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ISSN 20700482, Mathematical Models and Computer Simulations, 2013, Vol. 5, No. 4, pp. 379–393. © Pleiades Publishing, Ltd., 2013. Original Russian Text © E.A. Nosova, A.A. Romanyukha, 2013, published in Matematicheskoe Modelirovanie, 2013, Vol. 25, No. 1, pp. 45–64.

Mathematical Model of HIVInfection Transmission and Dynamics in the Size of Risk Groups E. A. Nosovaa and A. A. Romanyukhab a

Federal Scientific Methodical Center (FSMC), Ministry of Health, Moscow, Russia b Institute of Numerical Mathematics, Russian Academy of Sciences, Russia email: [email protected], [email protected] Received September 22, 2012

Abstract—The research is aimed at developing and studying the distribution model of the human immunity deficit virus (HIV) that includes dynamics in the formation of risk groups. Most of the HIV distribution models assume that the risk of infection does not change over the individual’s lifetime. This work, in contrast, proposes a model of virus transmission in a population with a dynamic risk. The risk dynamics are described by the model of the formation of groups of individuals with alcohol and drug dependence, which are the main factors influencing the spread of HIV in Russia. Keywords: mathematical model, HIV, risk groups, estimation of parameters DOI: 10.1134/S207004821304011X

1. INTRODUCTION One of the priority problems of health services is to efficiently control the transmission of socially dan gerous infections (including those induced by the human immunity deficit virus (HIV)) consisting in developing and implementing a set of measures that allow maximally limiting to number of new cases of infection. The scale of the HIV epidemic is relatively low, estimated at about 70 million throughout the world1. The number of new cases of the disease (up to 3 million per annum), the impossibility of eliminating the virus from an infected organism and preventing the development of the acquired human immunity deficit syndrome (AIDS), the shortening of the lifetime down to 7–10 years (without supporting treatment) at the average age of infection of 20–30 years, the high cost of supporting the quality of life of infected peo ple, and the economic losses caused by the reduction of the ablebodied population illustrate the impor tance of the designated mission. The analysis of publications has shown that the current approach in the fight against HIV infection is not entirely effective. There is no generally accepted and sound approach in estimating its efficiency, since the indicator of efficiency cannot be directly measured. What should be taken as the criterion of efficiency: reduction of the incidence of the disease, a longer length and better quality of life of infected individuals, or some other indicator? Another problem existing in some countries (including Russia) is the extrapolation of the results obtained on homogenous populations to the populations with a complicated structure of risk of infection. The recommendations and forecasts of the leading international organizations (WHO, ON) in con structing programs aimed at counteracting the transmission of HIV infection in Russia are based on esti mates obtained from mathematical models developed for Africa. The level of the relative incidence of the disease in Africa and East Europe (0.08 and 0.09 new cases per annum per an infected individual accord ing to the UNAIDS2 data) and welldeveloped model based on the data from African studies of HIV infection transmission allow the international experts to extend the results to other developing countries. However, some kinds of actions, formed in this way, have failed to yield the expected results [3,4]. Here, we give the results from the study of the HIVinfection transmission in Russia. The analysis of the data [1, 2] has allowed us to classify Russia’s regions, based on the formation of groups at risk to HIV 1 For comparison: 2 billion people are infected by mycobacteria of tuberculosis in the world. 2 The Joint United Nations Programme on HIV/AIDS.

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Users of injected narcotics

Childrenorphans Men with homo and Migrants 10–20 % bisexual orientation Hemodependents Commercial sex workers

10–20 % Shift workers

General population 60–80 %

Fig. 1. Structure of the population by HIV transmission risk [6]. The hue shows the core group, the dotted line shows the groupbridge. Percentages show the shares of the appropriate subpopulations in total number; thickness of arrows shows the estimated intensity of the virus transmission.

infection, as well as formulate and investigate the model of virus transmission, after taking these differ ences into account. 2. BRIEF DATA ON THE EPIDEMIOLOGY OF HIV INFECTION A person is a source and reservoir of HIV at all stages of the disease. The germ is a virus transmitted in one of the following three ways: sexual, prenatal (by blood), and vertical (utero from infected mother to the child, in childbearing or breastfeeding). In Russia HIVinfection is transmitted primarily among people aged 15–49 years. The infection rate in the rest of the population is less than 2% of the total number infected. Definition 1. Behavior when the individual does not use measures to prevent infection is called risky. Definition 2. The core group is the set of individuals whose risky behavior is sufficient to generate viable3 chains of disease transmission and whose absence leads to the total disappearance of infection in the population [5]. Definition 3. Individuals who practice risky contacts both with members of the core groups and the rest of the population form a bridge group [5]. The set of individuals that are outside of the both the core or the bridge group will be called the basic population. Figure 1 shows the structure of the population with respect to the probability of HIV infection ordi narily used in studies. Table 1 gives the structure of the population in accordance with the probability of disease propagation in Russia and some numerical characteristics of the infection risk based on expert estimates. Obviously, according to the social and economic conditions and the cultural tradition, the structure and size of the subpopulations may change. Thus, one task of the work is to investigate how this structure is being implemented in today’s Russia and how it is influenced by regional specific features. The described structure leads to more complicated dynamics of the process of the epidemic, to the appearance of epidemic phases, and the transfer between them. Definition 4. A phase in an epidemic is the state of a population at which the disease is transmitted mostly in the same part or in a few of its parts with a different specific risk of infection. According to the classification proposed by UNAIDS, three basic phases have been identified for HIV infection. The lowlevel phase is the period over which the rise of incidence is negligible. Then, over a shorter time period, a sharp growth in the number of infected is observed due to saturation of the infection 3 Capable of existing a long time and developing.

