AIDS Infection Spread

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

CONTROL OF THE HIV/AIDS INFECTION SPREAD G. Dzemyda, V. Saltenis and V. Tie~is Institute of Mathematit;s and Informatics. Akademijos St. 4. 2600 Vilnius. LithUiJniLl

Abstract. The model of the HIV/ AIDS infection spread characterizing any risk group by three differential equations is proposed and investigated. These equations describe the dynamics of active susceptible, active infected, and passive infected individuals. The paper deals with some specific features of the disease spread at the initial stage. The evaluation of parameters from demographical and medical data is discussed. The package for the investigation of infection is presented, and possibilities to control the infection are shown. Two general directions of control may be distinguished: the HIV / AIDS blood tests and the publicity and availability of protective means. Key Words. Biomedical; predictive control; process models; infection control; HIV / AIDS infection; simulation

1. INTRODUCTION

infection may be stopped. All calculations are made on the data from Lithuania.

Transmission models of the HIV / AIDS epidemic help to explore both the nature of disease spread in a population and the efficiency of control strategies, as well as to predict the extent of infection in the near future.

2. THE MODEL An important assumption of DEM models is that the choice of partners is independent of the former choices. The partnership is assumed to be rather short. Also, it is assumed that the population may be divided into some risk groups, which differ in parameters of the sexual behavior. This leads to the model similar to that given by Van Druten et al. (1990), but an additional equation is added. It describes the growth in number of infected individuals, who usually don't spread the infection (e.g., impotents and a part of persons indicated as infected). Such individuals make a significant part among the infected persons at low growth of infection. The following model was investigated:

Three main approaches are used in the recent scientific literature: extrapolation of observed data (Karon et al., 1989), the models based on differential equations (DEM) (Kaplan, 1990; Van Druten et al. , 1990; Sattenspield and Castillo-Chavez, 1990; Kault and McLeond, 1991; Jacquez and Simon, 1990; Dzemyda et al., 1992), and stochastic models (SM) (Cairns, 1991). The paper concerns some specific features of the disease spread at the initial stage, i.e., when the infection extent is small enough and a plentiful statistics is absent, so only two last approaches (DEM and SM) are suitable in this case. The DEM model is investigated in this paper. The evaluation of parameters from demographical and medical data is discussed. The software package for the investigation of infection is presented. The investigations showed under what conditions the HIV / AIDS

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The parameters of the model (1) - (3) may be evaluated directly or determined from demographical and medical parameters: is the number of individuals recruited into the group per year; is the per cent of infected among Uf; - Ut is the per cent of infected individuals, which do not spread the infection, recruited into the group per year; is the per cent of birth per year; M i are the per cent of mortality per year because and not of AIDS, correspondingly; - M~ is the per cent of individuals excluded out of the group per year for any reasons other than death; is the per cent of persons examined on the AIDS tests per year; is the per cent of infected persons faIled ill with AIDS per year; is the per cent of individuals, using protective means during sexual contacts; is the per cent of individuals, terminating the infection spreading after diagnoses of infection per year.

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Xi(t) is the number of susceptible individuals in the i-th group; Yi(t) is the number of infected individuals, who take part in the infection spread; Zi(t) is the number of infected individuals, who do not spread the infection; is the number of groups; Y'! and Z,! are the initial values; uf, ut are the numbers of susceptible, active and passive individuals recruited into the group per time unit; is the birth rate; Ji-l are the mortality rates not due to AIDS and due to AIDS, correspondingly; is the rate at which an individual leaves the group because of any reasons other than death (e.g., old age, impotence); is the rate of AIDS examination; is the rate of transition from HIV infected to AIDS; is the averaged number of contacts of an individual in group j with persons in group i per unit time; is the probability of infection from an infective person in group j to a susceptible one in group i per contact.

Let the dynamics of the number of individuals from the i-th group be determined by one of these parameters (e.g. M i ). The number of active infected in (2) must be Yi = Y'! . (1 - M i 1100) in a year. It is a solution of the equation dy;/ dt = -In(l M i /100)Yi. So, Ji-i = -In(1 - M i 1100) ~ M'/100. The parameters Ji-i and Ji-~ are determined analogously. This difference equation may be solved by iterative procedure dividing the year into a fixed number m of equal parts Ill. The numbers of infected individuals at the fixed moments Yi, Yt,···, yi may be calculated by a recurrent formula: Yt = y~-l. (1- M i 1100)~t. The rate of AIDS examination is equal k i = vi J(i 110000.

The balance of contacts must be satisfied: (Xj + Yj)bij = (Xi + Yi)bji .

The individuals from the same group can have various rates of contacts. The percent of individuals Nf has rate bt. It is shown by Jacquez and Simon (1990), that at the initial stage of infection the following averaged bij may be used:

3. THE PARAMETERS All the models of HIV / AIDS transmission require to solve the problems of statistical estimation of the initial extent of infection, and of various rates: mortality, partnerchanging, AIDS examination and other ones. The main problem encountered is the extent of compromise between a complexity of model and a possibility to estimate unknown parameters: the growth of complexity increases the number of parameters and decreases the possibilities of their good estimation.

