Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 552 https://internationalpubls.com Analyzing the Dynamics of HIV/AIDS Phobia with Stability and Awareness Strategies 1 Naresh Kumar Jothi, 2 Vadivelu. V, 3 Deepa. S, 4 Vivekanandan. T, and 5 Ramkumar. C 1, 2 Department of Mathematics, Vel Tech Rangarajan Dr.Sagunthala, R&D Institute of Science and Technology, Avadi, Chennai, Tamilnadu, India. 3 Department of Mathematics, Vel Tech High Tech Dr.Rangarajan Dr.Sakunthala, Engineering College, Avadi, Chennai, Tamil Nadu, India 4Department of Mathematics, Vel Tech Multi Tech Dr.Rangarajan Dr.Sakunthala, Engineering College, Avadi, Chennai, Tamil Nadu, India 5 Department of Mathematics, Bharath Institute of Higher Education and Research, Selaiyur, Chennai, Tamilnadu, India 1nareshsastra@yahoo.co.in, 2shrivadivelu@gmail.com , 3vsadeepa7682@gmail.com , 4mtvivek2017@gmail.com, 5scpram@gmail.com 1. Introduction: Many people are afraid of "cancer" and "AIDS." Even if those worries could be quite reasonable, what would happen if they started to rule your life? "HIV phobia is an extreme fear of contracting HIV." especially the anxiety of getting infected with the virus, even if your risk is relatively minimal. Put differently, the worry is excessive and unfounded. Before very effective antiretroviral medication was available, in the 1980s and 1990s, there was a lot of documentation and description of AIDS phobia. These kinds of phobias are not well known in human behaviour. According to some mental health Article History: Received: 01-10-2024 Revised: 29-11-2024 Accepted: 07-12-2024 Abstract: One of the risks to global health is sexually transmitted diseases (STDs). Usually, diseases do not cause death, but 50% of deaths are caused by fear of disease. This causes policymakers and experts studying diseases to become increasingly concerned. In this section we discuss about HIV/AIDS phobia, a particular type of nosophobia is an excessive and illogical dread of contracting HIV/AIDS. It is still a major global health concern due to its high death rate, and in most African nations as well as other places, it is the main source of HIV/AIDS anxiety. We develop a five-component deterministic model to examine how phobia affects HIV/AIDS dynamics within a particular population. The system's HIV/AIDS-free equilibrium is considered asymptotically stable when the effective reproduction number ℜ0< 1, and unstable in other cases. Additionally, we used to investigate the endemic equilibrium's stability by using the Lyapunov function, positivity, boundedness, Lipschitz condition, and the conditions for the existence of uniqueness are discussed. The outcome demonstrates that everyone must have awareness about HIV/AIDS. Unfortunately, those affected with HIV/AIDS don’t fear; take healthy food, and medicines; and exercise regularly; the counselling about HIV/AIDS will prolong life and reduce the phobia of HIV/AIDS in the population. The HIV/AIDS phobia epidemic model is analyzed by way of Matrix Laboratory. Keywords: Epidemic Model, Sexually transmitted diseases (STDs), HIV/AIDS phobia, HIV free equilibrium (HFE), HIV endemic equilibrium (HEE), Stability, Lipschitz and Lyapunov. mailto:nareshsastra@yahoo.co.in mailto:shrivadivelu@gmail.com mailto:vsadeepa7682@gmail.com mailto:mtvivek2017@gmail.com mailto:scpram@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 553 https://internationalpubls.com professionals, the reason could be inherited —a predisposition to acquire phobias as a result of your inherited composition. HIV-phobias frequently believe they have the virus, and even negative tests cannot make them feel any less afraid. Some people think that traumatic experiences and incidents in a person's life are the cause of phobias. For instance, knowing someone who drowned may cause someone to develop a phobia of the water. Similarly, knowing