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Table 1. Structure of population and quantitative characteristics of the risk of HIV infection, Russia [7] Frequency of alternation Probability of infection of partners (partners per per contact person per annum) Core group Users of injection narcotics Groupbridge Commercial sex workers Men practicing sex with men Other vulnerable categories of citizens Basic population

5–10 50–80 3–5 5–10 0.5–1

Share in the total population

0.7–0.9

1–5%

0.001–0.02 0.003–0.05 0.003–0.05 0.001–0.02

1–5% 1–5% 5–10% 75–90%

in the core group. This is followed by the concentrated phase of with the slow penetration of the infection into the bridge group. With the saturation of this group, the propagation is increased by the penetration of the virus into the basic population; it is the beginning of the generalized phase. 3. MATHEMATICAL MODELS OF HIV DISSEMINATION The most advanced method of mathematical modeling of the dissemination of infectious diseases is that based on population groups. The population (compartmental or averaged) approach is based on the consideration of classes of individuals by their response to infection: receptive, infected, immune, etc. Further research is being done in terms of the numbers of these classes described by determined or sto chastic processes. The earliest attempts at forecasting the transmission of HIV infection were of actuary importance and carried out in terms of the simplest population models of the transmission of infectious diseases [8]. The results were obtained as early as in 1984 and had a highly unfavorable forecast, i.e., an explosive growth of the AIDS epidemic, implying that over a third of the population would be infected by the beginning of the 1990s. Soon it became clear that the theoretical assumptions of the models were insufficient to describe the dynamics observed in practice. Before the 2000s, the development of population simulation of the transmission of HIV was carried out in two directions: aggregated and structural models. The basic objective of the aggregate approach is the investigation of the role of the individual progress of the illness in the epidemic process and the possibilities of controlling HIV transmission by supporting therapy. Structural models were developed to investigate the role of various aspects of the inhomogeneity of the population (age of carriers, virulence of strains, etc.) in HIV transmission.The development of these two approaches is closely linked to Anderson and May, CastilloChavez [9], Longini and Satten [10], Blower [11], and several other authors. In order to solve the problems of controlling HIV infection, such as identifying those infected and developing vaccines, models were developed on the basis of a combined approach. The analytical study of the combined models was carried out in full in the papers by Hyman, Li, and Stanley [12]. After 2005, the development of population models of HIV transmission was aimed at assessing the role of the corebridge structure of the population in the dynamics of the epidemic. The works of Yu.H. Hsieh are the most significant in this area [13]. Imitation modeling, i.e., investigating the integral characteristics of the population as a result of a set of local interactions, has been gradually acquiring popularity also in the application to the problems of the analysis of the dynamics of the HIV epidemic in the works of Van De Vijver [14], as well as Sloot and Boukhanovsky [15]. Among the drawbacks of the existing works in the field of projecting HIV transmission, the following should be noted: identifying the parameters of most of the models is carried out on the results of sampling studies of individual parts of the population; the risk of infection of an individual is supposed to be con stant; the entry of receptive individuals into the risk groups and of the infected ones into the basic popu lation (risk dynamics) are not taken into account. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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Table 2. Values that are linearly and nonlinearly correlated with the incidence of HIV infection [1] Symbol

Name

Dimensions

λS λD λA λex

ISTI (measured by the incidence of syphilis)4 Speed of drug addiction occurrence Speed of alcoholism occurrence Ratio of yearly consumer expenditures to the yearly cost of consumer basket

Pmin Pst

Yearly subsistence minimum per person related to regional average Portion of undergraduates in the structure of total population

