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Научный журнал КубГАУ, №108(04), 2015 года
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УДК:510: 61
01.00.00 Математические науки
О МАТЕМАТИЧЕСКОЙ МОДЕЛИ ВЛИЯНИЯ НЕСОБЛЮДЕНИЯ ПРОФИЛАКТИЧЕСКИХ МЕР ПО ПРЕДУПРЕЖДЕНИЮ РАСПРОСТРАНЕНИЯ ВИЧ/СПИДА СРЕДИ ГЕТЕРОГЕННОГО НАСЕЛЕНИЯ
Басси Басси Эченг РИНЦ SPIN-код: 2099-2088 аспирант - Кафедра математических и компьютерных методов
Лебедев Константин Андреевич
д. ф.-м. н., профессор кафедры вычислительной
математики и информатики
РИНЦ SPIN-код: 6744-1690
Кубанский государственный университет,
Краснодар, Россия
В данной работе рассматривается математическая модель влияния несоблюдения профилактике ВИЧ/СПИДа среди гетерогенного населения, основанная на известную модель Kimbir et al (2006). Эффективность использования презервативов и последствия несоблюдения населением с профилактической меры (презерватив) являются целью данной научной работы. В этой работе, с определенными коэффициентами, нелинейных используется модель, которая состоит из системы шести дифференциальных уравнений для различных групп населения (шести группам населения) для получения модельных уравнений. По сравнению с существующей моделью Kimbir, предлагаемая модель с большой степени учитывает рождаемость изучаемого населения. Численное моделирование уравнений модели показывает, что сокращения скорости передачи ВИЧ/СПИДа могут быть эффективно достигнуты в течение определенного времени, и только там, где сравнительно высокая степень презерватив эффективность и высокий уровень соблюдения по восприимчивы и зараженным наблюдаются.
Из полученных результатов мы видим, что контроль ВИЧ/СПИДа в гетеросексуальной популяции зависит от чистой соответствие и эффективность рекомендованных профилактики (использование презервативов). В качестве рекомендации, модель ориентирована на интенсивное обучение и продолжающейся кампании по повышению информированности населения со стороны правительственных и неправительственных учреждений по эффективному использованию презерватива
Ключевые слова: ВИЧ / СПИД, ПРАКТИКА
UDC:510: 61
Mathematical and Computer Sciences
ON MATHEMATICAL MODEL OF THE IMPACT OF NON-COMPLIANCE WITH PREVENTIVE MEASURES FOR THE PREVENTION OF THE SPREAD OF HIV/AIDS AMONG HETEROGENEOUS POPULATION
Bassey Bassey Echeng SPIN-code: 2099-2088 postgraduate student of the Department of Mathematical and computer methods
Lebedev Konstantin Andreyevich Dr.Sci.Phys.-Math., professor of the Department of computational mathematics and Informatics SPIN-code: 6744-1690 Kuban state university, Krasnodar, Russia
In this article we consider a mathematical model of effect of non-compliance with the prevention of HIV/AIDS among a heterogeneous population based on known model by Kimbir et al (2006). The effectiveness of a condom use and implications of non-compliance with a population of preventive measures (condoms) are the aim of this research work. In this work, with definite coefficients, nonlinear model is used, which consists of system of six differential equations for different population groups (six groups of the population) to obtain the model equations. Compared with the existing model by Kimbir, the proposed model to a large extent, takes into account the birth rate of the studied population. Numerical simulation of the model equations shows that reducing the rate of transmission of HIV/AIDS can be effectively achieved within a certain time, and only where relatively high condom efficacy and high compliance by susceptible and infected are observed. From the obtained results, we can see that the control of HIV/AIDS in the heterosexual population depends on the net compliance and effectiveness of the recommended prevention (condom use). As a recommendation, the model focuses on intensive training and ongoing campaigns to raise the awareness of the population by governmental and non-governmental agencies on the effective use of the condom
Keywords: HIV/AIDS, CONDOM USE, NON-
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ИСПОЛЬЗОВАНИЯ ПРЕЗЕРВАТИВОВ, COMPLIANCE, EFFICACY, HETEROGENEOUS,
НЕСОБЛЮДЕНИЕ, ЭФФЕКТИВНОСТЬ, IMPACT, TRANSMUTE
ГЕТЕРОГЕННЫЙ, ВОЗДЕЙСТВИЕ,
ТРАНСМУТИРОВАТЬ
1. INTRODUCTION
The basic routes of the Human Immune Deficiency Virus (HIV) transmission between persons are now well understood and widely known, but the non-adherent to the application of recommended preventive measures which leads to HIV prevalence and trends among population has remained an area of debate and scientific research [1]. Diseases can be transmitted either horizontally or vertically. In the case of HIV, horizontal transmission occurs from direct contact between an infected individual and susceptible persons. Vertical transmission is as a result of direct transfer from mother-to-child or newborn offspring [2], as over 40% of all HIV cases result from mother-to-child transmission.
