strategies when it is important to observe particular age groups separately or when the force of infection is different for particular age groups. Simulations were made for 18 different age groups but results are presents in summed five age groups (0–6, 7–14, 15–19, 20–29, P 30 years) for easier understanding. Because of the difference in the force of infection, some age groups have to be separately defined in the model, but are not as important for the rubella dynamics. Partition of population by sex is necessary because of sex dependant vaccination strategies. For the purposes of simulation, epidemiological parameters related to epidemiological classes such as the duration of stay in them and coefficients of transfer were data taken from the literature. When literature data was given as intervals, the means of the interval was used. When multiple sources indicated different values, a mean of available values was used. k;s Sixteen epidemiological classes (X0k;s –X15 ) related to rubella, with duration of stay in each class are presented in the flow chart (Fig. 3).
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Maternal immunity (6 months) X2
Newborns X0 Congenital rubella (10 months) X3
Passive immunity (6 months) X11
Susceptible X1
Non -infective incubation (10 days) X5
Chronicle illness (life long) X4
Complications due to rubella (3 months) X9
Deaths due to Rubella X14
Legend:
Vaccinated (18 days) X12
Infectious incubation (8 days) X6
Active immunity (13 years) X13
Clinical illness (8 days) X8
Subclinical illness (8 days) X7
Resistant (life long) X10
Deaths due to other causes X15
Non-infectious
81
Resistant
Infectious
Fig. 3. The flow chart of the dynamics of rubella.
Where k ¼ 1; . . . ; 18 number of age group and s ¼ 1; 2 sex dependence. The flow of the population through epidemiological classes is presented on the flow chart (Fig. 3). The inter-relationship between epidemiological classes and population movement through epidemiological classes is indicated by arrows. Epidemiological classes X0 ––newborns, X2 ––maternal immunity and X3 –– congenital rubella-infectious are related only for the first age group (< 1 years).
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For all other age groups, the entry in the particular age group is equal to the exit from the previous age group. Exit from the last age group (P 40 years) is at the same time exit from the entire population. Exit from all epidemiological classes except these denoted on diagram have exit in the epidemiological class X15 ––deaths due to other causes. 3.2. Mathematical formulation of the model The mathematical relationship between epidemiological classes of population, shown in the flow chart (Fig. 3), is expressed by the non-linear system of differential equations: k ¼ 1; 2; . . . ; 18; s ¼ 1; 2 k;s k;s k;s k;s k;s k;s k;s k;s k;s k;s k;s s X_ 1k;s ¼ Rk;s 0;1 X þ R2;1 X2 þ R11;1 X11 þ R13;1 X13 ðR1;5 þ R1;11 þ R1;12 þ R1;15 ÞX1 0
k;s k;s k;s s X_ 2k;s ¼ Rk;s 0;2 X0 ðR2;1 þ R2;15 ÞX2 k;s k;s k;s k;s s X_ 3k;s ¼ Rk;s 0;3 X0 ðR3;4 þ R3;14 þ R3;15 ÞX3 k;s k;s k;s k;s X_ 4k;s ¼ Rk;s 3;4 X3 ðR4;14 þ R4;15 ÞX4 k;s k;s k;s k;s X_ 5k;s ¼ Rk;s 1;5 X1 ðR5;6 þ R5;15 ÞX5 k;s k;s k;s k;s k;s X_ 6k;s ¼ Rk;s 5;6 X5 ðR6;7 þ R6;8 þ R6;15 ÞX6 k;s k;s k;s k;s k;s X_ 7k;s ¼ Rk;s 6;7 X6 ðR7;9 þ R7;10 þ R7;15 ÞX7 k;s k;s k;s k;s k;s X_ 8k;s ¼ Rk;s 6;8 X6 ðR8;9 þ R8;10 þ R8;15 ÞX8 k;s k;s k;s k;s k;s k;s k;s X_ 9k;s ¼ Rk;s 7;9 X7 þ R8;9 X8 ðR9;10 þ R9;14 þ R9;15 ÞX9 k;s k;s k;s k;s k;s k;s k;s k;s X_ 10 ¼ Rk;s 7;10 X7 þ R8;10 X8 R9;10 X9 R10;15 X10 k;s k;s k;s k;s k;s X_ 11 ¼ Rk;s 1;11 X1 ðR11;1 þ R11;15 ÞX11 k;s k;s k;s k;s k;s X_ 12 ¼ Rk;s 1;12 X1 ðR12;13 þ R12;15 ÞX12 k;s k;s k;s k;s k;s X_ 13 ¼ Rk;s 12;13 X12 ðR13;1 þ R13;15 ÞX13 k;s k;s k;s k;s k;s k;s X_ 14 ¼ Rk;s 3;14 X3 þ R4;14 X4 þ R9;14 X14 k;s X_ 15 ¼
13 X
k;s Rk;s i;15 X15
i¼1