1/t

dimensionless

Table 3. Classification of Russia’s regions by HIV epidemic factors

Designation of class

Lo Socially adapted people in the population

CA Alcoholism

CS Promiscuity

Ge Socially disadapted people in the population

Supposed phase of epidemic

Low level

Concentrated

Concentrated

Generalized

Speed of transmission Alcoholism ISTI

Low Low

High Low

Low High

High High

HIV epidemic factors Epidemic expan sion Inhibition

Drug addiction, ISTI Alcoholism

Drug addiction Drug addiction, ISTI alcoholism Alcoholism ISTI none

4. ANALYSIS OF THE DATA ON HIV TRANSMISSION IN RUSSIA The results of the analysis of the statistical data are given in [1, 2]. This work supplies the basic results. Definition 5. The vigor of infection λ is the specific rate of contagion of a receptive individual [16]. This indicator has dimensionality of 1/t and is the proper characteristic of the risk of infection. Assum ing that the properties of the germ are the same in all regions and do not change within the period of obser vation, the intensity of infection can be estimated using the data of disease incidence and prevalence: ΔI (t) 4 λ= , N (t) − I (t) where ΔI is the morbidity, the number of persons infected over time period (t; t +1); N is the size of the population; and I is the incidence of the number of infected people at the beginning of period t. The lack of immunity to HIV and the low level of people’s contamination make this approximation quite reliable. In Definition 5 contagion is understood only as the change in a person’s condition but not the mech anism of this change. According to this, the notion of the intensity of infection can be used to describe the transition into the state of alcoholism and drug addiction. In the result of the statistical analysis, the intensity of HIV infection can be bound with the values given in Table 2. The obtained estimate was named by us as the index of risk of infection by HIV: (1) λ = 0.81λ D λ S λ A + 0.85λ ex Pst (1.6 − 0.33Pmin + exp{1 − Pmin }). In (1) 69% of the interregional variation of the HIV infection’s intensity is accounted for by the first term. Definition 6. Social disadaptation is the maladjustment of individuals to the social environment. Factors (indices) of social disadaptation of the population are named by us as the values λS, λD and λA in (1) and in Table 2. In substance, what is common in these indices is that they characterize different forms of antisocial behavior that are signs of social disadaptation at the individual level. The degree of an individual’s social disadaptation may considerably vary, change with time, and influence the epidemic characteristics of the individual: the time of contagion, the diagnostics of the disease, and the quality of medical treatment. At the level of the population, the socially disadapted people are included in the core and the basic part of the 4 Measured by the incidence of syphilis.

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HIVinfection bridge group. Thus, it is important to estimate the risk of contagion for the socially adapted population. The dependence of the intensity of HIV infection on these indices is dubious. Using this observation as a criterion, it is possible to classify Russia’s regions by the ways the risk of HIV infection is formed (Table 3). The algorithm of the classification and the classes themselves are described in detail in [1, 2]. By summing the results of the data analysis with classification of the epidemic phases, it is possible to identify corre spondences. What is essential is that for Russia there are two versions of the concentrated phase: narcotic addiction (CA) and sexual (CS). Drug addiction and alcoholism have a contradictory effect on the HIVinfection epidemic process in some regions (Lo and CA). They compete with the population forming the core group. At the same time, in the CS group, both processes lead to an increase of the epidemic. This phenomenon fails to be included in any of models on the spread of HIV known to us. We should also note the existence of the Ge group of regions, where the factors of epidemic contain ment are absent, and the infection transmitted sexually is a growth factor. This allows us to assume the African scenario of the development of the HIVinfection epidemic in these territories. Thus, the process of alcoholism is likely to considerably affect the dynamics of the size of the core group of HIVinfection. This is typical of Russia and must be taken into account in the construction of the model. 5. MATHEMATICAL MODEL OF SEXUALLY TRANSMITTED INFECTIONS (STI) WITH THE DYNAMICS OF THE RISK GROUPS The spread of sexually transmitted infections (including HIV) is ordinarily described by the SImodels [5]. Let a population sized N(t) have n risk groups, each with Ii (t) infected and Si (t) = Ni (t)−Ii (t) receptive individuals. Then, the dynamics of the population size can be described by a system of equations of the following type n ⎧dS i S = f i − μ i S i − ωi pij I j S i , ⎪ ⎪ dt j =1 ⎨ n ⎪dI i I ⎪ dt = ωi pij I j S i − μ i I i , i = 1, n. j =1 ⎩



(2)



The number of nonresistant individuals in the ith group can increase due to the inflow into it of new individuals from the outside of the population fi (for example, by teenagers’ growingup) and decrease by the death of individuals with coefficient µ iS and infection. The number of infected persons in the ith group increases as a result of infection of receptive people and decreases as a result of the death of individuals with coefficient µ iI . The term describing the contamination consists of two components: the matrix of probabilities of the formation of P couples and the vector of probabilities of effective contacts ω. At the level of the popula tion, each component of matrix P characterizes the shared distribution of the risky interactions of each group with other parts of the population, and each component of vector ω characterizes the specific rate of contamination. A fairly large number of works have been devoted to this class of models and a large number of analyt ical results have been given. Averaged models of type (2) assume that the time since the moment of infection to the end result of the disease (patient’s recuperation or death) and the time between infections is short enough for the person to manage to change his behavior. For the case of HIV infection, it is reasonable to take into account the pos sibility for the person to migrate from one risk group to another. n n ⎧dS i S = fi + α ij S j − ωi pij I j S i , ⎪ ⎪ dt j =1 j =1 ⎨ n n ⎪dI i I α ij I j , i = 1, n. ⎪ dt = ωi pij I j S i + j =1 j =1 ⎩







(3)



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Table 4. Model variables: names of groups, basic route of virus transmission Group

