Mathematical Analysis of Side effects of HIV / AIDS Medication

Introduction: HIV/AIDS is one of the greatest threats to the mankind of the world. The immune system of our body protects us from various kinds of infections and diseases. HIV virus suppresses the immune system and any small infection can cause severe health problem even death. From 1981, more than 25 million people were killed with this disease. Five millions people are becoming infected with this virus every year and presently 40 million people have HIV that causes Aids. HIV/Aids acquired through unprotected sexual contact, blood transfusion, syringes and during child birth and breast-feeding through mother to child.


Mathematical Analysis of Side effects of HIV/AIDS Medication
Tirmizi SRH 1 , Khan N 2

Introduction:
HIV/AIDS is one of the greatest threats to the mankind of the world.The immune system of our body protects us from various kinds of infections and diseases.HIV virus suppresses the immune system and any small infection can cause severe health problem even death.From 1981, more than 25 million people were killed with this disease.Five millions people are becoming infected with this virus every year and presently 40 million people have HIV that causes Aids.HIV/Aids acquired through unprotected sexual contact, blood transfusion, syringes and during child birth and breast-feeding through mother to child.Antiretroviral toxicity is becoming an important subject in the management of HIV-infected persons.HIV infection is a chronic disease, and the drugs are being used in the patients for longer period of time.A study indicates that although the mortality and morbidity rates have been minimized with advanced Combination antiretroviral drugs treatment but major adverse effects is being increased and up to 25% of patients stop their initial HAART (Highly active antiretroviral therapy) regimen because of lethal 1 .The drugs, which are available in the world market, can be divided into five groups and their table is given below with mode of action.

Table I: (Drugs available in the market with mode of action)
The following data is taken from AIDS info 'A service of the U.S. Department of Health and Human Services', which is helpful in our study.These are some most frequent reported serious side effects with HIV medication.

ARV drugs class
Abbreviations First approved HIV treatment (year)  As the problem faced by them we feel it can be modeled using fuzzy matrix theory to identify the intensity of adverse effects in different intervals of time of treatment.We transform the collected data into matrix form by taking side effects along rows and duration of treatment along the column.The average time dependent matrix (ATD) is obtained by dividing each entry of the matrix by the time period and taking the average µ j and standard deviation of each column of the ATD matrix.We select a parameter from the interval [0,1] and form the refined Time dependent matrix by using the formula Where aij are the entries of the ATD matrix.In this way, we form the Refined Time Dependent (RTD) fuzzy matrix.The entries of this matrix are -1, 0, or 1. Entries and the sum of rows gives the matrix of intensity of adverse effects.We also combine these matrices for different values of select from the interval [0,1], in this way we get the Combined Effective Time Dependent Data (CETD) matrix and the row sum is obtained for CETD matrix.Conclusions are derived based on the row sums.Its graphical presentation is very simple and could be understood by a layman.

ESTIMATION OF SIDE EFFECTS OF HIV/AIDS MEDICATION USING 4X8 FUZZY MATRIX
On the basis of history of patients we have taken the following eight major side effects (A 1 ,A 2 ,A 3 ,A 4 ,A 5 ,A 6 ,A

Average Time Dependent Data (ATD) Matrix
Table IV: The Average and the Standard Deviation of the above ATD Matrix The RTD matrix for = 0.1 the row sum matrix Hepatotoxicity are associated with antiretroviral agents PIs in a potential cohort study it was observed that patients who were treated with ritonavir (RTV) and also the patients who were receiving saquinavir, nelfinavir (NFV) or indinavir (IDV) were faced severe hepatotoxicity 2 .Severe hepatotoxicity was also observed in patient using Ritonavir.Hepatotoxicity incidence is observed in other treatment groups, i.e., nucleoside analogs, nelfinavir, saquinavir,and indinavir as well 3 .In another study, highly active antiretroviral therapy (HAART) containing protease inhibitors (PIs) and ritonavir with saquinavir, drug-induced hepatotoxicity was observed with administration of its full dose) 4 .
HIV-infected patients also show risk factors for cardiovascular problems.Most recently, studies showed that NRTI, NNRTI and PI drug classes (alone and in combination) are increasing the risk of heart attack in HIV patients 8 .The Food and Drug Administration (FDA) has approved the DAD study (Data collection on Adverse events of anti-HIV Drugs), which indicates that those HIV patient treating with abacavir and didanosine have a maximum chance of heart attack patients on another medications 5 .Drug-induced renal failure is a common incident in patients with HIV.The antiviral like acyclic nucleoside phosphonates cidofovir and adefovir causes worse nephrotoxicity 6 .A study showed that acute renal failure developed in a patient with HIV using tenofovir 25 .A study showed that Urological complications were associated in majority of patients with the indinavir treatment because of elevated indinavir plasma concentration 1 .
Some special metabolic complications like osteopenia and osteoporosis are associated with protease 27 .
In the opinion of some researchers lactic academia and proposed mitochondrial toxicity are associated with analog (NRTI) which become the main cause of withdrawal from treatment.The PI therapy can cause cumulative incidence of hyperglycemia, hypercholesterolemia, hypertriglyceridemia, and lipodystrophy 7 .Antiretroviral medications may cause pathogenesis of hyperglycemia in HIV-infected patients 8,9,10 .In vitro evidence, it is mentioned that ritonavir can caused both insulin resistance 5 and impaired -cell function 11 .In some studies, it is also reported that administration of ritonavir-containing regimens can cause glucose homeostasis 8,9,[10][11][12][13][14][15][16] .
Acute pancreatitis is a life-threatening problem 3 .The annual incidence of acute pancreatitis in the US HIV population is considerably higher than in the general non-HIV-infected population 5 .Some studies show that the use of hydroxyurea with didanosine should probably be discouraged because the risk of pancreatitis is increased 17 .
Although, with all complications, we can not ignore the importance of medication.Different numerous studies have reported that there is always a need of treatment with vision of risk-benefit.The net median survival time after infection with HIV is estimated to be 9 to 11 years, depending on the HIV subtype 18 and the median survival rate after diagnosis of AIDS in resource-limited settings where treatment is not available ranges between 6 to 19 months, depending on the study 19 .But HAART therapy reduced the death rate by 80%, and raised the life expectancy for a newly diagnosed HIV-infected person to about 20 years 20 .

Conclusion and Suggestion:
We have observed from the above graphs that adverse effects start from the first day of the treatment and getting peak level after 5-10 years during the treatment.Then some supporting medication is needed to maximize the adherence with the drug.As we know efficacy and tolerability, these are the factors which make a drug 'the drug of choice.The efficacy of combination ARV regimen is increased but adverse effects of these therapies may cause worse effects on the body organs, we can say its mix blessing because of prolonged treatment and less tolerability, it is very difficult for the HIV positive person to continue the treatment that results in treatment failure.There is a need to continue the efforts in the improvement of medication and better understanding the side effects of ARV regimen for HIV Specialist and physician who care for HIV/AIDS patients.

Fig 1 :
Fig 1: The graph depicting the side effects of HIV/ AIDS medication with the passage of time for different parametric values.

Table II : (Drugs with their brand names and possible serious side effects) Description and Methodology:
University and Hospital and Baqai Medical University and Hospital Karachi (Pakistan).These are very selective patients who are already under regular treatment with antiretroviral medications and facing severe adverse effects of HIV/AIDS medication.

Table III : Initial data based on history of patients (n=195)
Side effects of HIV/AIDS Medication 120