those who have succumbed to the illness or been critically ill could also cause someone to develop an HIV fear. Some people, despite their obviously irrational behaviour, will stop at nothing to prevent contracting HIV. African countries account for eight of the top 10 countries with the highest number of AIDS-related deaths worldwide. It makes reasonable that someone with severe anxiety problems would mistake early HIV infection symptoms for other; more frequent illnesses because they are so similar to them. These symptoms exacerbate the worry because they are common to many ailments. Furthermore, even if they are not physically ill, their dread has the ability to cause them to become so consumed by their worry about infection that they are able to physically manifest these symptoms in their minds. Lerman refers to this as "Pseudo-AIDS." An HIV-phobic individual assumes they have HIV whenever they have comparable symptoms. HIV phobia typically presents as physical symptoms, according to Phillips. These symptoms—which include nausea, fever, rapid weight loss, night sweats, exhaustion, diarrhea, mouth or vaginal sores, and headaches—can resemble those that some people have experienced after recently becoming infected with HIV. An HIV phobia is frequently greatly influenced by culture. Phobias stem from biological and social elements as well as experiences. The subject of how HIV fear arises has no universally accepted explanation. After engaging in one condom-free sexual encounter, an individual may start to fear that they might be infected with a STD. At least one HIV test should be taken by anyone between the ages of 13 and 64, according to the CDC [Centers for Disease Control and Prevention]. This may often be done at your yearly physical examination. Ask your doctor about it if you haven't had the test. Tests should be performed more frequently if you are at a higher risk; ideally, every three or six months. However, Pantalone notes that a further factor contributing to the lack of testing is people's misconception that the high risk of the illness "fits within an identity" when in fact it's a virus spread by normal human conduct, such as having sex. You need to get tested for HIV if you've ever engaged in condom-free sexual activity. Pantalone advises against doing anything infrequently, even if it seems little risk. According to the CDC, you have an increased chance of contracting HIV if you can say "yes" to any of the following questions: 2. Have you had sex with an HIV-positive person, either anally or vaginally? 3. Since your last HIV test, have you dated anyone more than once? 4. Have you given anyone access to needles, inject able medications, or other injection supplies? 5. Have you ever traded drugs or cash for sex? 6. Have you ever engaged in sexual activity with someone whose past you are unaware of? 7. Have you received medical attention or a diagnosis for any other STD? 8. Do you identify as a male who has slept with another man? We note that health behaviour theory needs to address the mechanisms relating many levels of influence on behaviour and offer useful direction for multi-level behaviour modification treatments tailored to particular settings. We develop a mathematical model to examine the causes of the sexually Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 554 https://internationalpubls.com transmitted disease (STD) outbreaks and the best ways to prevent or lessen them. A model that represents the effect of phobia on HIV/AIDS interaction with disease transmission in the population is considered in this research. We start by describing some basic properties of the system, including equilibrium and its fundamental reproduction value. Additionally, we investigate if non-negative solutions to the given process exist and are unique. Mathematically, the concepts of difference equations are used