The Human Immune Deficiency Virus (HIV) is known to be the causative agent of the deadly disease called the Acquired Immune Deficiency Syndrome (AIDS), the extensive spread of which appears to have commenced in the early 1980s, [3]. In sub-Saharan Africa, over 2.5 million children under the age of 15, have died of AIDS as a result of been exposed to HIV during labor or breastfeeding. Thus, the impact of HIV transmission has been felt mainly in Africa [4], where the level of literacy is low, the poverty level is very high and the quality of health services is generally very poor.
The dynamics of HIV/AIDS has moved beyond the virus and risk factors associated with its transmission to a more detailed understanding of the mechanisms associated with the spread, distribution and impact of any intervention on the population [5]. Transmission modes of HIV/AIDS are basically via heterosexual contacts, blood transfusion, and contaminated injecting equipment [6].
Due to non-availability of any known medical cure for HIV/AIDS, therapeutic treatment strategies appear promising for retarding the progression of HIV-related diseases. In other words, prevention has remained the most
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effective strategy against the HIV/AIDS epidemics [7]. Control and intervention programs are focused towards educating the people on behavioral change. Other method of control include barrier contraceptives (i.e. condom use), which is a single strategy in the prevention of HIV/AIDS. Condom use, as a strategy to halt the HIV epidemics is considered in terms of its efficacy and compliance. The study of these two factors has been more rewarding through mathematical modeling.
Mathematical models of the transmission of HIV have proven to be useful in providing a logical structure within which to incorporate knowledge and test assumptions about the complex HIV epidemic [8]. Through mathematical modeling, the control of HIV/AIDS have been formulated as far back as 1987, when [9], developed simple function for the growth in the number of individuals who will develop AIDS and for the distribution of incubation period of those individuals [10].
Other models for the control of HIV epidemic include the following aspects: age-structure population, differential infectivity and stage progression; fully-integrated immune response model (FIRM), defense model, random screening, contact tracing, use of condom etc., for example, see [5,8,10,11,12, 13,14,15,16,17].
The motivating factor of this present work lies in the model [10], on a two-sex mathematical model for the prevention of the spread of HIV/AIDS in a varying population. In that study, the preventive measure (condom use) was, as well, introduced to the susceptible males among others. The model was aimed at studying the efficacy of the preventive measure which consequently leads to reduction of HIV infection within a given period of time. The impact or consequences of non-adherent to the usage of the preventive measure were conspicuously not treated.
New cases of HIV infection due to unprotected sex [18], indicate a high level of non-compliance in the application of preventive measure by both susceptible and infected individuals. In this present study, which is an extended version of the old model [10], we have developed a model for transmission
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dynamics of HIV/AIDS in a population of varying size with vertical and other demographical and epidemiological factors. Our purpose is to formulate a model for AIDS epidemic control measure that unveil and could transmute the impact of non-compliance and the trend of the efficacy of condom use in either horizontal or vertical HIV transmission. This work is devoted to the study of the impact of non-compliance by the population.
Consequently, in section 2, with improved coefficients and addition of novel features, we formulate a non-linear system of ordinary differential equations that model not only the efficacy of the condom but that which takes into account, the dynamics of non-compliance in the application of the preventive measure by both susceptible and infected males in a varying population. A transformation of the model equations into non-dimensionless proportions is carried out on section 3, followed by numerical simulation of the transformed equations in section 4. The last section covers a number of discussions, concluding remarks and recommendations.
2. THE MODEL EQUATION
In formulating the model equations, we consider figure 1(a & b) below:
B = c Bf-L
m mHf ^
P,Wm
Bf = Cf (Pm^T + P
N.
W
_m
*
)
(a) (b)
Fig. 1(a & b): A flow-chat of the model for male and female population for the prevention of HIV/AIDS in different groups of the population.