Receptive

Infected

Route of transmission

Socially adapted Increased risk of addiction Chronic alcoholism

SG SB SA

IG IB IA

Sexual

Drug addiction

SD

ID

Parenteral

Coefficients α, relating to noninfection dynamics in the population, fulfill the following conditions: (1) α iik ≤ 0, (2) ∀j ≠ i α ijk ≥ 0, ⎛ ⎞ (3) α iik = − ⎜ μ ik + ∑ α kji ⎟ , ⎜ ⎟ j ≠i ⎝ ⎠ k where µ i is the constant of outflow (death or aging) of persons from the population in a group with index i. Systems of type (3) are used for constructing models far less frequently. We have failed to find works generalizing the properties of this class of models. The existence, singularity, and continuous dependence of the solution of system (3) on the initial data and parameters can be easily testified by showing that the model equations satisfy the conditions of the theorem formulated in [17]. The model proposed by us to describe the process of the spread of the HIV infection in Russia is a spe cial case of system (3). Note that such models of epidemic processes present a rather rough approximation to reality because they ignore demographic, social, and economic factors that have been in force for more than 20–30 years. Changes in age and the social structure of a population and in behavioral preferences and social norms that take place within such time intervals may considerably change the values of param eters and the set of variables determining the dynamics of the epidemic process. Thus, the study of the asymptotic properties of the model solutions are not informative.

6. A MODEL OF THE SPREAD OF HIV DUE TO SOCIAL DISADAPTATION The population is structured in terms of substance abuse risk: drug addiction or alcoholism (Table 4). Groups in index A included persons diagnosed with chronic alcoholism; groups in D included people with drug addiction; groups in B included people for whom no substance abuse in the acute phase was found by the medical institution but the effect of risk factors were prevalent over the factors of protection; groups in G included socially adapted persons. It is assumed that the identification of drug addiction and alcoholism is constructed in such a way that groups D and A do not intersect. The model is a system of eight ordinary differential equations, each describing the dynamics of the sizes of groups. dS G = f b − (γ B + μ G )S G + β B S B − ωG ( pGG I G + pGB I B + pGA I A + pGD I D )S G , dt dS B = f m + γ B S G − (β B + λ A + λ D + μ B )S B + β AS A + β DS D dt − ωB ( pBG I G + pBB I B + pBA I A + pBD I D )S B , dS A = λ AS B − (β A + μ A )S A − ω A ( p AG I G + p AB I B + p AA I A + p AD I D )S A, dt dS D (4) = λ DS B − (β D + μ D )S D − ωD ( pDG I G + pDB I B + pDA I A + pDD I D )S D, dt dI G = ωG ( pGG I G + pGB I B + pGA I A + pGD I D )S G − (μ G + μ I )I G + β B I B , dt dI S = ωB ( pBG I G + pBB I B + pBA I A + pBD I D )S B − (μ S + μ I )I B + β A I A + β D I D, dt MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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385

SA Chronic alcoholism

SB High risk SD Drug addiction

ID Drug addiction IG Socially adapted

IA Chronic alcoholism

IB High risk

Coregroup

Bridgegroup Fig. 2. Scheme of the person’s transition states in the model. The dots show the core group and groupbridge; the gray hue identifies the submodel of social disadaptation.

dI A I = ω A ( p AG I G + p AB I B + p AA I A + p AD I D )S A − (μ A + μ + β A )I A, dt dI D I = ωD ( pDG I G + pDB I B + pDA I A + pDD I D )S D − (μ D + μ + β D )I D, dt Figure 2 shows a diagram of a person’s transitions between states that correspond to the equations of the model. The movement of individuals among states with a different level of social disadaptation is the submodel of social disadaptation. The influx of new individuals in the population is possible as the result of living up to the age of 15 years and inmigration from other territories but only of receptive individuals. Data from the Federal Scientific Methodical Center (FSMC)5 indicate a minor contribution of migration to the epidemic’s dynamics, and cases of survival of infected children under 15 years were not observed. The reduction of the population is the result of death, people leaving to go to prison, and migration to other areas. In the construction of the model we assumed that the process of social disadaptation can be described in most cases as sequential transitions between states with relatively close levels of social adaptation. At first the adapted individual may be found under the influence of risk factors to develop an addictive pathol ogy. Being in this state, the individual may be diagnosed as suffering from chronic alcoholism or drug addiction. At the same time, the individual may overcome the action of these factors and return to the group of socially adapted individuals. An individual from any group can be HIVinfected. Inside each group the individuals are indistin guishable by infection properties: virus transmissibility and the number of contacts. Differences are dis covered between members of groups with different levels of social disadaptation. In Russia among HIVinfected persons, a considerable number of measures are performed for their social adaptation, which may lead to considerable flows of infected individuals to more adapted groups. This process is the transition of the infection from the core to the basic population. 5 Federal Scientific Methodical Center for the fight against the spread of HIV/AIDS.