to develop the epidemic model of HIV/AIDS. The properties of the qualitative HIV/AIDS epidemic model are analyzed. The definite function of Lyapunov is derived from the equation of difference to stabilize the system of the HIV/AIDS epidemic model. HIV and HSV, the two main virus-caused STDs, are incurable and can have detrimental long-term effects on one's health. Lifelong HSV infection is also characterized by recurrent outbreaks at the sites of infection. HSV type 1 primarily affects the mouth and lips (cold sores), while HSV type 2 primarily affects the genital area, albeit genital HSV type 1 is on the rise. Despite the lack of a vaccine or cure, antiviral drugs can lessen the symptoms of HSV. There is a significant chance that a newborn will die or become severely disabled if HSV is spread during or after childbirth. HIV/AIDS risk is also increased by HSV type 2 infections. Since they were first used centuries ago to prevent STDs, male condoms have gained popularity as a key component of HIV/AIDS preventive strategies. Male condoms are a very efficient STD prevention tool that can cut the risk of infection by 80% when used correctly and consistently. In particular in Sub-Saharan Africa, the usage of female condoms as a method of female managed prevention has grown in popularity. Despite their effectiveness, female condoms are still not widely accepted by both women and their sexual partners. Safe sexual behaviour has numerous benefits for both your physical and emotional well-being. You might be able to safeguard your mental health by taking precautions against sexually transmitted diseases (STDs), such as HIV. Because mental health and HIV seem to be related issues, that can be challenging. In both HIV-positive and at-risk populations, mental health issues are more common. HIV risk is positively correlated with mental health. As a result of having HIV, some people may experience mental health issues, but others may develop them because they are afraid of contracting the virus. The diagnosis of a sexually transmitted infection (STI), particularly HIV, can also set off symptoms of other mental health conditions. One of the most prevalent mental health conditions affecting people living with HIV, for instance, is depression. Individuals living with HIV have twice the chance of developing depression compared to those who are at risk for HIV but do not yet have the virus. You might not be able to get tested, get results, seek or receive care, or continue your care if you have a mental health condition. By taking precautions to avoid contracting or spreading HIV, you may enhance your general mental well-being. Psychotherapy and medicine are two possible treatments for people with a crippling fear of HIV. Investigating the underlying causes of the anxieties may be more crucial than seeking all the information about the illness from a doctor or counsellor, even though doing so may be helpful. Many times, there will be no connection at all between the phobia and HIV. It typically helps to sit with a qualified mental health expert. Family counselling, group therapy, and individual therapy are all possible forms of treatment. Prescription medications such as Zoloft and Lexapro may be helpful for people who have been diagnosed with anxiety disorders. One's life need not be dictated by HIV fear. Returning to your best self, you can enjoy a life of fulfilment. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 555 https://internationalpubls.com 9. Epidemic model of HIV/AIDS: The HIV/AIDS epidemic model is mathematically portrayed by the system of equations with the following suppositions. ϖ =Existing populace Hs=No. of susceptible Hi=No. of infected Hc=No. of HIV patients attended Counselling Hn=No. of HIV patients didn't attend the Counselling Ha= Final stage of HIV/AIDS ϖ1=Rate of phobia about HIV ϖ2= Proper counselling about HIV and social phobia ϖ3=Not proper counselling about HIV and social phobia ϖ4=The rate of patient’s relief from HIV phobia and their lifetime extend ϖ5=The rate of patient’s not relieved from HIV phobia and their lifetime is easily reduced. υ =Normal death rate at all stages υ + a =The population death rate of infected HIV. Fig 1: HIV/AIDS phobia epidemic model of flow diagram Form epidemic model of HIV/AIDS is allowed by the accompanying arrangement of D.E. dHs d𝔱 = ϖ − (ϖ1 + υ) Hs (1) dHi d𝔱 = ϖ1Hs − (ϖ2 + ϖ3 + υ) Hi (2) dHc d𝔱 = ϖ2Hi − (ϖ4 + υ) Hc (3) dHn d𝔱 = ϖ3Hi − (ϖ5 + υ) Hn (4) dHa d𝔱 = ϖ4Hc + ϖ5Hn − (υ + a) Ha (5) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 556 https://internationalpubls.com Where Hs(0) > 0 ,Hi(0) > 0,Hc(0) > 0 ,Hn(0) > 0, and Ha(0) > 0 are initial condition 10. Determination of Fixed Points To find the equilibrium points(Hs̃,Hĩ,Hc̃,Hñ ,Hã) of the system Equations (1-5), we set the derivatives equal to zero. So, at equilibrium states, we get ϖ− (ϖ1 + υ)Hs̃ = 0 ϖ1Hs̃ − (ϖ2 + ϖ3 + υ)Hĩ = 0 ϖ2Hĩ − (ϖ4 + υ)Hc̃ = 0 (6) ϖ3Hĩ − (ϖ5 + υ)Hñ = 0 ϖ4Hc̃ + ϖ5Hñ − (υ+ a)Hã = 0\ 4. Equilibrium states 4.1 HIV free equilibrium (HFE) state The HIV free equilibrium for the HFE, we replace the variables as 𝔈0 = (Hs̃,Hĩ,Hc̃,Hñ ,Hã) = (Hs 0 ,Hi 0 ,Hc 0,Hn 0,Ha 0) is characterized as the place where no disease is available in the populace. Every one of the contaminated classes will be equivalent to zero. dHs d𝔱 = ϖ− (ϖ1 + υ) Hs = 0 ϖ− ϖ1Hs − υHs = 0 But, ϖ1 = 0 ϖ− υHs = 0 Hs = ϖ υ Thus, the HIV free equilibrium satisfies 𝔈0 = (Hs 0 ,Hi 0 ,Hc 0,Hn 0,Ha 0) = ( ϖ υ , 0,0,0,0) 4.2 HIV endemic equilibrium (HEE) state For the HEE, we replace the variables as 𝔈1 = (Hs̃,Hĩ,Hc̃,Hñ ,Hã) = (Hs 1 ,Hi 1 ,Hc 1,Hn 1,Ha 1) is defined as areas where the population has HIV. In this case, all affected classes do not equal zero. The point at which the disease exists within the susceptible population referred to as the endemic equilibrium. We set the equations resulting from our model is zero. From dHs d𝔱 = ϖ− (ϖ1 + υ) Hs Making Hs the subject of the equation we get; ϖ− (ϖ1 + υ) Hs = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 557 https://internationalpubls.com Hs 1 = ϖ (ϖ1+υ) From dHi d𝔱 = ϖ1Hs − (ϖ2 + ϖ3 + υ) Hi Making Hi the subject of the equation we get; ϖ1Hs − (ϖ2 + ϖ3 + υ) Hi = 0 Hi 1 = ϖ1 (ϖ2+ϖ3+υ) Hs 1 From dHc d𝔱 = ϖ2Hi − (ϖ4 + υ) Hc Making Hc the subject of the equation we get; ϖ2Hi − (ϖ4 + υ) Hc = 0 Hc 1 = ϖ2 (ϖ4+υ) Hi 1 From dHn d𝔱 = ϖ3Hi − (ϖ5 + υ) Hn Making Hnthe subject of the equation we get; ϖ3Hi − (ϖ5 + υ) Hn = 0 Hn 1 = ϖ3 (ϖ5+υ) Hi 1 From dHa d𝔱 = ϖ4Hc + ϖ5Hn − (υ+ a) Ha Making Hathe subject of the equation we get; ϖ4Hc + ϖ5Hn − (υ+ a) Ha = 0 Ha 1 = ϖ4 (υ+a) Hc 1 + ϖ5 (υ+a) Hn 1 By substituting the value of Hs 1 in Hi 1 we getting, Hi 1 = ϖ1 (ϖ2+ϖ3+υ) ϖ (ϖ1+υ) = ϖϖ1 (ϖ1+υ)(ϖ2+ϖ3+υ) By putting the value of Hi 1 in Hc 1 we have, Hc 1 = ϖ2 (ϖ4+υ) ϖ1 (ϖ2+ϖ3+υ) ϖ (ϖ1+υ) = ϖϖ1ϖ2 (ϖ1+υ)(ϖ2+ϖ3+υ)(ϖ4+υ) By substituting the value of Hi 1 in Hn 1 we getting, Hn 1 = ϖ3 (ϖ5+υ) ϖ1 (ϖ2+ϖ3+υ) ϖ (ϖ1+υ) = ϖϖ1ϖ3 (ϖ1+υ)(ϖ2+ϖ3+υ)(ϖ5+υ) By substituting the values of Hc 1 and Hn 1 in Ha 1 we have, Ha 1 = ϖ4 (υ+a) ϖ2 (ϖ4+υ) ϖ1 (ϖ2+ϖ3+υ) ϖ (ϖ1+υ) + ϖ5 (υ+a) ϖ3 (ϖ5+υ) ϖ1 (ϖ2+ϖ3+υ) ϖ (ϖ1+υ) Ha 1 = ϖϖ1ϖ2ϖ4(ϖ5+υ)+ϖϖ1ϖ2ϖ5(ϖ4+υ) (υ+a)(ϖ4+υ)(ϖ5+υ)(ϖ2+ϖ3+υ)(ϖ1+υ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 558 https://internationalpubls.com Hence, the HIV Endemic equilibrium satisfies 𝔈1 = (Hs 1 ,Hi 1 ,Hc 1,Hn 1,Ha 1) = ( ϖ (ϖ1+υ) , ϖϖ1 (ϖ1+υ)(ϖ2+ϖ3+υ) , ϖϖ1ϖ2 (ϖ1+υ)(ϖ2+ϖ3+υ)(ϖ4+υ) , ϖϖ1ϖ3 (ϖ1+υ)(ϖ2+ϖ3+υ)(ϖ5+υ) , ϖϖ1ϖ2ϖ4(ϖ5+υ)+ϖϖ1ϖ2ϖ5(ϖ4+υ) (υ+a)(ϖ4+υ)(ϖ5+υ)(ϖ2+ϖ3+υ)(ϖ1+υ) ) 5. Basic reproduction number (𝐑𝟎) In this segment, we get the basic reproduction number (ℜ0) which is defined as the quantity of secondary infection created by a single infected person when introduced into a totally susceptible populace. To get the