From figure 1(a), the functions (parameters) used in the model for the
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male populations are
Sm(f) -
4(0 -
Wm(f) -
Um(f) -
Nm(f) -
Bm(f) -
bNm -
V -
a -
P1 -
P2
S1
S2
ftm
в mm*
as defined below:
Number of susceptible males at time f;
Number of infected males at time f;
Number of infected males who use the condom at time f; Number of susceptible males who use the condom at time f;
Total population of males at time f;
The rate at which males are infected per unit time (incident rate);
Natural birth rate of male population, b > 0;
Natural death rate, m > 0;
AIDS - related death rate, ^ > 0;
The proportion of infected males who use the condom per unit time;
The proportion of infected males who initially use the condom and then decline per unit time;
The proportion of susceptible males who use the condom per unit time;
The proportion of susceptible males who initially use the condom and then decline per unit time;
Average number of contacts by males with females per unit time;
Probability of transmission by an infected males; and Probability of transmission by an infected males who use the condom.
Where
Nm (f) - Sm (f) + 4 (f) + Wm (f) + Um (f)
Using figure (1b), we also derive the functions (parameters) used in the model for the female population:
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Sf(t) - Number of susceptible females at time t;
If(t) - Number of infected females at time t;
Nf(t) - Total population of females at time t;
bNf - Natural birth rate of female population, b > 0;
Bf(t) - The rate at which females are infected per unit time
(incident rate);
^ - Natural death rate, m > 0;
a - AIDS - related death rate, a> 0;
cf - Average number of contacts by females with males per
unit time; and
ftf - Probability of transmission by an infected females.
Where
Nf (t) = Sf (t) + If (t)
The assumptions in this model follow from those made in the previous model [15], except that here, we assume that:
(i) susceptible individuals also use the condom, with a varying degree of
efficacy;
(ii) a number of susceptible males that use the condom initially, decline
thereafter, per unit time;
(iii) infected males who use the condom initially, decline thereafter per unit time; and
(iv) the impact of b, (natural birth rate), is only in the susceptible population.
In constructing the model, coefficients are considered positive, and the signs before the terms are taken with a plus if the arrow enters the area and with a minus sign, if the arrow goes out of scope. As can be seen, the model takes into account the birth rate b, natural population decline m and death from disease,a.. In this study, from the flow-chat (fig. 1 (a)), the model is governed by
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the following systems of non-linear ordinary differential equations:
S'm = bNmNf - Bmsm -msm - SSm + a2vm;
t'm = BmSm + PlWm - (P + a + P\)1m ';
K = p1Im- (r+m+a)Wm;
Um = SSm - (S + P)Um ';
(2.1)
(2.2)
(2.3)
(2.4)
From figure 1 (1b), we also derive the functions (parameters) used in the
model for the female population:
s' = bNfNm - BfS f-mSf (2.5)
If = BfSf - (m + a)If (2.6)
Differential equations is supplemented by algebraic equations
Nm = Sm + Im + ^m + Um (2.7)
Nf = Sf + If (2.8)
The coefficients Bm and Bf are expressed in terms of the relative I f I W
— and—^, —- of HIV-infected individuals:
Nf Nm Nm
B = c pf^~
m m^f N
IW
Bf = Cf (bmNT + bm Y-)
(2.9)
(2.10)
If p1 = p; o1 = o; p2, o2, and pm* are zero, then we obtain the model as in [15] and if only p1 = p; o1 = o; p2, o2 are zero with pm* Ф 0, we obtain the model as in [10]. Therefore, these parameters and the present equations (2.4) and (2.10), make the difference between models [10, 15] and the new model. The detail parameters are summarized as in table 1, below:
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Table 1: The original data are summarized in tabular format:
Symbols Expression for the males Symbols Expression for the females
Sm number of Susceptible males at time t; Sf number of susceptible females at time t;
Im number of infected males at time t; If number of infected females at time t;
Wm number of infected males who use the condom at time t; Nf Sf(t) + If(t) - total population of females at time t;
Um number of susceptible males who use the condom at time t; bNf natural birth rate of females population, b > 0;
Nm Sm(t) + Im(t) + Wm(t) + Um(t) - total population of males at time t; Bf the rate at which females are infected per unit time (incident rate);
Bm the rate at which males are infected per unit time (incident rate); M natural death rate, m > 0;
bNm natural birth rate of male population, b > 0; a AIDS - related death rate, a > 0;
M natural death rate, fi> 0; Cf average number of contacts by females with males per unit time;
a AIDS - related death rate, a > 0; Pf Probability of transmission by an infected female.
P1 the proportion of infected males who use the condom per unit time;
P2 the proportion of infected males who initially use the condom and then decline per unit time;
s the proportion of susceptible males who use the condom per unit time;
02 the proportion of susceptible males who initially use the condom and then decline per unit time;
cm average number of contacts by males with females per unit time;
Pm Probability of transmission by an infected males;
Pm* Probability of transmission by an infected male who use the condom.
