Introduction
One of the most severe epidemics of modern society, which is often underestimated against the background of other diseases, is the human immunodeficiency virus (HIV). For several decades, HIV has been a major threat to the health of millions of people around the world. The virus, by entering the body through blood or sexual contact, effectively infects immune lymphocytes (Del Amo et al., 2020). As a consequence, a person’s immune ability to resist other infections drops dramatically. Such a patient, if not adequately treated, dies not from HIV but from an opportunistic infection. For a long time, HIV was thought to be a disease unique to homosexuals and drug addicts. However, years of research on the virus have shown that anyone, regardless of racial, sexual, or even cultural preference, has the same chance of contracting the infection. Understanding the epidemiological nature of HIV has forced clinicians to seek mechanisms to contain the disease quickly. One of the most apparent avenues was prevention education, where people were informed about the disease and told how to protect themselves from it. The next level of containment was the development of antiretroviral therapy in 1996, which allowed already sick people to effectively mask their diagnosis and live as if they were healthy (Forsythe et al., 2019). HIV therapy has a very important property, namely, the containment of new cases. An HIV-positive patient on therapy is unlikely to transmit the infection to his or her sexual partner or future child. As a consequence, such therapy is effective in curbing epidemiological growth. This paper evaluates the possibility of using mathematical modeling to identify an equation suitable for the spread of HIV. With knowledge of such an equation, it becomes possible to predict new cases. The first thing to say is that in any epidemiological disease, there is a distinction between new cases and current cases. In the case of HIV, it should be understood that the infection with this disease is chronic, which means that the patient will live with the diagnosis for life. Rare cases have been reported in which patients have fully recovered from HIV, so this is not addressed in current work (Molteni, 2021). Only new cases of infection are studied in this simulation since this is what allows us to assess the dynamics of the epidemic. To determine the equation describing the HIV epidemic, it was chosen to look at historical data. Specifically, Roser and Ritchie provided data on the global spread of new HIV cases over the past thirty years, since 1990 (Roser & Ritchie, 2019). This data set contains thirty rows and is presented in Appendix A; all data are in millions. Using MS Excel to graph these data, it became clear that the form of the appropriate equation had to be polynomial because the graph behaved dynamically. A built-in regression function was then used with a choice of trend line type. It is worth saying that regression is a statistical model that allows us to determine the curve equation that best fits the current data set. The indicator of the fit of the constructed model to the data set is the parameter R2, the coefficient of determination (LibreTexts, 2021). This coefficient reflects how much variance of all data was covered by the current model. Accordingly, the higher this coefficient, the better the equation fits the data set. Specific Equation The built-in regression function with polynomial trend line selection produces a specific regression equation, as shown below. One can perfectly see that this equation is polynomial in that it contains five terms of different orders. Furthermore, this equation cannot be simplified any further, which is also a property of polynomial functions. The greatest degree of the terms in this equation is four, and the other terms are also present, including the free coefficient. We can also see that some of their coefficients contribute positively to the overall equation, while three of the terms contribute negatively.
Graphical Representation
A graphical representation of the simulated equation was constructed using MS Excel. The figure below shows both the initially collected data set and the polynomial line. In addition, the regression equation is plotted, and the value of the R2 coefficient is shown. As noted earlier, this coefficient corresponds to the degree of fit between the model and the data set. The value of 0.9869 shows that about 98.69 of the total variance of the initial data set was covered by this equation (LibreTexts, 2021). In other words, this is an extremely good result that shows the reliability of the equation.
Forecast
As can already be seen from the above graph, the HIV epidemic was gaining momentum until 1996, after which a slow decline in new cases began. One cannot assume that this is due to the low availability of medicine because, on the contrary, an increasing number of potential patients have been receiving medical care over the past decades. Consequently, an important epidemiological event was accomplished in 1996 to reverse the trend in the spread of HIV. As already mentioned, this was due to the discovery of antiretroviral therapy, which was the salvation of humanity (Del Amo et al., 2020). Fewer and fewer people have been infected with HIV, even in contact with the patient. Mathematical prediction makes it possible to predict what development of the disease should be expected in the following years. In particular, it is well seen that the trend has been downward since 1996, which means that if the current trend continues, we can expect the number of new cases to continue to decrease in the coming years. Increased access to medicine, the expansion of antiretroviral therapy, and the development of new HIV vaccines will also contribute to this decrease. Interestingly, substituting years before 1990 also results in a decrease in the total number of new cases, as the graph shows. From this boundary, however, it is inappropriate to talk about an increase in the availability of medicine. On the contrary, the decrease in the number of cases over time shows that medicine was increasingly less developed, and diagnostic methods were not as widely used. Substituting the value of 2022, the current year, into the above equation yields several 1.33 million, which is the projected number of new HIV cases this year. Subsequent extrapolation reveals that the equation goes into negative numbers by 2026. Hence, the model predicts that there will be no more new HIV cases in the world by 2026 because therapy will be ubiquitous. This extrapolation is too unambiguous and does not take into account additional factors, but it allows us to trace the general trend in the development of the epidemic.