A Dynamical Model of the Spread of HIV/AIDS and Statistical Forecast for HIV/AIDS Population in India

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1 Advances in Alied Mathematical Biosciences. ISSN Volume, Number 1 (011), International Research Publication House htt:// A Dynamical Model of the Sread of HIV/AIDS and Statistical Forecast for HIV/AIDS Poulation in India Anulekha Taadar 1, Koushik Ghosh and Kriasindhu Chaudhuri 3 1 Deartment of Mathematics Pailan College of Management and Technology Plot: -6, Sector-1, Phase I, Bengal Pailan Park Amgachia Road, Joka, Kolkata , India anulekha.ju@gmail.com Deartment of Mathematics, University Institute of Technology University of Burdwan, Burdwan , India koushikg13@yahoo.co.uk 3 UGC Emeritus Fellow, Deartment of Mathematics, Jadavur University Kolkata , India chaudhuriks@gmail.com Abstract Each year a large number of eole all over the world die from HIV/AIDS. Although there are many comlicating factors behind the sread of HIV, we still believe that relevant mathematical models can rovide a good insight of the dynamics of the sread of it. If we can rovide a satisfactory rofile of this dynamics, it will certainly hel government officials to make timely remedial actions. In the resent work, we have roosed a mathematical model of the sread of HIV. We have made a search for equilibrium oints for the system and discussed about their stabilities. On the basis of extensive analysis, relevant comments are made on mutual co-existence of the grou infected by HIV and the grou not infected by that. An effort is also made to evaluate the arameters involved in our model in the context of India. This ractical study reveals a forecasting rofile of the roortion and mass of HIV/AIDS infected eole in India in coming years. Keywords: HIV/AIDS, equilibrium oints, stability, mutual co-existence, HIV/AIDS oulation in India.

2 6 Anulekha Taadar et al Introduction The Human Immunodeficiency Virus or HIV belongs to the family of Retroviruses, whose genetic material is RNA. In 1984, researchers discovered the rimary causative viral agent, the human immunodeficiency virus tye 1 (HIV-1). In 1986, a second tye of HIV, called HIV- was discovered in West Africa, where it may have been resent decades earlier. Both HIV-1 and HIV- have the same modes of transmission and are associated with similar oortunistic infections. In ersons infected with HIV-, immunodeficiency seems to develo more slowly. Comared with ersons infected with HIV-1, those with HIV- are less infectious early in the course of infection [1]. HIV is transmitted by direct inoculation during intimate and unsafe sexual contact, esecially associated with the mucosal trauma of recetive rectal intercourse, transfusion of contaminated blood or blood roducts and sharing of contaminated needles or translacental or ostartum transmission from an infected mother to the fetus (by cervical or blood contact at delivery and in breast milk). HIV is not transmitted by casual household or social contact []. Human CD4 + T, lymhocytes, macrohages, microbial, dendritic and langerhans cells are believed to be targets for HIV-1 infection. The main target is CD4 + T heler cell, a tye of T cell; T-cells are an imortant art of the immune system because they hel to facilitate the body s resonse to many common but otentially fatal infections. By ways that are not yet comletely understood, HIV s life cycle directly or indirectly causes a reduction in the number of T-cells in the body, eventually resulting in an increased risk of infections. Over time there are not enough T-cells in the body. At this stage, a erson is said to have Acquired Immuno Deficiency Syndrome or AIDS, and becomes suscetible to infectious that a healthy immune system could deal with. The time in between the first infection and initiation of antibody synthesis is 6-1 weeks. The medium time to receive an AIDS diagnosis among those infected with HIV is 7-10 years [1]. We have strong evidences [3] that infectivity of HIV infected is not constant, rather it deends on the stage of infection and viral load. On the basis of most of the studies on this toic the transmission robability is highest at the early stage of infection.considering the onset time of AIDS from HIV, Mukandavire et.al [4] took into account the time delay in incubation eriod and exressed incubation eriod as a function of time t. Kalan [5], Greenhalgh [6], Lewis and Greenhalgh [7] and Bobashev et al. [8] develoed mathematical models of the sread of HIV by sharing of contaminated needles. In the resent work, we have worked in a larger ersective by constructing a mathematical model of the sread of HIV taking into account unsafe sex, transfusion of contaminated blood or blood roducts or sharing of contaminated needles and birth and death in the grou not infected by HIV as well as in the grou infected by HIV which is an advancement of the model roosed by Taadar and Ghosh [9]. In addition to this, here we concentrate on creating a ractical rofile by using the rior data to estimate the osterior results in the context of India. Rao [10] identified that there is a lack of scientific means of gathering information about the infectious eriod of HIV. Godbole and Mehendale [11] resented a vivid rofile of sread of

