The Department of Biomedical Engineering, South-central University for Ethnic Communities, Minyuan Road 5, Wuhan, 430074, P.R. China.
Sirthe model constructed by Yangxi Zhang1 does not strictly capture the processes involved in the epidemiological pattern under study but it is now cited by some other researcher2,3 to fit and analyse the epidemiological data and has been introduced by some monographs,4,5 so we think it is necessary to correct the error in his model to avoid the spreading of inaccurate results.
Yangxi Zhang1 combined a reversible catalytic model with a two-stage catalytic model, and proposed that a compound catalytic model with both reversible and two-stage types should be used to analyse the distribution characteristics of the age-specific infection rate of certain parasites. According to some characteristics of the data surveyed, by means of simplification he constructed the model and estimated parameters. Based on Yangxi Zhang's proposal,1 we attempt to reconstruct a compound catalytic model with both reversible and two-stage types in order to exactly describe the epidemic processes.
Denote by A the susceptibles, by B the people with infective indications and by C the people whose infective indications have gone and who will not be infected any longer. A can covert into B at the rate of a, and B into A at the rate of b and into C at the rate of c. Only B can reverse into susceptibles. This is a compound relationship that a reversible situation coexists with a two-stage situation, which can be illustrated as follows
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Consider a population in which all people are susceptibles at time t = 0. Denote by x the fraction of the population who are infected, including individuals in categories B and C and by (1 x) the fraction of those who are susceptible at time t. Denote by z the fraction of those who have been infected but have now lost infective indications and will not be infected any longer, and by y = x z the fraction of those who, at time t, have been infected and are still seropositive.
To describe the processes in which the reversible situation coexists with the two-stage situation, Yangxi Zhang1 proposed the differential equation
![]() | (1) |
Generally, (bx) is taken as the middle item of the right-hand side in the above equation, which cannot really reflect the epidemic processes. As pointed out before, only B is reversible, but taking (bx) as the middle item of the right-hand side amounts to tacitly approve that both B and C are reversible. Thus, the
rate produced by y should be and that
produced by Substituting x = y + z into and combining them, we get a reversible and two-stage type compound catalytic model
![]() | (2) |
The general solution of which is
![]() | (3) |
where C1 and C2 are arbitrary constants and
![]() | (4) |
Substituting the initial conditions, y|t = 0 = 0 and z|t = 0 = 0 into (3), we get
![]() | (5) |
If we write
![]() | (6) |
it is not difficult to see from equations (4) that the following
![]() | (7) |
are tenable, and equation (5) can be written as
![]() | (8) |
We used the above model to fit the age prevalence of hookworm infection in Changshou County in 1981. First, residual method6 was used to estimate the initial values of D, and ß, then the method of non-linear least squares regression6 was performed to calculate the terminal values of D,
and ß, and finally the values of a, b and c could be obtained from equation (7). Thus, a compound catalytic model with both reversible and two-stage types for infection rate of hookworm in Changshou County was obtained
![]() | (9) |
The parameters we got are a = 0.11, b = 0.007, c = 0.003, which have shown that per year the people who became infectives were 110 of 1000 susceptibles, those who reversed into susceptibles were 7 of 1000 infectives and those who lost infective indications were no longer infected were 3 of 1000 infectives. Here the index of correlation R2 = 0.957.
To a certain extent, the results obtained by Yangxi Zhang differ from ours, but on the whole, they are very close. According to equation (1), the results are R2 = 0.955, a = 0.110, b = 0.006 and c = 0.003. This situation appears to be inevitable as well as coincidental. Though in equation (1) all infectives were mistaken for the people who were reversible and (by) was taken as (bx) after all, equation (1) reflected the property of the two stages of epidemic processes and the solving process of the equation was similar to that of a two-stage catalytic model. So the solution of equation (1) is also a double-exponential function
![]() | (10) |
If we write it is not difficult
to test and verify that there are relationships between D,,ß and a,b,c which have been shown in the first and last formulas of equations (7). When b and c are very small compared with a, the second formula of equation (7) is approximately tenable,
because In addition, no matter what
method is used to estimate parameters, its target is to make actual observed values and estimated values as consistent as possible. During the iterations, D, and ß above are inevitably approximated to the results we obtained. Therefore, it is no wonder that Yangxi Zhang's results are very similar to ours. It is in advance known that compared with a, b and c are both very small, the model proposed by Yangxi Zhang can be thought to be a simplified model. But in terms of the properties of the data studied, we would rather build a more perfect one so as to achieve a more accurate conclusion. Furthermore, we would like to use the modelling as a tool to study more problems. The model derived by Yangxi Zhang cannot strictly capture the processes involved in the epidemiological pattern under study, so when the parameters b and c are not very small compared with a, the parameter value c obtained from Yangxi Zhang's model is not correct. Therefore, it is necessary to have a correct mathematical form of the compound catalytic model with both reversible and two-stage types so that we will have something to go by when solving a new problem. It is necessary to note that because the people B with infective indications may reverse into the susceptibles A or convert into C whose infective indications have gone, we cannot expect that the loss rates b and c of B in the compound catalytic model with both reversible and two-stage types are just like the loss rate b of B in the simple
two-stage which roughly inversely pertains to the lifespan of infection.
There is the similar error in the compound catalytic model with both simple and reversible types derived by Yangxi Zhang7 and we will correct it elsewhere.
References
1 Yangxi Zhang. A compound catalytic model with both reversible and two-stage types and its applications in epidemiological study. Int J Epidemiol 1987;16:61921.[Abstract]
2 Ziyuan Liu et al. Application of a compound catalytic model with both reversible and two-stage type on leptospira infection in epidemiology. J Math Med 1995;8:16870.
3 Yaohu Jiang. The biological effects of Bacillus Calmette-Guerin evaluated with compound catalytic model. Chin J Health Stat 1990;7: 4647.
4 Yangxi Zhang. The application of mathematical statistics in the planned immunity. Beijing: People's Health Press, 1990.
5 Guanyi Geng. Epidemiology. Vol. 1. Beijing: People's Health Press, 1995.
6 Huaiwu Zhou. Mathematical Medicine. Shanghai: Shanghai Scientific and Technical Press, 1983.
7 Yangxi Zhang. A compound catalytic model with both simple and reversible types and its applications in the surveillance of diseases. Chi J Epidemiol 1985;6:11518.[Medline]