3. TRANSFORMATION OF MODEL EQUATIONS TO NONDIMENSIONAL FORM
It is essential that for easy handling and to initiate the biological meaning of the proportions of infected individuals, which defines the prevalence of
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infection, we transform the model equations into proportions. Let,
Nm (0) = Nm, Nf (0) = Nf and m(t) = Nm .N f, f (t) = Nf .Nm
Then,
Sm = S,,/N,, , (3.1)
Sf = Sf/Nf , (3.2)
}’m = I„/Nm , (3.3)
Уf = l,/N,, (3.4)
«„ = U,,/Nm, (3.5)
»m = Wm/Nm (3.6)
Clearly, in terms of proportion,
m(t) = Nm (tVNm = Sm + ym + Wm + Um (3 .7)
and
f(t) = Nf (t)/Nf = Sf + yf (3.8)
The coefficients (2.9) - (2.10), in the dimensionless form are obtain as
Bm = C mff У f / f (t) (3.9)
Bf = Cf (fmym +fm*Wm )/m(t) (3.10)
Then equations (2.1) - (2.4) can be rewritten as
/
Sm = bm(t) - BmSm ~№m ~ S1 Sm +S2Um (3.11)
y'm = BmSm + PiWm ~ (m+tf+PXVm \ (3.12)
/
Wm = Pym - (P2 +P+a)Wm (3.13)
U'm = SSm - (S +P)Um (3.14)
Similarly, equations (2.5) and (2.6) can as well be rewritten as
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sf = bf(t) - Bfsf -msf (3-15)
/
yf = BfSf -(m + a)yf (3.16)
Thus, the changes in the various groups of the population are discussed on the basis of equations (3.11) - (3.16) as summarized below:
Table 2: Proportions of the dynamics of HIV transmission
Group Derivatives Eqn. no.
Sm / sm = bm(t) - Bmsm - ^m - SPm +S2Um (3.11)
1m у m = Bmsm + P2 Wm - (P + a + P1 ^m (3.12)
Wm / Wm = P1 ym - (P2 +m+a)Wm '; (3.13)
Um um = s1sm - (s2 + m)um m 1 m \ 2 Л^/ m (3.14)
Sf / sf = bf (t) - Bfsf -msf (3.15)
h / yf = Bfsf - (m + a)yf (3.16)
4. NUMERICAL SIMULATIONS AND ANALYSIS OF THE MODEL
The model represents systems of 6 differential equations concerning 6 considered groups of the population, sm, ym, wm, um, sf, yf with initial values and
parameters given in the table 3.
TABLE 3: T able of parameter values for Variants 1 -5
Variant b cc ffl cr2 cm cf дЗ Д.
i[*i 0.02 0.1 0 03 0 0.9 0 C. 5 0.1 0.14 0.14
2 QJQ2 0.04 0 0 0 0 0 C. 5 0.1 0.1 0.2
3 0.02 0.04 0.1 0 0 0 0 C. 5 0.1 0.1 0.2
4 0J02 0.04 0.1 02 0 02 0 C. 5 0.1 0.1 0.2
5 0)02 0,04 0.1 02 0.1 02 0.1 C. 5 0.1 0.1 0.2
V1 -Sm-Sm= J*(0) = 0.7
-'*=>■*<<>) = o.i
V4~UK -Um=*v(0') = 0.1
-,T* = 'M0)=0-1
V; — s j. — S f. j ДО) = o.s V- — у f —If. уf (01) = 01.2
Where initial valves of all characteristics
V - sm - Sm , sm (0) = 0.7
V - ym - Im , ym (0) = 0.1
V - Um - Um , Um (0) = 0.1
V - Wm - Wm , Wm (0) = 0.1
are constant.
V6 - sf - Sf, sf (0) = 0.8 V7 - yf - h,yf (0) = a2
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In the program we shall designate sm, ym, wm, um, sf, yf through a vector Y = {Y1,Y2, Y„ Y4, Y5, Yt }T ^ Y = {Yi }‘,1 ' Ym = f Y, and Yf = f Y, .
i=1
i=5
Then the system of differential equations can be written as
Г dY
— = f (Y), t e [0, T]
< dt .
Y (0) = Y0
To integrate the system, one can use different numerical methods. The simplest, is the Euler’s first order of accuracy. However, in MATHCAD, there is built-in function “rkfixed” realizing Runge-Kutter method of accuracy of order 4.