basic reproduction number (ℜ0), we take on the same methodology utilized in where the authors utilized ℱ and 𝒱 of two matrices, representing the remaining terms of transfer as 𝒱 and the new infection terms as ℱ. Our model has only five compartments with only one infective compartment. i. e., dHi d𝔱 = ϖ1Hs − (ϖ2 + ϖ3 + υ) Hi We have that ℱ = [ϖ1Hs] and 𝒱 = [(ϖ2 + ϖ3 + υ) Hi] At the equilibrium of disease free, the matrix ℱ becomes ℱ = [ϖ1 ϖ υ ] At the equilibrium of disease free, the matrix 𝒱 becomes 𝒱 = (ϖ2 + ϖ3 + υ) The inverse matrix of 𝒱 becomes 𝒱−1 = 1 (ϖ2 + ϖ3 + υ) Let ℜ0 = ℱ𝒱−1 Then ℜ0 = [ϖ1 ϖ υ ] [ 1 (ϖ2+ϖ3+υ) ] ℜ0 = ϖ1ϖ υ(ϖ2 + ϖ3 + υ) Using the characteristic equation |ℱ𝒱−1 − ΛI| = 0. From equation of ℜ0 we compute the eigenvalues as follows |ℱ𝒱−1 − ΛI| = 0 implies |ℜ0 − ΛI| = 0, where I is an identity matrix. | ϖ1ϖ υ(ϖ2+ϖ3+υ) − Λ| = 0 ϖ1ϖ υ(ϖ2+ϖ3+υ) − Λ = 0 Λ = ϖ1ϖ υ(ϖ2+ϖ3+υ) . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 559 https://internationalpubls.com Hence, ℜ0 of basic reproduction number is given as ℜ0 = ϖ1ϖ υ(ϖ2+ϖ3+υ) < 1. Epidemiologically, for widely utilized infection models when: ℜ0 < 1; this implies that the pathogen will be prevented from entering the population. ℜ0 = 1; this implies that the pathogen will be constant. ℜ0 > 1; this implies that the infection can begin to spread throughout the populace Considering these criticisms of our model, we have If ϖ1ϖ<υ(ϖ2 + ϖ3 + υ), the occurrence of the infection and phobia will decrease. If ϖ1ϖ=υ(ϖ2 + ϖ3 + υ), the infection and phobia occurrence will be constant. If ϖ1ϖ>υ(ϖ2 + ϖ3 + υ), the frequency of infection and fear will rise.. As a result, we also came to the following results. 6. Uniqueness and Existence We can write the system (1–5) like this: d𝔛 d𝔱 = 𝔉(𝔛), 𝔛 = 𝔛0, (7) In the case where 𝔛 = (Hs,Hi ,Hc ,Hn, Ha) 𝒯 the state variables are contained in a column vector in ℛ5, which also defines a mapping from [0, +∞) to ℛ5. But still, 𝔉(𝔛) = (𝔉1(𝔛), 𝔉2(𝔛), 𝔉3(𝔛), 𝔉4(𝔛), 𝔉5(𝔛)) 𝒯 ∈ ℛ5, is a component-based vector valued function from ℛ5 to ℛ5 . 𝔉1(𝔛) = ϖ− (ϖ1 + υ) Hs 𝔉2(𝔛) = ϖ1Hs − (ϖ2 + ϖ3 + υ) Hi 𝔉3(𝔛) = ϖ2Hi − (ϖ4 + υ) Hc 𝔉4(𝔛) = ϖ3Hi − (ϖ5 + υ) Hn 𝔉5(𝔛) = ϖ4Hc + ϖ5Hn − (υ+ a) Ha Lipschitz continuous in 𝔛 is the function 𝔉 in (7). Therefore, the nonlinear system (7) has a unique solution according to the existence and uniqueness theorem. If this solution is bounded and non- negative, it will have greater biological significance and be more grounded in reality. In the next Theorems 1 and 2, we illustrate these essential characteristics. 7. Positivity and boundedness Theorem 1. Let Hs(0) = Hs0 , Hi(0) = Hi0 , Hc(0) = Hc0, Hn(0) = Hn0and Ha(0) = Ha0 be the initial values of the state variables. If Hs0 , Hi0 , Hc0, Hn0and Ha0are positive then it implies thatHs(𝔱), Hi(𝔱),Hc(𝔱),Hn(𝔱), and Ha(𝔱)are positive for all time t > 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 560 https://internationalpubls.com Proof: Let Hs0 ,Hi0,Hc0 ,Hn0, and Ha0 and be positive, it is our goal to demonstrate the positivity of the state variables. According to system (1), we have: dHs d𝔱 = ϖ − (ϖ1 + υ) Hs Then, dHs d𝔱 + (ϖ1 + υ) Hs = ϖ Since ϖ ≥ 0 it follows that, dHs d𝔱 + (ϖ1 + υ)Hs ≥ 0 Now, we have: dHs Hs ≥ −(ϖ1 + υ)d𝔱 (8) By integrating equation (8) we have: In(Hs) ≥ −∫(ϖ1 + υ) dt+ c Let 𝔸(𝔱) = −∫(ϖ1 + υ) dt , it suggests that In(Hs(𝔱)) ≥ 𝔸(𝔱) + c (9) at 𝔱 = 0 we have In(Hs(0)) ≥ 𝔸(0) + c (10) We subtract equations (10) from (9) to haveIn ( Hs(𝔱) Hs(0) ) ≥ 𝔸(𝔱) − 𝔸(0) By calculating the exponential of either side, we get: Hs(𝔱) ≥ Hs(0)e 𝔸(𝔱)−𝔸(0) ≥ Hs(0) for 𝔱 ≥ 0 Since Hs(0) = Hs0is positive, that is, Hs0> 0 for 𝔱> 0 it suggests that Hs(𝔱) ≥ Hs(0) > 0 for 𝔱> 0 We haveHs(𝔱)> 0 for all 𝔱> 0 and we determine that Hs(𝔱) is non-negative for all 𝔱> 0. Therefore, Hs> 0. The other state variables also hold true in this manner. This demonstrates that Hs(𝔱),Hi(𝔱) ,Hc(𝔱),Hn(𝔱)and Ha(𝔱)are positive for all time 𝔱> 0. Theorem 2. Within the invariant region, the nonlinear system (1–5) has bounded solutions. 