Initial conditions: X1k;s ðt0 Þ > 0 X3k;s ðt0 Þ þ X6k;s ðt0 Þ þ X7k;s ðt0 Þ þ X8k;s ðt0 Þ > 0;
k ¼ 1; 2; . . . ; 18
All equations in this system represent the age ðk ¼ 1; . . . ; 18Þ and sex ðs ¼ 1; 2Þ dependent change in time in the epidemiological classes. A change of an epidemiological class in time t ðXik;s ðtÞÞ is defined by input and output from a particular class. Input in particular class is a sum of all entries from classes
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from the same age and sex group from which a transfer in a particular class exists (see Diagram) and the identical type of class from previous age group ðk1;sÞ ðXi ðtÞÞ. Output is sum of all exits of the same age and sex group for which the transfer exists (see Diagram) and from the same class to the next age group. Input from one class to another is equal to the product of coefficients of transfer and class inputs. Output from one class to another is equal to the product of coefficients of transfer and class outputs. For k ¼ 1; 2; . . . ; 18 and s ¼ 1; 2 Ri;j ¼ Ri;j ðtÞ Xi ¼ Xi ðtÞ d X_ ik;s ¼ Xik;s ðtÞ functions of time dt 15 X Xik;s ðtÞ ¼ N k;s ðtÞ size of the kth age group of sex s i¼1 18 X
N k;s ðtÞ ¼ N s ðtÞ size of male or female population
k¼1
N 1 ðtÞ þ N 2 ðtÞ ¼ N ðtÞ size of population The non-linearity of the system of differential equations arises from the coefficient Rk;s 1;5 for k ¼ 1; . . . ; 18 and s ¼ 1; 2, Rk;s 1;5 ¼
2 X 18 f k;s X ðX i;j þ X7i;j þ X8i;j Þ N j¼1 i¼1 6
This is coefficient of transfer from class X1 (susceptible) to X5 (non-infectious incubation). It is product of two probabilities (force of infection f k;s and contact rate). On the assumption that population is homogeneous, contact rate is equal to proportion of infectious persons for all age groups and sex in the entire population: contact rate ¼
2 X 18 1 X ðX i;j þ X7i;j þ X8i;j Þ N j¼1 i¼1 6
3.3. Estimation of model parameters The force of infection is estimated by the simulation of incidence of rubella on Tresnjevka community and it is age and sex depended (Tables 1 and 2). The force of infection was estimated by iterative simulation until the average
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Table 1 Coefficients of transfer 6¼ 0 R0;1 ¼ 2:7 103 R0;2 ¼ 1:2 102 R0;3 ¼ 3:3 104 R1;5 ¼ depends on force of infection (Table 2) R1;11 ¼ depends on vaccination strategy R1;12 ¼ depends on vaccination strategy R2;1 ¼ 1 R3;4 ¼ 0:8 R3;14 ¼ 0:2 R4;14 ¼ 5 102 R5;6 ¼ 1
R6;7 ¼ 0:4 R6;8 ¼ 0:6 R7;9 ¼ 1 105 R7;10 ¼ 9:9 101 R8;9 ¼ 2:5 104 R8;10 ¼ 9:9 101 R9;10 ¼ 0:8 R9;14 ¼ 0:2 R11;1 ¼ 1 R12;13 ¼ 1 R13;1 ¼ 1
Table 2 Force of infection for non-epidemic and epidemic years by age groups and gender in Tresnjevka municipality Age groups (in years)
Females
Males
Non-epidemic
Epidemic
Non-epidemic
Epidemic
0 1 2 3 4 5 6 7 8–11 12 13 14 15 16 17–19 20–29 30–39 P 40
6.5 · 104 8.2 · 104 5.8 · 104 8.5 · 104 2.7 · 104 4.5 · 104 2.8 · 104 4.5 · 104 4.4 · 104 3.5 · 104 3.1 · 104 3.1 · 104 7.9 · 106 7.9 · 106 7.9 · 106 2.0 · 105 1.0 · 106 1.0 · 106
3.6 · 103 7.7 · 103 9.4 · 103 1.1 · 102 1.9 · 102 1.8 · 102 1.8 · 102 2.4 · 102 2.2 · 102 1.5 · 102 1.3 · 102 1.3 · 102 3.5 · 103 3.4 · 103 3.5 · 103 6.6 · 104 3.7 · 104 2.1 · 105
5.0 · 104 1.2 · 103 5.0 · 103 5.0 · 103 4.1 · 104 4.2 · 104 3.5 · 104 6.5 · 104 4.7 · 104 1.6 · 104 1.6 · 104 1.6 · 104 4.0 · 105 4.0 · 105 4.0 · 105 5.0 · 106 5.0 · 106 1.0 · 106
1.9 · 103 8.2 · 103 8.8 · 103 1.3 · 102 1.5 · 102 1.6 · 102 1.8 · 102 2.1 · 102 1.9 · 102 1.2 · 102 1.2 · 102 3.0 · 103 4.0 · 103 3.8 · 103 3.9 · 103 2.9 · 104 1.1 · 104 1.0 · 106
observed incidence in non-epidemic and epidemic years was reproduced within a computational error. The maximum relative aberration for the size of age group is 0.03%. The force of infection is age and sex dependent. Another values were estimated as mean or weighted mean value taken from literature [12,19–22].