Importantly, it is essential to note that in analyzing the results of the computer simulations that follows, our interest is in the corresponding changes that are likely to occur in proportions of infected males and females (ym, yf), for
each of variant (1-5), as in table 3. The efficacy of the condom are determined by the behavior ofsyrand b*mwhile the non-compliance situation are determined by s2and ^respectively. These do not mean a play-down on other parametric factors. So that if Vw, represent the time interval t, (in years) and
from Table 2, letting V /[2},Vi?) ,Vw, V15 ,V <6and V1' 7,equal sm, ym,
m ’ J m ’ Wm , Um , S f , У f
and
/
yf respectively, then for bm* =bm1, the corresponding graphs obtained from Table 3, are represented as in figures (4.1) - (4.5), below:
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From figure 4.1 above, we investigate the simulation of the old model
[10], where natural birth rate were not defined in relation to the existence of interaction between the male and female population as presented by equations (2,1) and (2.5) of the present model. For compatibility, the only variation in that model is the birth rate (i.e. b = 0.02,ф 0.5). With high average number of contacts by both sexes (cm,cf); 50% - 90% condom use and compliance level not accounted for (i.e.s2 = 0 = p2), it is seen that the entire susceptible became infected within 10years, leading to rapid extinction of infected (without condom use) population within 20 - 30 years. Prolonged lifespan and increase in population were experience by the proportion of the males that used the condom. The entire female population was seen contaminated by the infection, leading to slow death after 50 years. The result indicates that condom use was not efficacious and coupled with non-compliance situation.
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а = 0.04,6 = 0,02,bm 1 = 0.1,bm = 0.1,b = 0.2. We observed initial increase in
infection by both male and female infected population, consuming the susceptible population after an average of 30 - 40 years. With no preventive measure (condom use), the situation is seen to graduate to slow death of the infected population.
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Here, if the situation in fig. 4.2, persist with natural death rate of 10% (i.e.a = 0.1), as seen in fig 4.3 above, then within a space of 10 years interval, both susceptible male and female population are seen to be consumed by infection which then translate to gradual extinction of the entire population after 30 years’ of time interval.
Furthermore, in fig.4.4 below, we investigate the effect of condom use (i.e.^ = 0.2 = r), alongside with other induced parameters as in fig. 4.3. We observed a drastic eradication of infection within 18 - 30 years for both the males and female population. The interval taken for complete eradication indicates the low level of condom application and variation of the efficacy of the condom.
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Finally, from fig. 4.5 above, we allow the observation as in fig. 4.4, to hold but introduce a non-compliance factor rate of 10% by both males without the condom and those using the condom (i.e.s2 = 0.1 = p2). It is seen that due non-compliance to effective use of the condom, decline of infection from both sexes were slow, taking about 30 years for complete eradication of infection. However, in about ten years on the average, the susceptible had been contaminated by the infection. The situation signified a wholesome noncompliance in condom use and portrays the varying level of condom efficacies.
5. DISCUSSION AND CONCLUSION
In this paper we have extended the model [10] by incorporating in the transmission of HIV, the study of the impact of non-compliance to condom use by both susceptible and infected male population. A non-linear mathematical model was used with the assumption that the population is a two sex and the condom is of varying degree of efficacy. Importantly, in order to appreciate the level of impact of non-compliance of the preventive measure, the model also took into account, natural birth rate b, natural death rate mand AIDS-related death ratea.We also established the fact that efficacy and non-compliance condom use is dependent on the general behavior of the parameters s, rand b*m for efficacy while non-compliance situation are determined by s2 and p2.
Using MATHCAD program, results obtained from the numerical simulation of the model equations indicate that eradication of HIV infection cannot not be achieve in a population whereby both the susceptible and infected exhibit a high level of non-compliance in the use of condom as the most effective control program. From the graphical analysis, it is shown that if the rate of compliance is high, the spread of the disease can be reduced significantly. This work is exploratory in nature, for the fact that, it is an improved model that can be used as a correction and conduct of relevant research works, as is in the case of [10] and as well allow the comparison of
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results.
Furthermore, it is evident from the numerical simulation that, for any effective program, the efficacy and compliance of the condom use should be sufficiently high. Achieving these findings, we recommend intensive and continuous education of the population through all and including media publicity by both government and non-governmental agencies on the effective use of the condom. Any other control measure, ranging from creation of adequate awareness programs, especially in the developing countries and including the study of female condom use, will be highly recommended.
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