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Table 5. Parameters and initial data of the model: classification, names, variables Symbol

Name

N

Total population

NA

Chronic alcoholics

ND kB

Drug addicts

fb fm μG μB μA μD μI

γB λA λD βB βA βD ωG ωB ωA ωD pGB pAB pDB

Interval

Dimension

Initial data (number) [1–3] × 106 [2–8] × 104

persons

[0.5–2] × 104 0–1 dimensionless Flows of individuals from the outside Natality flow [2–6] × 104 persons/year Migration flow [0–1] × 104 Individuals’ outflow from the population (speed constants) Deaths from all causes other than AIDS, as well as alcohol and drug [5–11] × 10–3 abuse Outflow in the group of high risk [3–4] × 10–2 Outflow of patients with chronic [4–16] × 10–2 1/year alcoholism Outflow of drug addicts Deaths caused by AIDS

[4–11] × 10–2 [4–11] × 10–2

Source of data Rosstat6 [18] Russia’s Ministry of Health and Social Development (MHSD)7 [19] Estimate (5) Rosstat [18]

Rosstat [18] Russia’s Ministry of Health and Social Development (MHSD)[19] FSMC (Federal Scientific Methodical Center)[7]

Parameters of social disadaptation (constants of velocity) Of social adaptation [0.1–6] × 10–1 1/year Estimate (5) 3 Russia’s Ministry of Health Development of chronic alcoholism [1–6] × 10 persons/year and Social Development Development of drug addiction 50–1000 (MHSD)[19] Parameters of social adaptation (speed constants) Of social adaptation [0–12] Estimate (5) –2 Remission of chronic alcoholism [1–7] × 10 Russia’s Ministry of Health 1/year and Social Development Remission of drug addiction [0.5–1] × 10–2 (MHSD) [19] Parameters of contagion (specific speeds of contagion) Socially adapted individuals 5 × 10–8 Receptive individuals [2–20] × 10–6 According to the data (persons/year)–1 of Table 1 and estimates (6) Chronic alcoholics 5 × 10–4 3.5 × 10–4 Parameters of the bridgegroup’s boundary penetrability With general population (G) 0–1 With alcoholics (A) dimensionless Estimate (6) With core group (D) Drug addicts

67

7. ESTIMATING PARAMETERS BY THE REAL DATA Table 5 provides classification, names, and some characteristics of the model parameters which we have used in the numerical solution of the Kochi problem for system (4). 6 Federal State Statistics Service of the Russian Federation. 7 The Ministry of Health and Social Development of the Russian Federation.

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From Table 5 it is seen that the parameters of the submodel of social disadaptation, kB, γB, and βB, have no source of data, while those parameters whose values can be obtained from the annual statistical state ments in the most complete form are given for 2009 and 2010 because the forms for the collection of infor mation have undergone a number of changes and supplements. In order to obtain estimates for these parameters, we have considered a model of the processes of social disadaptation without HIV infection into account: dN G = f b − (γ B + μ G )N G + β B N B , dt dN B = f m + γ B N G − (β B + λ A + λ D + μ B )N B + β A N A + β D N D, dt dN A = λ A N B − (β A + μ A )N A, dt

dN D = λ D N B − (β D + μ D )N D, dt The analysis of the data allows us to assume that out of 40 regions, in which statistics on drug diseases are prepared, at least in 7 of them in the past 3–5 years socialeconomic stability has existed, and was expressed in the quasistationary distribution of individuals by groups A and D. Then, the lacking param eters of the submodel of social disadaptation can be estimated by minimization of the difference between the analytical and observed solutions.

⎧ ( AT A)−1 AT f − N 0 → min, ⎪ ⎪0 < kB < 1, ⎨ −1 ⎪0.03 < γ B < 12 (year) , ⎪ −1 ⎩0.03 < β B < 12 (year) ,

(5)

where

0 0 βB ⎛ −(γ B + μ G ) ⎞ ⎜ ⎟ γB −(μ B + β B + λ A + λ D ) βA βD ⎟, A=⎜ λA −(μ A + β A ) 0 0 ⎜ ⎟ ⎜ ⎟ 0 λ 0 − ( μ + β ) ⎝ D D D ⎠ f = [ f b f m 0 0]T , N 0 = [(1 − k B )N − N A − N D kB N N A N D ]T is estimated by the observable distribution of individuals among groups. The specific velocities of individuals’ contagion were estimated by expert data. The group of increased risk (B) consists of population categories whose specific risk of contagion is substantially different (see Table 1). The value ωB significantly affects the dynamics of the epidemic in the model population. Thus, instead of the expert estimate this parameter was included in the optimization task (6). For reasons of symmetry (the number of contacts of group i with group j must be equal to the number of contacts of group j with group i), as well as the requirement that the total percentage of each group inter actions with others is 1, matrix P in (4) is completely determined by the three entrance parameters of the bridge group (Table 5) and has the form

pGB 0 0 ⎞ ⎛1 − pGB ⎜ p c p c + p AB c A + pDB cD p AB c A pDB cD ⎟ ⎜ GB G 1 − GB G ⎟ P = ⎜ cB cB cB cB ⎟ . ⎜ 0 1 − p AB 0 ⎟ p AB ⎜ ⎟ 0 1 − pDB ⎠ pDB ⎝ 0 MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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Table 6. Parameters of permeability and specific rates of transmission of the groupbridge, share of the size of the bridgegroup, alcoholics and the core population for the regions in Russia Subject of the Russian Federation