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3.4. Validation of the model The model needs to be validated to show if simulations adequately fit the real life data, and possibly, to modify the model if needed. The model was validated by simulating the course of rubella in Tresnjevka municipality in Zagreb from 1961 to 1991 (Fig. 4). The data received from the Municipal Health Centre Tresnjevka does not precisely represent the real situation for more reasons. One of them lies in the method of registering the rubella cases. This model assumed that 60% of clinical illnesses were registered. While in some cases, if the epidemic occurred in some children institution like school, kindergarten or scholarÕs house number of registered cases is very high (near 100%), sometimes in such cases some susceptive to rubella are registered too (Fig. 4). The simulations were made with an average natural birth-rate of 1.5% and mortality for particular age groups. The natural flow of the disease was simulated from 1961 to 1973, and after that vaccination strategies applied in Croatia were used: from 1974 vaccination of girls at age 14, and from 1977 vaccination of both girls and boys at age 1 and girls at 14. The simulations were made for the fixed effectiveness of vaccine of 95%, but for different immunisation coverage (60%, 70%, 80% and 90%) comparing the best fit with real data. The best fit was obtained for simulation
INCIDENCE PER 100.000 POPULATION
2000
1500
1000
500
0 1960
1965
1970
1975
1980
1985
1990
OBSERVED SIMULATED
YEARS
Fig. 4. Observed and simulated rubella incidence per 100,000 population in Tresnjevka municipality from 1961.
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with immunisation coverage of 70% for age group from 1 to 2 years old girls and boys, and for 80% for age group from 14 to 15 years old girls (Fig. 4). It was also simulated that until 1973, an epidemic occurred every 3–4 years, which is obvious from the observed data (Fig. 4). Fig. 7 shows that the epidemics that occurred in 1972 and 1973 with low intensity was in fact one epidemic which occurred at the end of 1972 and at the beginning of the 1973. 3.5. Simulations The model was used to simulate the changes in the dynamics of the disease that occurred with three different vaccination strategies. The effects of three different vaccination strategies were simulated assuming vaccination during 60 years with immunisation coverage of 75% and vaccine effectiveness of 95% [23]. The natural flow of rubella was simulated for the first 5 years, and different vaccination strategies were simulated in years 6–60: • G1,14B1 Vaccination of girls at ages 1 and 14, boys at age 1. • G1,7,14B1,7 Vaccination of girls at ages 1, 7 and 14, boys at ages 1 and 7. • G1,12B1,12 Vaccination of both and girls at ages 1 and 12. The strategy G1,14B1 is the strategy applied in Croatia till 1994. The strategy G1,7,14B1,7 is applied in Croatia from 1994. The strategy G1,12B1,12 is strategy applied in Croatia from 1997. All simulations are graphically represented according to age groups for both sexes and for the whole population. Simulation for age groups are represent as incidence per 10,000 population while simulation for the whole population are represent as incidence per 100,000 population. The figures do not contain the first 5 years of natural flow of rubella because it is the same for all vaccination strategies and it is not very important for comparisons of the strategies (Figs. 5–17). The incidence of rubella decreased in all age groups after vaccination, regardless of the vaccination strategy as shown in Figs. 5–17. While interpreting Figs. 5–17, one should take into account that local variations of less than 5 per 10,000 (i.e. 0.05%) in certain graphs are of no epidemiological importance whatsoever being the result of erroneous approximations and inaccurate database entry. The simulations indicate that all three strategies would have similar effects and would barely differ in the incidence for the next 23 years. The vaccination strategy here marked as G1,14B1 would result in comparable incidence decrease for the first 8–10 years, until the first epidemic that would occur sooner and at a greater scale than with the vaccination strategies G1,7,14B1,7 and G1,12B1,12 (Fig. 5). A close look at the simulation results shows that such difference in incidence between the vaccination strategy G1,14B1 and the others could primarily be attributed to the incidence in the
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140
INCIDENCE PER 100.000 POPULATION
120
100
80
60
G1,12 B1,12 40
G1,7,14 B1,7
20 0
10
20
30
40
50
G1,14 B1
YEARS
Fig. 5. Simulation of three vaccination strategies against rubella on whole population in Tresnjevka municipality for period of 55 years.