𝚽 = {(𝐇𝐬, 𝐇𝐢 , 𝐇𝐜 , 𝐇𝐧, 𝐇𝐚) ∈ 𝓡+ 𝟓 : 𝟎 < 𝓝𝐇(𝖙) ≤ 𝛡 𝛖 ;𝐇𝐬, 𝐇𝐢 , 𝐇𝐜 , 𝐇𝐧, 𝐇𝐚 > 𝟎} Proof: The total population for human is given by 𝒩H(𝔱) = Hs(𝔱) + Hi(𝔱) + Hc(𝔱) + Hn(𝔱) + Ha(𝔱) Such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 561 https://internationalpubls.com d𝒩H(𝔱) d𝔱 = dHs(𝔱) d𝔱 + dHi(𝔱) d𝔱 + dHc(𝔱) d𝔱 + dHn(𝔱) d𝔱 + dHa(𝔱) d𝔱 By simplification we have d𝒩H(𝔱) d𝔱 = ϖ− υ(Hs + Hi + Hc + Hn + Ha) − aHa But 𝒩H = Hs + Hi + Hc + Hn + Ha So, we have d𝒩H(𝔱) d𝔱 = ϖ− υ𝒩H − aHa ≤ ϖ− υ𝒩HIn the absence of HIV death in the population (a = 0) d𝒩H(𝔱) ϖ−υ𝒩H ≤ d𝔱 Integrate the two forms 𝔱 = 0 and 𝔱 = 𝔱α , we have ∫ d𝒩H(𝔱) ϖ−υ𝒩H 𝔱α 0 ≤ ∫ d𝔱 𝔱α 0 we have [− 1 υ In (ϖ− υ𝒩H)] 0 𝔱α ≤ [𝔱]0 𝔱α This gives us In (ϖ− υ𝒩H(𝔱α)) − In (ϖ− υ𝒩H(0)) ≥ −υ 𝔱α Then In ( ϖ−υ𝒩H(𝔱α) ϖ−υ𝒩H(0) ) ≥ −υ𝔱α Taking exponential of both sides we have: ϖ−υ𝒩H(𝔱α) ϖ−υ𝒩H(0) ≥ e−υ𝔱α ϖ− υ𝒩H(𝔱α) ≥ (ϖ− υ𝒩H(0))e −υ𝔱α which gives υ𝒩H(𝔱α) ≤ ϖ− (ϖ− υ𝒩H(0))e −υ𝔱α 𝒩H(𝔱α) ≤ ϖ υ − (ϖ−υ𝒩H(0)) υ e−υ𝔱α (11) Taking the inequality (11) limit as 𝔱α → ∞, we get 𝒩H(𝔱α) ≤ ϖ υ Thus, we have 𝒩H(𝔱) ≤ ϖ υ . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 562 https://internationalpubls.com It follows that 0 < 𝒩H(𝔱) ≤ ϖ υ . As a result, all the solutions are bounded in a feasible region Φ. If 𝔛0 = 0, the answer for the simple case stays 𝔛 = 0 for all t > 0. Furthermore, for any t > 0, every solution trajectory with an initial condition in ℛ+ 5 will remain in the region Φ. Thus, all of the solutions in ℛ+ 5 are drawn to the region Φ. As a potential solution set of state variables for our system (1–5), it follows that the closed set Φ is positively invariant in ℛ+ 5 . Therefore, the system (1–5), or equivalently (7), is well-posed mathematically and epidemiologically. Theorem 3. The HIV free equilibrium point,𝔈0, is stable if ℜ0 < 1, whereas 𝔈0 is unstable if ℜ0 > 1. Proof. The Jacobian matrix associated to the system (1-5) at a given point (Hs,Hi ,Hc ,Hn,Ha) is calculated as follows: 𝒥(𝔈0) = ( −(ϖ1 + υ) 0 0 0 0 ϖ1 0 0 −(ϖ2 + ϖ3 + υ) ϖ2 ϖ3 0 −(ϖ4 + υ) 0 0 0 −(ϖ5 + υ) 0 0 0 0 0 ϖ4 ϖ5 −(υ+ a)) and the characteristic polynomial is|𝒥(𝔈0) − λI| = 0. Solving this polynomial, the Eigen values are given by |𝒥(𝔈0) − λI| = ( −(ϖ1 + υ) − λ1 0 0 0 0 ϖ1 0 0 −(ϖ2 + ϖ3 + υ) − λ2 ϖ2 ϖ3 0 −(ϖ4 + υ) − λ3 0 0 0 −(ϖ5 + υ) − λ4 0 0 0 0 0 ϖ4 ϖ5 −(υ+ a) − λ5) = 0 We get λ1 = −(ϖ1 + υ) < 0 λ2 = −(ϖ2 + ϖ3 + υ) < 0 λ3 = −(ϖ4 + υ) < 0 λ4 = −(ϖ5 + υ) < 0 λ5 = −(υ+ a) < 0 All eigenvalues are negative. Hence, 𝔈0 is stable.Thus biologically it reduces HIV phobia. Theorem 4. If ℜ0 > 1, then the system (1-5) has a unique HIV point of endemic equilibrium. Proof. Using the Lyapunov function, the endemic equilibrium point's locally stable analysis is examined. For this, the following is defined: 𝔏(Hs,Hi ,Hc ,Hn,Ha) = 1 2 (Hs 2 +Hi 2 + Hc 2 + Hn 2 + Ha 2) At the endemic equilibrium point, the function 𝔏 equals zero and is bigger than zero. When we differentiate the function in relation to time, we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 563 https://internationalpubls.com d𝔏 dt = 1 2 (2Hs dHs dt + 2Hi dHi dt + 2Hc dHc dt + 2Hn dHn dt + 2Ha dHa dt ) Applying model (1–5) to the derivative values, we obtain d𝔏 dt = Hs(ϖ− (ϖ1 + υ)Hs) + Hi(ϖ1Hs − (ϖ2 + ϖ3 + υ)Hi) +Hc(ϖ2Hi − (ϖ4 + υ)Hc) + Hn(ϖ3Hi − (ϖ5 + υ)Hn) +Ha(ϖ4Hc + ϖ5Hn − (υ+ a)Ha) By solving this, we get d𝔏 dt = −([ϖ1 + υ]Hs 2 + [ϖ2 + ϖ3 + υ]Hi 2 + [ϖ4 + υ]Hc 2 + [ϖ5 + υ]Hn 2 + [υ + a]Ha 2) + [[ϖ + ϖ1Hi]Hs + [ϖ2Hc + ϖ3Hn]Hi + [ϖ4Hc + ϖ5Hn] Ha ] From model (1-5), we know that ϖ,ϖ1,ϖ2,ϖ3,ϖ4,ϖ5, υ, and a are all non-negative, and the values of Hs,Hi,Hc,Hn, and Ha are all non-negative. As a result, it is clear that every term enclosed in a closed bracket is non-negative, and every term enclosed in an open bracket is