3 A Dynamical Model of the Sread of HIV/AIDS 63 HIV/AIDS in India and about the reventive measures taken. Nagelkerke et al. [1] described a dynamic comartmental simulation model for HIV/AIDS eidemics in Botswana and India develoed to identify the best strategies for reventing sread of HIV/AIDS. Chaturvedi [13] roosed a Neuro-Fuzzy aroach to develo dynamic model of HIV oulation in Agra region in India. Though major risk grous have been identified but still the actual icture of the dynamics of HIV in India is not transarent. We have, in our hand, the details of the sread of HIV/AIDS in India for last few years [14, 15, 16 17, 18, 19, 0, 1]. This data is used carefully to forecast the total scenario of the resent interest for the coming few years. Formulation of the Model To construct the model we assume the following: I = Number of ersons carrying HIV at time t S = Number of ersons not carrying HIV at time t P 1 = Mass of the oulation, initially not infected by HIV, going for unsafe sex P = Mass of the HIV infected oulation going for unsafe sex P 3 = Mass, initially not infected but getting infected by HIV due to transfusion of contaminated blood or blood roducts or by sharing of contaminated needles b = Birth rate er individual in the grou not infected by HIV d = Death rate er individual in the grou not infected by HIV b = Birth rate er individual in the grou of HIV infected d = Death rate er individual in the grou of HIV infected. We consider the governing equations [9] ds = αp1p βp3 + bs ds (1) dt di = αp1p + β P3 + bi di dt where α, β are the arameters characterizing the sread of HIV. Again we have, P 1 = Q 1 S P = Q I () P 3 = Q 3 S where Q 1, Q and Q 3 are corresonding roortionality factors such that Q i (0, 1) for i=1,,3. Initially let there be n individuals not infected by HIV and a individuals infected by HIV, i.e., S (0) = n, I (0) = a. So we get,

4 64 Anulekha Taadar et al ds = α. QSQI 1. β. QS 3 + ( b d) S dt and (3) di = α. QSQ 1. I+ β. QS 3 + ( b d ) I dt with S (0) = n, I (0) = a ds or, SI qs rs dt = + and (4) di = SI + qs + r I dt with S (0) = n, I (0) = a where = αq 1 Q, q = βq 3, r = b d and r = b d. Search for Equilibrium Points For equilibrium oint, we must have ds dt di = 0 and = 0 dt i.e., SI qs + rs = 0 (5) si + qs + r I = 0 Solving this system, we get the oints of equilibrium as (0, 0) and r ( q r q, r. The second equilibrium oint exists only if r > q and r < 0 to maintain the non-negative identity of both S and I. Analysis of Stability of Equilibrium Points An equilibrium oint is considered to be stable if the system always returns to it after small disturbances. If the system moves away from the equilibrium after small disturbances, then the equilibrium is unstable. The number of eigen values is equal to the number of state variables. In our case, there will be two eigen values. If both the eigen values are real, then the equilibrium oint is said to be a node and if they are conjugate comlex numbers, then it is said to be a focus. If both the eigen values are