160
INCIDENCE PER 100.000 POPULATION
140
120
100
80
60
40
G1,12 G1,7,14 20 0
10
20
30
40
50
G1,14
YEARS
Fig. 6. Simulation of three vaccination strategies against rubella for female population in Tresnjevka municipality for period of 55 years.
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INCIDENCE PER 100.000 POULATION
160
140
120
100
80
60
40
B1,12 B1,7
20 0
10
20
30
40
50
B1
YEARS
Fig. 7. Simulation of three vaccination strategies against rubella for male population in Tresnjevka municipality for period of 55 years.
INCIDENCE PER 10.000 POPULATION
13
12
11
10
9
G1,12 G1,7,14 8 0
10
20
30
40
50
G1,7
YEARS
Fig. 8. Simulation of three vaccination strategies against rubella for age group 0–6 years female population in Tresnjevka municipality for period of 55 years. All vaccination strategies have the same effect.
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16
INCIDENCE PER 10.000 POPULATION
14 12 10 8 6 4 2 0
G1,12 G1,7,14 0
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20
30
40
50
G1,7
YEARS
Fig. 9. Simulation of three vaccination strategies against rubella for age group 7–14 years female population in Tresnjevka municipality for period of 55 years.
3.5
INCIDENCE PER 10.000 POPULATION
3
2.5
2
1.5
1
0.5
0
G1,12 G1,7,14 0
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20
30
40
50
G1,14
YEARS
Fig. 10. Simulation of three vaccination strategies against rubella for age group 15–19 years female population in Tresnjevka municipality for period of 55 years.
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INCIDENCE PER 10.000 POPULATION
8
6
4
2
0
G1,12 G1,7,14 0
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20
30
40
50
G1,14
YEARS
Fig. 11. Simulation of three vaccination strategies against rubella for age group 20–29 years female population in Tresnjevka municipality for period of 55 years.
3.5
INCIDENCE PER 10.000 POPULATION
3
2.5
2
1.5
1
0.5
0
G1,12 G1,7,14 0
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20
30
40
50
G1,14
YEARS
Fig. 12. Simulation of three vaccination strategies against rubella for female older than 30 years in Tresnjevka municipality for period of 55 years.
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20
INCIDENCE PER 10.000 POPULATION
18
16
14
12
10
8
B1,12 B1,7 6 0
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20
30
40
50
B1
YEARS
Fig. 13. Simulation of three vaccination strategies against rubella for age group 0–6 years male population in Tresnjevka municipality for period of 55 years. All vaccination strategies have the same effect.
20
INCIDENCE PER 10.000 POPULATION
18
16
14
12
10
8
B1,12 B1,7 6 0
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20
30
40
50
B1
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Fig. 14. Simulation of three vaccination strategies against rubella for age group 7–14 years male population in Tresnjevka municipality for period of 55 years.
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INCIDENCE PER 10.000 POPULATION
140 120 100 80 60 40 20 0
B1,12 B1,7 0
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20
30
40
50
B1
YEARS
Fig. 15. Simulation of three vaccination strategies against rubella for age group 15–19 years male population in Tresnjevka municipality for period of 55 years.
7
INCIDENCE PER 10.000 POPULATION
6
5
4
3
2
1
0
B1,12 B1,7 0
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30
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50
B1
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Fig. 16. Simulation of three vaccination strategies against rubella for age group 20–29 years male population in Tresnjevka municipality for period of 55 years.
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1.4
INCIDENCE PER 10.000 POPULATION
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1
0.8
0.6
0.4
0.2
0
B1,12 B1,7 0
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Fig. 17. Simulation of three vaccination strategies against rubella for men older than 30 years Tresnjevka municipality for period of 55 years.
male population (Fig. 7). Namely, strategy G1,14B1 ceases to protect the boys after reaching the average age of 13. The epidemic has a greater effect when the vaccination is first introduced, while after continued vaccination treatment for 10–15 years, this effect tends to weaken notably as the general population becomes well protected. As for the girls, the simulations show comparably good protection with all three strategies (Figs. 6,8–12). The rubella incidence in women aged between 20 and 29 (part of the reproductive age), however, does not depend on the vaccination strategy (Fig. 11).
