Distribution of hepatitis B infection among the crowd of mathematical simulation
【Key Words】 crowd
Modern medicine, by revealing the amount of disease, the relationship between the various factors to achieve a qualitative description of the disease from the general development of the quantitative research is to look for and the establishment of mathematical models describing the main laws, one of the important objectives, we can achieve the simulation, prediction and inspection purposes. The assumption
that groups of all ages, hepatitis B infection rate of infectious diseases with the same conditions, the logistic function in this study, based on the principle and analysis of the process of hepatitis B infection, through the distribution
of the disease in the population study of the actual situation and put forward different models, attempt to hepatitis B in the population the number of infections at different times and different age distribution of infection in the number of paper
proposes a simulation method to reach the actual decision-
making have some effect.
1 Principles and Methods
logistic function is to observe the value of the probability of an event has occurred and the relationship between the statistical model generally applies to a number of symptoms of the disease through some sort of indicators to predict the severity of the disease. However, after practice, data analysis, we found that logistic function in the prediction and simulation of a number of time-related laws in
the incidence of frequently-occurring disease, its
effectiveness is also very obvious. Hepatitis B is a common one of the diseases, according to the actual data has been obtained, the establishment of hepatitis B occurred in a
mathematical model to simulate the occurrence and distribution of an area to reduce the uncertainty of future projections is the use of simulation methods is an important aspects. After the study, we propose the following logistic function to simulate and predict the expression of hepatitis B occurred in the number of people is an ideal and convenient: y = L / (1 eβe-x / α) where α, β, L are through the original data to determine the constants, x is Toshitsugu, y is the number of the rate of occurrence. The formula x great time, ex / α
tends to zero, then y nearly constant, L, namely L for a certain region of hepatitis B occur in the future the number of asymptotic values of the peak. Peak for the fixed value, can be interpreted as the crowd in order to shift a constant
rate of infection after infection, as time increases, the number of infections has also increased, but always less than the maximum because of the spread of hepatitis B can cause disease while , but can also be caused by immunization. Through the raw data to estimate α, β, L values, can use the
following method: First calculate y-1, namely, y-1 = k abx,
where k = 1 / L, a = eβ / L, b = 1/ae. In this way, based on past hepatitis B occur annually collected data on the number of data, to determine k, a, b, after, we arrive at the number
of cases have occurred in an area of analog formula that Toshitsugu to predict the value of x substituted into the formula, you can get y -1 values, the number of cases of
hepatitis B occurred the year y shall be the number of trend
value; L = k-1 is a hepatitis B in the region may reach the highest place in the future the number of approximations. Pairs a, b of the following methods are available to determine given the data will be collected in chronological order, divided into the number of years equal to three segments, the number of years of n, calculated the number of cases of each piece of data in the aggregate the sum of the number of y-1i
(i = 1,2,3), then according to the formula: b = (1y3-1 y2) /
(1y2-1 y1) 1/na = 1y2-1y1b-1 (bn-1) 2k = 1y1-(bn-1 b -1) a / n
In estimating the number of annual occurrence, the further we have to simulate the age distribution of hepatitis B occur laws, you can place the actual process of hepatitis B were analyzed. Generally believe that Hepatitis B is caused by the hepatitis B virus infection, its pathogenesis should be to attack from the immune cells from the infected liver cells start, and can be changes in the status of liver cells and hepatitis occur in mathematical language to describe the
process as "normal cells into infected cells, the rate of change proportional to the number with the virus, hepatitis infected cells into cells and cellular immunity is proportional to the rate of change. " In this way, the incidence of hepatitis B, in theory could be interpreted as follows: populations to infection to infection characteristics of a positive person, you may also become infected with the rate of b features was negative. To consider the following expressions can be used to simulate the processes occurring
hepatitis B: dy dt = dxdt-dzdt = a (1-x)-by where x for a
certain age have been infected with the ratio, y as to have been infected and the infection was characteristic positive rate, z to have been contaminated, but has lost some of
infection characteristics, t is time.
Solvable by the expression may simulate the formula: y = a ab (e-bt-e-at) of its theoretical maximum value of time tm = lna-lnb ab.
As the simulation time variable directly into the formula, it can better carry out a certain period of hepatitis B occurrence in different age simulation. Reposted elsewhere in the paper for free download http://
Cases collected from a certain area the past 18 years the number of hepatitis B patients in Table 1, to facilitate the simulation find the formula, the data were divided into three groups according to demand, and at the same time find the number of cases of each group of data combined. Table 1 in an area the past 18 years the number of hepatitis B table
Therefore, the number of cases of hepatitis B occurred simulation formula 1 y = 1.424 × 10-4 1.169 × 10-4 × 0.931x
= (1.424 1.169 × 0.931x) × 10-4 asymptotic peak of L = 1 / k
= 7022 (one )
Simulation of hepatitis B occurred in the logistic
formula: y = L 1 eβe-x / α = 7022 1 0.821 × (0.931) x if x
= 5 Shi, y ? 4461; x = 9, y ? 4906. Simulation is still a
high degree of approximation, it is necessary to conduct future of the region of hepatitis B with a few predictions,
such as the first 19 years in the region of hepatitis B occurrences (x = 18), Based on the type, availability of y ?
5727 (people), the occurrence of subsequent years can also be a similar number of calculated and predicted.
Cases of two investigations in an area, get a certain year, hepatitis B patients with different ages data distribution, via the estimated parameters a, b, available in the region, the age prevalence of the analog equation: y = 180.6 (e-0.044te -0.069t)
The data shown in Table 2, where t is age (by age group, the value of dollars), according to the theoretical value of the projected simulation are presented in Table last column, it is clear that the method of fit is still better. Table 2 occurred in an area of hepatitis B prevalence by age
distribution of data (omitted)
According to the formula, the highest incidence of hepatitis B in the area of age should be t = lna-lnb ab = 18
According to the simulation method given in this study, a
combination of both, we can quantitatively estimate of hepatitis B in an area of "infectious", quantitative prediction of the region overall incidence of hepatitis B in a
certain time, the number of various age groups with a few of the laws of in order to achieve pre-control effect of
prevention and evaluation of the role and serve as a basis for decisions. Of course, time-related medical process is often
very complex, but also diverse, the incidence of hepatitis B purely down to the results of a random process, only a form of reaction to this process, only from the The mere occurrence information to simulate the number of hepatitis B has happened, and attempts to reflect the inherent relationship
between the process, predict the future. Therefore, we think it may well be an incident of hepatitis B and can be repeated one of the methods. In addition, the need to explain that the text just put a simple process and can roughly simulate actual
simulation formula; also features the introduction of more
indicators, consider the multi-factor formula to create a
simulation, but sometimes the system processes or factors taken into account complex, parameter dependence relationship
between the complexity of the model will show a certain degree of complexity to the method involving a question of competency. Therefore, the design of a simplified simulation of the formula to prevent too many side issues, will enable the realization of simulated target, the more smoothly.
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