5 A Dynamical Model of the Sread of HIV/AIDS 65 ositive, then the equilibrium oint is an unstable node; if both the eigen values are negative, then it is a stable node and if one is ositive and the other is negative, then it is a saddle oint. For comlex eigen values if the real art is ositive, then the equilibrium oint is an unstable focus and if the real art is negative, then it is a stable focus. For the resent system, we have the characteristic equation as I q + r λ S = 0 I + q S + r λ For the equilibrium oint (0, 0), we have q+ r λ 0 = 0 q r λ which gives, λ = r, r q. If r > q, r > 0 then both the eigen values are ositive. In this case, the equilibrium oint (0, 0) is an unstable node. If r > q, r < 0, then it is a saddle oint. If r < q, r > 0 then it is again a saddle oint. If r < q, r < 0, then it is a stable node. r ( q In the domain r > q, r < 0 for the equilibrium oint, r we have the characteristic equation as r ( q λ r = 0 rq r λ r which gives λ = rq r q ± + 4 r ( r q) r r r q If the discriminant + 4 r ( r q) 0 both the eigen values are negative and in r r q that case, it reresents a stable node. Again if the discriminant + 4 r ( ) < 0 r then the equilibrium oint reresents a stable focus.

6 66 Anulekha Taadar et al We have framed the series solution of (4) by the following way: S = n + ( na qn + rn) t + 1 [(q n + qan r an arn + a n a n qn ] t + (6) and I = a + (na + qn + r a) t + 1 [arn + qrn + r an + qr n + r a q n qn a n + a n + qn ] t + Mutual Co-existence of the HIV not-infected and HIV infected individuals: From (4), we get di SI + qs + r I = ds SI qs + rs (7) Here we restrict ourselves in the domain r > q, r < 0. In most of the cases, it is exected that the birth rate er individual is less than the death rate er individual in the grou of HIV infected eole. So it looks good to assume r < 0. Also the net rate of increase (rate of birth - rate of death) er individual in the grou not infected by HIV is exected to be greater than the rate er individual of getting infected by HIV due to transfusion of contaminated blood or blood roducts or by sharing of contaminated needles in the same grou. So, r > q looks consistent in most of the cases. di Case A: 0 ds > di ds Subcase (i): > 0 and > 0 i.e., SI + qs + r I > 0 and SI qs + rs > 0. In dt dt this case, we have I <. We consider I = h for h > 0. From the inequality SI + qs + r I > 0,we have 1 r ( ) S > r h ( r h) r r rovided h <. But as I > 0, f < <, So the above range for S is a consistent range for S in this case. 1 r ( ) We take f( h) = r h r h

7 A Dynamical Model of the Sread of HIV/AIDS 67 rq Therefore, f ( h) = < 0 ( r h) since r < 0 So, f (h) is a decreasing function of h. Hence f (h) < f (0) for h > 0, i.e., 1 r ( ) r ( q rh < r h r Hence in this case, we fail to get any secific range for S. di ds Subcase (ii): < 0 and < 0, i.e., SI + qs + r I < 0 and SI qs + rs < 0. In dt dt this case, we have I >. We consider I = + h for h > 0. From the inequality SI + qs + r I < 0, we have 1 r ( q S < r h ( r + h) We take 1 r ( q g( h) = r h ( r + h) rq Therefore, g ( h) = > 0 ( r+ h) since r < 0. So, g (h) is an increasing function of h. Hence g(h) > g(0) for h > 0, i.e., 1 r ( q r ( q rh > ( r + h) r Hence in this case also we fail to get any secific range for S. di Case B: 0 ds < di ds Subcase (i): < 0 and > 0, i.e., SI + qs + r I < 0 and SI qs + rs > 0. In dt dt this case, we have I < and we take I = h where h > 0. From the inequality SI + qs + r I < 0, we get

8 68 Anulekha Taadar et al 1 r ( ) S < r h ( r h) r r q r rovided h <. But as I > 0, h < <. So the above range for S is a consistent range for S in this case. By revious argument in Case A : subcase (i),we have 1 r ( ) r ( q rh < ( r h) r Hence, S < r ( q r di ds Subcase (ii): > 0 and < 0, i.e., SI + qs + r I > 0 and SI qs + rs < 0. In dt dt this case, I > and we take I = + h where h > 0. From the inequality SI+qS+r I > 0, we get 1 r ( q S > r h ( r + h). Now by revious argument in case A : subcase (ii), we have 1 r ( q r ( q rh > ( r + h) r Hence, r ( q S >. r