Class

Nizhni Novgorod oblast

pGB

pAB

pDB

ωB ×10–6 (year/person)–1

kB

kA

kD

0.01

0.01

0.5

0.84

0.05

0.03

0.002

0.9

0.3

0.8

2.6

0.23

0.08

0.008

0.4

0.1

0.6

5

0.14

0.05

0.003

0.01

0.01

0.01

0.74

0.22

0.02

0.008

0.02

0.07

0.01

0.76

0.27

0.03

0.004

0.1

0.01

0.9

0.24

0.42

0.02

0.005

Ge Irkutsk oblast Ulyanovsk oblast

CA

Samara oblast CS Omsk oblast Krasnodar krai

Lo

In order to estimate the contagion parameters, problem (6) was formulated to find such values of the unknowns which minimize the sum of relative deviations of the number of the infected in system (4) from the observed values



0 0 ⎧ε = I (t i ) − I (t i ) I (t i ) → min, ⎪ i ⎪ ( p c ⎨ GB G + PAB c A + pDB c D ) c B ≤ 1, ⎪0 ≤ kiB < 1, i ∈ {G, A, D}, ⎪ −4 −1 ⎩0 ≤ ωB ≤ 0.1 × 10 (person s year) ,

(6)

where I 0 are the observed values of the number of HIVinfected people. The time moments ti correspond to 2000–2009 and are taken with an interval of 1 year. The utility function of problem (6) is given gridlike and the analytical expression is unknown. In order to obtain the solution in the first approximation, an algorithm of the gridlike minimization with a fraction of the grid step in the minimum neighborhood was used. The results for six regions of Russia are given in Table 6. In Table 6 it is possible to trace the following relation of the region’s classification and model (4) parameters. Class Lo is characterized by low values of the specific velocity of contagion in the bridge group and of the parameter of permeability with the base population. In class CS the bridgegroup is relatively isolated from the rest of the population. Class CA has high values of all parameters of permeability and of the specific rate of contagion in the groupbridge with relatively low sizes of the core and bridgegroups. The percentage of the group with alcoholism is rather high. Class Ge regions have high values of the per meability parameter for the core group. 8. MODELING RESULTS Considering the data on the spread of HIV infection in different countries (Fig. 3), we can distinguish cer tain patterns. In Canada and Portugal, HIV infection is spreading slowly and has not yet overcome the lowlevel phase. In Cambodia and India, despite the different scale of the epidemic, a similar phenomenon of exhaustion can be seen for 1995–2000 when the number of cases coincided with the size of the population at risk of infection. The subsequent decline was the result of the limited size of this population due to nat ural conditions or the influence of therapeutic measures. For the Russian Federation, in Fig. 4, it can be seen that in 2000 there was a phase transition from a low level to a concentrated epidemic. For the study of the behavior of the model solution on the real data, for Russia, the following regions were taken into consideration: Samara, Nizhni Novgorod, Ul’yanovsk, Irkutsk and Omsk oblasts, and Krasnodar krai. The choice was based on the quality of the available statistical data and on their closeness to the assumption about the stability of the model parameters. At first the calculations were made on the assumption that the initial distribution of receptive individ uals among the groups of social disadaptation corresponds to the 2010 data of observation. This means MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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0.8 0.7

Spread, %

0.6 0.5 0.4 0.3 0.2 0.1 0

1990

1995

2000

2005

t, years The Russian Federation

Canada

India

Cambodia

Portugal

Fig. 3. Dynamics of HIVinfection transmission in countries estimated by UNAIDS. (Incidence is measured at % of HIVinfected to total population).

that the core group at the moment of the appearance of the virus in the population had been already formed. The results given in Fig. 4 show that under this assumption the model poorly describes the data in four out of the six selected areas. The following reasoning has helped improve the quality of approximation (Fig. 5). At the initial stage of the spread of HIV in Russia (the 1980s), the virus was circling mostly within the homosexual population and was spread sexually. This group, small in number (Table 1), was greatly socially isolated from both the main population and from the drug addicts. Thus, before the 1990s, the effective size of the core group can taken as zero. In Samara oblast and Krasnodar krai, the first case of HIV infection was recorded before 1990. We assumed that the initial number of the core (of drug addicts) was zero. Later (in the 1990s), the infection was carried to the group of patients with drug addiction, and HIV began to spread parenterally. In Omsk and Nizhni Novgorod, the first case of HIV infection was recorded after 1990. We assumed that in these regions at that moment the core group was the half the size it had grown to in 2010. For these four areas, problem (6) was recalculated. Ultimately, the parameters and initial data have the values shown in Table 7. The permeability parameters in Omsk oblast, a typical member of the CS class, suggest the high inten sity of infection transmission from the bridge to the base population through sexual contacts. In combi nation with the high permeability of the bridgecore boundary, the conditions from rapid transition from the phase of concentration to the generalized phase are created. Samara oblast is found to be the closest to the criterion of classification and has the properties of all the classes, which is reflected in the values of the permeability parameters. The results of the estimate of the permeability parameters for the unharmed class Lo (Krasnodar krai) also correspond to the assumption about the high degree of isolation of the base population from the core of the epidemic in this class of territories. The variants of the epidemic’s dynamics in different countries (Fig. 3) are also observed (Fig. 5) in the result of the numerical calculations of the model for the regions of Russia. The character of the dynamics of the total spread is determined by the parameters of the spread of infection in particular components of the population (core group, bridge group, base population). Of the regions from classes CS, CA, and Lo, it is typical that as of 2000, the priority in epidemic dynamics has movesd from the core group to the bridge group. After 2000, in the regions from class Ge, MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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NOSOVA, ROMANYUKHA Samara oblast