4. Discussion This paper presents the simulation model for rubella only. There is a possibility to make simulation models for specific age groups such as women in reproductive age. Such specific models require highly reliable/precise data input that includes both the female population belonging to the age group and congenital syndrome incidence [6,8,9,13]. It is worth mentioning that all parameters and epidemiological classes in this model may be modified according to intentions and knowledge on the development course of rubella. This model can provide useful information in the choice of appropriate protection strategy, and in the evaluation of actual results.
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Rubella follow-up shows that the epidemics occur every 4 years, and that about 1500 of 100,000 people are affected by the disease. After the vaccination, the incidence decreased in all age groups regardless of the vaccination strategy (Figs. 5–17). All simulated strategies are indirect, that is, women are protected by halting the expansion of the virus within the population. All simulations start from the same premises: 75% immunisation coverage, 95% vaccine efficiency, and sex/age dependent data for infection force. The simulations for the general population show that all three strategies would have similar effects and would barely differ in the incidence for the next 23 years. The vaccination strategy here marked as G1,14B1 would result in comparable incidence decrease for the first 8–10 years until the first epidemic that would occur sooner and at a greater scale than with the vaccination strategies G1,7,14B1,7 and G1,12B1,12 (Fig. 5). A close look at the simulation results shows that such difference in incidence between the vaccination strategy F1,14B1 and the others could primarily be attributed to the incidence in the male population (Fig. 7). Namely, strategy G1,14B1 ceases to protect the boys after reaching the average age of 13. The epidemic has a greater effect when the vaccination is first introduced, while after continued vaccination treatment for 10–15 years, this effect tends to weaken notably as the general population becomes well protected. As for the girls, the simulations show comparably good protection with all three strategies (Figs. 6,8–12). The rubella incidence in women aged between 20 and 29 (part of the reproductive age), however, does not depend on the vaccination strategy (Fig. 11). Strategies G1,7,14B1,7 and G1,12B1,12 show comparable effects for all age groups and both sexes (Figs. 5–17). Their differences in the incidence move around 10 per 100,000, and are even smaller within individual age groups (Fig. 5). The difficulties related to the rubella vaccination cost-benefit analysis are quite complex when it comes to different strategies. Part of the difficulties relates to unreliable, inaccessible, or incomplete input data. Besides, an exhaustive cost-benefit analysis would require data on post-vaccination and post-rubella complications, as well as on the number of children born with congenital malformations due to foetal infection with the rubella virus. It is sad, however, that no such information is recorded. The existing records refer to connate malformations, yet without reference to their cause. The data on post-vaccination or post-rubella complications have been taken from the published papers and further elaborated in our study. Due to unavailable or partial data, the cost-benefit analysis is based on the immunisation cost exclusively. It is rather important to stress that rubella is associated with high expenses due to CRS [24]. The US researchers have estimated that the cost of healthcare services amount to US$ 200,000.00 for a single child with CRS during her/his lifetime [25]. The opinion prevailing in some authors [24,26] is that selective immunisation of adolescent girls is more cost-effective than the general
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immunisation since, in the prevention of congenital syndrome, the invested funds return far sooner with the selective immunisation [24,26]. Secondly, however arbitrary the cost estimations of the early rubella vaccination, they can hardly indicate at the financial gain, and thirdly, if the immunity of adolescent girls results lower than 100%. The theories of other authors (Knox, Dietz) rely exclusively on CRS incidence model that is estimated to correlate with the rubella incidence in women between 25.0 and 25.1 years of age. These models show that the vaccination limited to 14-year-old girls results more yields higher results and financial benefit than the vaccination of all 2-year-olds if the immunisation coverage is lower or equal to 80%. As we mentioned earlier, a complete cost-benefit analysis, that is, an exhaustive estimation was not possible due to missing data for the Tresnjevka municipality. The purpose of this study relying on comparison of estimated cost and rubella incidence in the general population was to see or let the reader decide which of the three simulated strategies would yield more benefit. The meaning of cost-benefit, if one reviews the literature, seems to shift from country to country. A good example is Island where the cost-benefit analysis resulted to favour previous sera testing of 14-year-old girls and vaccination of those who had never had rubella than undertaking general vaccination of 2-year-olds [26]. The question is which of the strategies protecting the boys to chose. Strategy G1,7,14B1,7 achieves better immunisation coverage as the boys who were not covered at year 1 have a fair chance to be covered at the age of 7. Strategy G1,12B1,12 achieves almost identical effect while the immunisation cost is lower.
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