9 A Dynamical Model of the Sread of HIV/AIDS 69 Keeing in view all the above facts we can have the following table : Sign of di/ds + Serial No. Sign of ds/dt Sign of di/dt Range of S No secific range for S No secific range for S 3 + r ( q S < r 4 + r ( q S > r Range of I I < I > I < I > Situation 3 is analogous to the model where the grou not infected by HIV increases its number, and the grou infected by HIV decreases its number. On the other hand, situation 4 is very alarming where the grou not infected by HIV reduces its number as well as the grou infected by HIV exands with time. This situation can be termed as critically eidemic situation. Results Estimation of arameters in the context of India We work to find out the best fitted values of the arameters, q, r and r resent in the model in the context of India. For this, we have taken into account the existing statistical database at Indian context [14, 15, 16, 17, 18, 19, 0, 1]. Previously it was thought that around 5.7 million eole were living with HIV in India in the year more than in any other country. Better data including the results of a national household survey led to a major revision of the revalence estimate in July 007 [17, 18]. It was estimated then that around.3 million eole in India were living with HIV. This entire rocess leaves some controversies. The roblem arises when sentinel surveillance data is used to estimate a country s HIV burden. Sentinel surveillance is not designed to make estimates, but it has been used to rovide rough estimates of the HIV burden for many years, for want of a better aroach. There are various biases in the existing samling rocess. However these biases are not ublicly acknowledged, as a result of which the ublic is misled. In reality, all sentinel sites are in government hositals, whereas the majority of the eole go to rivate hositals. Again the samles are from regnant women and grous with high risk. So no information is available for other men and women outside these grous. Finally if the condition is unevenly distributed in the oulation, any samle collected from it cannot reresent the entire oulation. So the latest calculation may give us a better rofile of the roblem, but they too have their limitations [19].

10 70 Anulekha Taadar et al So in the resent work for calculations, we have taken into account both these databases and furnished two different sets of values of the arameters involved in our model as well as two different sets of grahical rofiles. Using first set of data: We have the following table in this connection [14, 15, 16]: Year Indian Poulation HIV/AIDS oulation In India (in millions) (In millions) The calculation gives rise to the following best-fitted magnitudes of the arameters: 10 = , q = , r = and r' = These estimated values can be further used effectively to redict the number of HIV/AIDS infected in the context of India at near future instants. Using second set of data which Government of India accets now-a-days: We have the following table in this regard [19, 0, 1] Year Indian Poulation HIV/AIDS oulation In India (in millions) (In millions) The calculation gives rise to the following best-fitted magnitudes of the arameters: 10 = , q = , r = and r' = We have used these estimated values to redict robable HIV/AIDS oulation in India in coming future. Our Forecast for HIV/AIDS Poulation in India Using the first set of estimated values of arameters we have estimated the number of HIV infected and uninfected individuals in India uto 015 (007 onwards) and in this context we resent Figure 1 and Figure.

11 A Dynamical Model of the Sread of HIV/AIDS 71 S vs t Millions S t Figure 1 (Total number of uninfected versus time using first set of data) I vs t I t Figure (Total number of infected versus time using first set of data) Using the second set of estimated values of the arameters we find the robable values of S and I u to 015 (008 onwards) and lotting those values grahically here we resent Figure 3 and Figure 4.