Omsk oblast

2

Spread, %

Spread, %

3

1 0 1990

1995

2000 t, years Irkutsk oblast

2005

0.4

0.2 0 1990

2010

2005 2000 t, years Krasnodar krai

1995

2010

1995

2010

2.0 0.6 Spread, %

Spread, %

1.5 1.0 0.5 0 1990

1995

2000 2005 t, years Ulyanovsk oblast

0.4 0.2 0 1990

2010

2000 2005 t, years Nizhni Novgorod oblast

1.0

Spread, %

Spread, %

0.4

0.5

0.3 0.2 0.1

0 1990

1995

2000 t, years

2005

2010 Model

0 1990

1995

2000 t, years

2005

2010

Real data

Fig. 4. The model solution behavior in 1990–2010. The values of the incidence for all the curves are given in % to the total population. The core was formed before the beginning of the epidemic.

the main contribution to the number of the HIVinfected people has been made by the base population, and the role of the bridge group in the epidemic’s dynamics is not high over the whole period of observa tions. Despite the considerable improvement in the quality of approximation of the real data to those of the model, we note the systematic underestimate of the scale of the epidemic at the end of the period of observations. This can be due to different causes; for example, to an underestimate of the sizes of the core in the submodel of social disadaptation. The optimization task (6) can be considerably improved by supplementing it with the analysis of the data for a longer time period. We have had only two points accessible to us. We also note that the real data for the period before 2000 cause some doubt as to their correspondence with the processes in the population. At that moment, the system of discovery of HIV infection had not yet been established in Russia. The quality of approximation could be improved by the introduction of weight coefficients for the data relating to the period before and after 2000. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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MATHEMATICAL MODEL OF HIVINFECTION TRANSMISSION Samara oblast

Omsk oblast 0.4 0.3

2

Spread, %

Spread, %

3

1 0 1990

1995

2000 2005 t, years Irkutsk oblast

0.2 0.1 0 1990

2010

1995

2000 2005 t, years Krasnodar krai

2010

1995

2000 2005 t, years Nizhni Novgorod oblast

2010

1995

2010

0.4 Spread, %

3 Spread, %

391

2 1

0.3 0.2 0.1

0 1990

1995

2000 2005 t, years Ulyanovsk oblast

0 1990

2010

0.4 Spread, %

Spread, %

3 2 1 0 1990

1995

Population

2000 t, years

2005

Core

2010

0.3 0.2 0.1 0 1990

General population

2000 t, years Bridge

2005 Real data

Fig. 5. The behavior of the model solution in 1990–2010 after the change of the initial conditions. The values of the inci dence for all curves are in % to the total population.

Finally, there is still another possible cause for the mismatch between the model solutions and the data, i.e., more frequent cases of misrepresentation of the data and duplication of the records of infected indi viduals due to the features of financing being dependent on the number of people infected. 9. CONCLUSIONS The objective of this work was to develop tools for the effective control of the spread of HIV in Russia. Our analysis of the data and of other accessible materials on this topic has allowed us to classify regions by factors of the formation of groups of risk for HIV infection that includes the dynamics of the individual risk of infection. The results of the numerical solution of the Kochi problem for the formulated model indicate the exist ence in the population dynamics of HIV infection of the phases of the epidemic, whose characteristics (maximum incidence, duration) depend on the socioeconomic features of regions and are associated with the constructed classification. The presented model and algorithms of its identification need further improvement. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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NOSOVA, ROMANYUKHA

Table 7. Data for model computations. T is the time interval from the moment the first case of HIV infection was recorded to 2010 Samara oblast T NG(0) NB(0) NA(0) ND(0) fb