The probability Pij depends on the probability of transmission per unprotected contact pij' the probability of failure of protection rij (rij = 0.1 using a condom) and

180

into account Xi/(Xi + yd ~ 1 we have for all j, that dyldt ~ 0 for any Xi and Yi, if and only if

the probability to use the protection Aj: + Aj(rij - 1)), Aj = a i 1100.

Pij = pij (1

Some parameters (the probability of infection Pii' the rate of transition from HIV infected to AIDS k a and the mortality rate due to AIDS Jll) must be treated as increasing in time, because the portion of individuals with later stages of HIV increases. The values of Pij, k a and Jli are minimal at the begining of infection spread and increse until some value. Provisory, the linear increasing function was used as an approximation for such increasing.

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I.e., the spread of HIV decreases in this case. In Fig. 1 the region, where (4) is satisfied, is marked by points for Lithuanian homosexuals at the beginning of the spread ( left) and at the widespread infection ( right ). The figures indicate that the possibility to stabilize the infection spread decreases in time.

4. THE SOFTWARE

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The computer software, the Package for Investigation of HIV / AIDS Infection (PIHI), realizing DEM is developed.

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Turbo Pascal (version 6.0) and its objectoriented application framework Turbo Vision, was used as a tool to write the PIHI, because the application needed a high-performance, flexible and consistent interactive user interface.

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Fig. 1. The regions, where HIV decreases

Pull-down menus were used for the selection of one of the possible models (singlegroup or multi-group) and for the selection of the parameters of the model. A radio button construction was used for input of the model parameters in the cases, when the set of possible data values was limited (for example, the year of the end of modelling). The most reliable values of the model parameters are set in the PIHI by default. The Turbo Vision windows were used for: output of results, the help texts, the error messages. A global context-sensitive help system is created.

5.2. Investigations of Control on the Basis of Real Initial Data The following values of the parameters for (1) - (3) were chosen on the basis of the world experience and the Lithuanian data: n = 3 (homo/bisexual men, promiscuous heterosexual men and women); Xl = 93000; x2 = 216000; x3 = 168000; Yl = 25; Y2 = 1; Y3 = 3; zl' = 9; z2 = 0; Z3 = 1. The spread of infection will be illustrated by varying the values of a i and J(i. a i = a, i = l,n. Y = 2:7=1 Yi in Fig. 2-4. An assumption is made, that the usage of protective means during sexual contacts grows and a converges to 100% at infinite time. al = 10% is the initial value of a. a2 is the forecasted value of a at the end of simulation period.

5. CONTROL OF THE INFECTION SPREAD The parameters ki and Pij may be controlled by means of prophylaxis. That may be done by the efforts of varying the values of f{i, vi and a i .

The following experiments are performed: 1. The values of J(i, i = G, are made all equal and take the following values: 10%, 15%,20%, 25%. The results are marked by 1, 2, 3 and 4, correspondingly (see Fig. 2 for a2 = 50% and Fig. 3 for a2 = 85%). 2. The values f{i = 20%, i = Dare fixed. a2 takes the following values: 75%, 80%,85%,90%. The results are marked by 1,2,3 and 4, correspondingly (see Fig. 4).

5.1. The Conditions when an Increase of Infection Stops y - 0 i Le t Y -- ",n L.i=l Yi, Ui an d Jlsi -- Jl i +Jlli +Jl2' Then, by summing (2) over i and taking

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6. CONCLUSIONS The investigations showed under what conditions the HIV / AlDS infection may be stopped. Two general directions may be distinguished: the HIV / AlDS blood tests and the publicity and availability of protective means. The last one is considerably cheaper and more effective. The infection spread stops when approximately 85% of individuals from the risk groups use preventive means and 20% HIV / AlDS blood tests are performed among them.

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7. ACKNOWLEDGEMENT

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We wish to express our appreciation to S.Chaplinskas and A.Trechiokas from AIDS Centre of Lithuania for their help.

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8. REFERENCES 87

Cairns, A.J.G. (1991). Model fitting and projection of the AIDS epidemic. Mathematical Biosciences, 107, 451-489. Dzemyda,G., V.Saltenis, V.Tiesis, S.Chaplinskas and A.Trechiokas (1992). HIV/ AIDS infection spread: simulation and control. Informatica, 3(4), 455-468. Jacquez, J.A., and C.P.Simon (1990). AIDS The epidemiological significance of two different mean rates of partner change. IMA Journal of Mathematics Applied in Medicine and Biology, 7, 27-32. Kaplan, E.H. (1990). An overview of AIDS modeling. New Directions for Program Evaluation, 46, 23-36. Karon, J.M., O.J.Devine and W.M.Morgan (1989). Predicting AIDS incidence by extrapolating from recent trends. In: Mathematical and Statistical Approaches to AIDS Epidemiology. Lect. Notes in Biomath., (C.Castillo-Chavez, Ed.), Vo1.83, pp.58-88. Springer, Berlin. Kault, D.A., and L.McLeod (1991). Modeling AIDS as a function of other sexually transmitted disease. Mathematical Biosciences, 103(1), 17-31. Sattenspiel, L., and C.Castillo-Chavez (1990). Environmental context, social interactions, and the spread of HIV. American Journal of Human Biology, 2, 395-417. Van Druten, J.A.M., and co workers (1990). HIV infection dynamics and intervention experiments in linked risk groups. Statistics in Medicine, 9, 721-736.

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