12 7 Anulekha Taadar et al S vs t s millions t Figure 3 (Total number of uninfected versus time using second set of data) I vs t I m illion t Figure 4 (Total number of infected versus time using second set of data) It is notable that although the years 009 and 010 were already ast but still we kee them in the estimation eriod. This is due to the fact that the corresonding factsheets for these two years are still unavailable. The arameters involved here are very delicate in nature. If we go for a long run, the values of r and r must be changed significantly. So here we have considered a comaratively small time range (u to the year 015) where it is very much exected that the magnitudes for these should have a very small amount of variation so that the result is not affected altogether. Discussion In the resent work, we have roosed a mathematical model of the sread of HIV and estimated the arameters involved in that model in the context of India in order to

13 A Dynamical Model of the Sread of HIV/AIDS 73 make the forecasting of number of infected eole in India ossible. These arameters can also be estimated for any other domain in the world using the corresonding data for that articular domain. Using the estimated arameters, one may redict what will be the total number of infected individuals by HIV/AIDS at near future for any geograhical domain. The resent statistical analysis has limitation that it cannot be used for a very large scale of future time because, in the rocess, values of r and r will certainly be changed and that will disturb the entire rocess. To overcome this roblem, we have to introduce additional equations in the model which sufficiently deicts the time-variation of these two arameters. Concentrating on this fact, further research can be carried to have the forecasting for a sufficiently large scale of time. References [1] S. Bhattacharjee: Why does HIV manifest into AIDS in humans, Everyman s Science, XLI (6), 401 (007). [] Causes of HIV/AIDS, htt:// [3] S. Cassels, S. J. Clark and M. Morris: Mathematical Models for HIV Transmission Dynamics: Tools for Social and Behavioral Science Research, Acquir. Immune Defic. Syndr, 47 (1), S34 (008). [4] Z. Mukandavire, W. Garira and C. Chiyaka: Asymtotic roerties of an HIV/AIDS model with a time delay: Journal of Mathematical Analysis and Alications, 330, 916 (006). [5] E. H. Kalan, Needles that kill: Modelling human immunodeficiency virus transmission via shared drug injection equiment in shooting galleries, Rev.Infect. Dis., 11, 89 (1989). [6] D. Greenhalgh: Effects of Heterogeneity on the Sread of HIV/AIDS Among Intravenous Drug Users in Shooting Galleries, Mathematical Biosciences, 136, 141 (1996). [7] F. Lewis and D. Greenhalgh: Three stage AIDS incubation eriod: a worst case scenario using addict-needle interaction assumtions, Mathematical Biosciences, 169, 53 (001). [8] G.V. Bobashev et al.: Geograhically-Enhanced Mathematical Models of HIV Dynamics, htt:// [resented at NIDA Symosium on AIDS, Cancer and Related Problems, May 6, 004, St. Petersburg, Russia]. [9] A. Taadar and K. Ghosh: Sread of HIV: A Mathematical Model, Proceedings of the 13 th Asian Technology Conference in Mathematics (ATCM 008) [Edited by Yang, Majewsk, de Alwis & Khairiree] (Held at the Suan Sunandha Rajabhat University, Bangkok, Thailand during December, 008), 03 (008). [10] A.S.R.S. Rao: Mathematical Modelling of AIDS eidemic in India, Current Science, 84 (9), 119 (003).

14 74 Anulekha Taadar et al [11] S. Godbole and S. Mehendale: HIV/AIDS eidemic in India: risk factors, risk behaviour & strategies for revention & control, Indian J. Med. Res., 11, 356 (005). [1] N.J.D. Nagelkerke et al.: Modelling HIV/AIDS eidemics in Botswana and India: imact of interventions to revent transmission, Bulletin of the World Health Organization, 80 (), 89 (00). [13] D.K. Chaturvedi: Dynamic Model of HIV/AIDS Poulation of Agra Region, Int. J. Environ. Res. Public Health, (3), 40 (005). [14] htt://data.unaids.org/ub/epislides/007/007_eiudate_en.df. [15] htt://content.nejm.org/cgi/content/full/356/11/1089. [16] htt:// [17] UNAIDS: Press Release:.5 million eole in India living with HIV, according to new estimates, 6 th July, 007. [18] UNAIDS: India: Country situation, 008. [19] M.P. Kumar: 0 million or million? htt://infochangeindia.org/ /agenda/hiv/aids-bigquestion/0million or million.html. [0] htt:// [1] htt://