non-positive. Thus, we have −([ϖ1 + υ]Hs 2 + [ϖ2 + ϖ3 + υ]Hi 2 + [ϖ4 + υ]Hc 2 + [ϖ5 + υ]Hn 2 + [υ + a]Ha 2) ≤ 0 [[ϖ+ ϖ1Hi]Hs + [ϖ2Hc + ϖ3Hn]Hi + [ϖ4Hc + ϖ5Hn] Ha ] ≥ 0 Thus, we can say that d𝔏 dt ≤ 0 for all non-zero vectors Hs,Hi,Hc,Hn, and Ha and non-negative parameters that satisfy the model (1-5). The endemic equilibrium point is locally asymptotically stable since the derivative of the Lyapunov function 𝔏 is non-positive. This finding suggests, epidemiologically, that HIV phobia will persist in the human population for a very long time. An explanation of the model We briefly provide basic property which will be useful in the next theorem. We define the Banach’s space Β = 𝔹 = 𝔈 × 𝔈 × 𝔈 ×…× 𝔈 and 𝔈 = ℭ[0, 𝒯], with the norm ‖𝒲‖ = ‖(𝓌1,𝓌2, … ,𝓌n)‖ = max 𝔱∈[0,𝒯] {|𝓌1(𝔱)| + |𝓌2(𝔱)| + ⋯+ |𝓌n(𝔱)|}, 𝓌i ∈ ℭ[0, 𝒯] for i = 1,2, … , n. Theorem 5. For each i ∈ {i = 1,… ,5}, the kernels 𝔤i(i = 1,… ,5) are 𝔏i-Lipschitz continuous. Furthermore, if 𝔏i< 1, for all i ∈ {i = 1,… ,5}, then the kernels define a contraction in ℭ[0, 𝒯]. Proof. Letϖ1,ϖ2,ϖ3,ϖ4,ϖ5, υ ∈ ℭ[0, 𝒯], and consider ‖ϖ1‖ < a , ‖ϖ2‖ < b , ‖ϖ3‖ < c , ‖ϖ4‖ < d , ‖ϖ5‖ < e, ‖υ‖ < f. Let Hs and Hs∗be given, then it holds that ‖𝔤1(𝔱,Hs) − 𝔤1(𝔱,Hs∗)‖ = ‖ϖ− ϖ1(𝔱)Hs(𝔱) − υ(𝔱)Hs(𝔱) − [ϖ− ϖ1(𝔱)Hs∗(𝔱) − υ(𝔱)Hs∗(𝔱)]‖ = ‖[ϖ1(𝔱) + υ(𝔱)][Hs∗(𝔱) − H s (𝔱)]‖ ≤ ‖ϖ1(𝔱) + υ(𝔱)‖‖Hs∗(𝔱) − H s (𝔱)‖ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 564 https://internationalpubls.com ≤ [𝔞 + 𝔣]‖Hs∗ − H s ‖ Consequently, ‖𝔤1(𝔱,Hs) − 𝔤1(𝔱,Hs∗)‖ ≤ 𝔏1‖Hs∗ − H s ‖, where𝔏1 = 𝔞 + 𝔣. For 𝔤1, it is evident that the Lipschitz condition is met. Additionally, 𝔤1 defined a contraction if 0 ≤ 𝔏1 < 1. In the same way, we can demonstrate that 𝔤2, 𝔤3, 𝔤4, and 𝔤5 have the Lipschitz and contraction properties. Hence, ‖𝔤2(𝔱,Hi) − 𝔤2(𝔱,Hi)‖ ≤ 𝔏2‖Hi ∗ − Hi‖ ‖𝔤3(𝔱,Hc) − 𝔤3(𝔱,Hc∗)‖ ≤ 𝔏3‖Hc∗ − Hc‖ ‖𝔤4(𝔱,Hn) − 𝔤4(𝔱,Hn∗)‖ ≤ 𝔏4‖Hn∗ − Hn‖ ‖𝔤5(𝔱,Ha) − 𝔤5(𝔱,Ha∗)‖ ≤ 𝔏5‖Ha∗ − Ha‖ Where 𝔏2 = 𝔟 + 𝔠 + 𝔣 , 𝔏3 = 𝔡 + 𝔣 , 𝔏4 = 𝔢 + 𝔣 , 𝔏5 = 𝔣 + a 11. Simulation of Numerical To solve the model equations using numerical methods, we utilized the MATLAB ODE algorithm for the constructed model. We generated graphs depicting each compartment of the model as a function of time, where the time interval extended from 0 to 365 days. Now we need to investigate the significance of each model parameter of the disease phobia. These analyses help us identify which parameters have the greatest influence on the phobia of the HIV/AIDS disease. By pinpointing the most sensitive parameters those with higher sensitivity indices we gain valuable insights into which factors play the most significant role in driving the phobia of the disease. Specifically, our sensitivity analyses focused on parameters ϖ = 0.4 , ϖ1 = 0.65 , ϖ2 = 0.2 , ϖ3 = 0.58 , ϖ4 = 0.03 , ϖ5 = 0.4 , υ = 0.4 , a = 0.005. Fig. (2): Time series of Population Dynamics of all stages The parameters values of Fig. (3) as follows: ϖ = 20 , ϖ1 = 0.65 , ϖ2 = 0.2 , ϖ3 = 0.58 , ϖ4 = 0.03 , ϖ5 = 0.4 , υ = 0.4 , a = 0.005. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 565 https://internationalpubls.com Fig. (3): Time series of Population Dynamics in the stages of Hs, Hi, Hc, Hn, Ha The parameters values of Fig. (4) as follows: ϖ = 100 , ϖ1 = 0.65 , ϖ2 = 0.2 , ϖ3 = 0.58 , ϖ4 = 0.03 , ϖ5 = 0.4 , υ = 0.4 , a = 0.005. Fig. (4): Time series of Population Dynamics The parameters values of Fig. (5) as follows: ϖ = 1000 , ϖ1 = 0.65 , ϖ2 = 0.2 , ϖ3 = 0.58 , ϖ4 = 0.03 , ϖ5 = 0.4 , υ = 0.4 , a = 0.005. Fig. (5): Time series of Population Dynamics The parameters values of Fig. (6a) and Fig. (6b) as follows: ϖ = 0.5 , ϖ1 = 0.65 , ϖ2 = 0.2 , ϖ3 = 0.58 , ϖ4 = 0.03 , ϖ5 = 0.4 , υ = 0.4 , a = 0.005. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 566 https://internationalpubls.com Fig. (6a): Comparison of the stages Hs vs Hn vs Ha of 3D Plot of Population Dynamics Fig. (6b): Comparison of the stages Hs vs Hn vs Ha The parameters values of Fig. (7) as follows: ϖ = 0.5 , ϖ1 = 0.65 , ϖ2 = 0.2 , ϖ3 = 0.58 , ϖ4 = 0.03 , ϖ5 = 0.4 , υ = 0.4 , a = 0.005. Fig. (7): HIV Model Compartment Index. The reason for this is that while there is no cure for HIV infection, it can be effectively managed. The graphs indicate that the prevalence of HIV phobia decreases as the percentage of Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 567 https://internationalpubls.com diagnosed individuals who are treated rises. From a biological perspective, this implies that the HIV/AIDS phobia spreads evenly throughout the community. 12. Conclusion HIV phobia can be diagnosed by a mental health professional, and standard treatment entails therapy in the form of exposure therapy or cognitive behavioural therapy (CBT). Dealing with HIV phobia can also be accomplished through therapeutic and care interventions, such as stress management training sessions. These therapies can enhance one's quality of life and mental health. To prevent HIV phobia, use condoms, avoid sexual activity, not share sterile needles, or think about taking a PrEP or PEP medication. Your mind may be at ease if you are certain of your safety. Seek advice from a medical practitioner or contact a local AIDS organization for recommendations to specialized providers if you or a loved one suffers from HIV phobia. Or, through your community HIV clinic or the 24-hour AIDS hotline that's available in most states, you might be able to connect with a local support group. Whenever researching the HIV/AIDS disease phenomenon, human safety is the most important consideration. Certain scholars have directed their attention on creating a model that takes preventive measures for the human population into account. In the human population, this preventive measure lowers the fear and infection risk. The HIV/AIDS is demonstrated beneath the idea of D.E. The model of the stability is assessed in view of the circumstances of the Lyapunov. The planning of the function of Lyapunov is advancement in the difference equation idea. Additionally investigated the Lipschitz condition, and the existence of uniqueness of the system of equations. For this HIV model, the boundedness, non-negativity, continuity results have been examined. Also, disease free equilibrium and equilibrium of endemic are analyzed. The D.E. point is asymptotically stable when the reproduction number ℜ0< 1 and unstable when ℜ0> 1. Subsequently, they can anticipate the results of HIV/AIDS. Also, the review will assist with the help to fight against HIV/AIDS-phobia by policy-makers, NGOs, and other concerned organizations. Funding details: No funding was received for the preparation of this paper. Disclosure statement:No potential conflict of interest was reported by the authors. Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study. Acknowledgments Author Commitments: All authors contributed in basically the same manner to this paper. All authors read and supported last variant of this paper. Conflicts of Interest: The author proclaims no conflict of interest. Information Accessibility: Everything information expected for this exploration is incorporated in this paper. 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Lakshmi, Senthil Kumar, P., Dayalan, S.K., Ramkumar, C., 2024. Exploring Tuberculosis: A Theoretical Framework for Infection Regulation and Eradication, Communications on Applied Nonlinear. 31(3S), pp. 559–579 javascript:; javascript:; javascript:; javascript:; javascript:; javascript:; https://doi.org/10.1063/5.0189284 https://www.scopus.com/authid/detail.uri?authorId=59202746300 https://www.scopus.com/authid/detail.uri?authorId=59202301500 https://www.scopus.com/authid/detail.uri?authorId=59202746300 https://www.scopus.com/authid/detail.uri?authorId=58943994400 https://www.scopus.com/authid/detail.uri?authorId=56622903500