Year

23

×106 person ×104 person ×104 person/year

fm μG

×10–3 1/year

μA μD

×10–2 1/year

μI γB γA γD βB βA βD

×10–2 1/year ×10–3 1/year 1/year ×10–2 1/year

1.2

0.6

0.15

0.48

0.16

1.8

0.46

4.8

8.7

3.6

2

7.9

2.4

0

0.24

1.3

0.45

0

0.35

3.3

3.4

3.6

1.6

5.4

2.6

0.56

0.35

0

0

1.8

0

4

7

5

4

3.4

3.7

3.3

3.2

11

3.5

3.9

4.1

3.8

16

7.9

6.1

6.5

4.1

4.4

11

5

4.7

4.1

3.3

3.8

3

4

5

7

11

62

1

3

2

23

13

11

2

2

2

3

4

3

1

1

0.07

0.6

1.95

0.007

0.018

0.019

4.6

1.7

7.2

2.7

1.3

1.7

0.9

1

1.1

1.6

0.5

0.4

4.5

2.6

5

3.1

0.4

0

0.9

0.4

0

1

0.1

0

0.3

0.1

0

0.08

0.9

0.08

0.8

0.6

1

0.9

9.3

ωA

5

ωD

×10–4 (person/year)–1

pDB

15

2.4

×10–7

Dimensionless

23

Omsk oblast

0.97

×10–6 (person/year)–1

pGB

23

Krasnodar krai

1.6

ωB

(person/year)–1

19

Ulyanovsk oblast

2.7

ωG

(person/year)–1

Irkutsk oblast

2

×10–8

pAB

19

7

μB

Nizhni Novgorod oblast

1.9

2.5

5 19

3.5

The obtained results show the need to take into consideration the social processes in the description of the epidemiology of such infections as HIV. If a society is in a far from stationary state and its structure is changing, this unavoidably influence the dynamics of socially dependent diseases. Thus, the model should describe the common dynamics of the social and epidemic processes. The model described here is one of the earliest examples in the implementation of this approach at the intersection of sociology and epide miology. ACKNOWLEDGMENTS The work was financially supported by the Russian Foundation for Basic Research, grant no. 1001 00779a. MATHEMATICAL MODELS AND COMPUTER SIMULATIONS

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REFERENCES 1. E. A. Nosova and A. A. Romanyukha, Regional index of HIV infection risk based on factors of social disadaptation (RJNAMM, 2009), pp. 325–340. 2. E. A. Nosova, O. V. Obukhova, and A. A. Romanyukha, “HIV transmission and social disadaptation of the pop ulation in Russia,” SPID, Rak i Obshch. Zd. 14 (2), 13–32 (2010). 3. S. Alistar, D. Owens, and M. Brandeau, “Effectiveness and costeffectiveness of expanding drug treatment pro grams and HIV antiretroviral therapy in a mixed HIV epidemic: an analysis for Ukraine,” SPID, Rak i Obshch. Zd. 14 (1)(23), 44 (2010). 4. S. V. KupriyashkinaMcHill, “The effect of the grants from the global foundation on the policy in HIV/SPID in Ukraine,” SPID, Rak i Obshch. Zd. 14 2(24), 7–12 (2010). 5. K. L. Cooke and J. A. Yorke, “Some equations modeling growth processes and gonorrhea epidemics,” Math. Biosci., No. 16, 75–101 (1973). 6. V. P. Malyi, HIV. AID. Recent Medical Handbook (Eksmo, Moscow, 2009) [in Russian]. 7. V. V. Pokrovskii, N. N. Ladnaya, E. V. Sokolova, and E. V. Buravtseva, “HIVinfection” (FNMC SPID, Moscow, 2009) [in Russian]. 8. S. Haberman, “Actuarial review of models for describing and predicting the spread of HIVinfection and AIDS,” JIA, No. 117, p. 319–405 (1990). 9. C. CastilloChavez, Mathematical and statistical approaches to AIDS epidemiology (Springer Verlag, 1989). 10. M. Longini, W. S. Clark, R. H. Byers, J. W. Ward, W. W. Darrow, G. F. Lenp, and H. W. Hethcote, “ Statistical analysis of the stages of HIV infection using a Markov model,” Stat. in Med., No. 8, 831–843 (1989). 11. D. P. Wilson, J. Kahn, and S. M. Blower, “Predicting the epidemiological impact of antiretroviral allocation strategies in KwasuluNatal: the effect of urbanrural divide,” PNAS 103 (38), 14228–14233 (1989). 12. J. M. Hyman, J. Li, and E. A. Stanley, Sensitivity studies of the differential infectivity and stage progression models for the transmission of HIV (LAUR992253, 1999). 13. Y.H. Hsieh and K. Cooke, “Behaviour change and treatment of core group and bridge population: its effect on the spread of HIV/AIDS,” IMA J. Math. Appl. Med. Biol., No. 17, 213–241 (2000). 14. N. Bacaer, C. Pretorius, and B. Auvert, An agestructured model for the Potential Impact of Generalized Access to Antiretrovirals on the South African HIV Epidemic (Bulletin of Mathematical Biology, 2010). 15. P. M. A. Sloot, S. V. Ivanov, and A. V. Boukhanovsky, D. Van De Vijver, and C. Boucher HIV, in European Con ference on Complex Systems (Population Dynamics on Complex Networks, 2007), p. 1–2. 16. R. M. Anderson and R. Ì. Ìay, Human Infection Diseases: Dynamics and Control (Mir, “Nauchny mir”, 2004) [in Russian]. 17. K. K. Avilov, “Mathematical modeling of tuberculosis transmission and case detection,” Avtom. Telemekh., No. 9, 145–160 (2007). 18. Population of Russia (statistical book), Ed. A. E. Surinova (2009) [in Russian]. 19. Form No. Data on Patients with Alcoholism, Drug Addiction, Substance Abuse, Federal statistical observation [in Russian].

Translated by D. Shtirmer

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