The theoretical mathematical model with the Social Determinants of Health has been developed in stages.

### Social determinants of health

For the selection of SDH that were incorporated into the model, a narrative literature review was conducted focused on peer-reviewed articles, in PubMed/Medline and Scopus databases between 2010 and 2020. This review included key words HIV, AIDS and Social Determinants. A total of 31 SDH were obtained in the review, divided into four groups: Individual Factors, Socioeconomic Factors, Social Participation and Health Services (Fig. 1). In the end, four determinants were selected for incorporation into the model: Education, Employment, Use of Drugs and Alcohol abuse and Condoms Use. The choice of these SDH was based on the recurrence; recognized importance in the literature and the capacity for measurement and quantification, in order to be incorporated into the equations of the Compartmental Model. Although these four chosen determinants are among the most studied and important in the field of HIV/AIDS, they do not have the capacity and robustness to represent all other determinants and groups.

### Education

Education is recognized as an important social determinant of health, being directly associated with the socioeconomic development and well-being of individuals. Different educational levels in a society lead to economic disparities and especially health inequities. Individuals with higher educational levels, in general, have better job opportunities, a lower unemployment rate, better economic conditions and, consequently, better psychosocial conditions for decision-making about their own health. Thus, education, within society, has an influence and impact on life expectancy and its morbidities^{24}.

For HIV/AIDS, education plays an important role in reducing its incidence and prevalence, especially in low- and low-income countries. As many of the young people in developing countries who attend school have not yet had sexual intercourse, education for these young people is critical to the success of HIV/AIDS prevention programs. At this time, it is possible to guide young people to delay the beginning of their sexual activities, as well as encourage the use of protective methods such as condoms^{25}. Petifor et al. (2016)^{26} noted that social protection programs, such as conditional cash transfer to school attendance, made young women stay in school longer, reducing the risk of acquiring HIV. Furthermore, education helps people to access and adhere to ART, as well as helping to reduce stigma, discrimination and gender inequality^{27}.

### Poverty and employment

Between the years 2020 and 2021, largely as a result of the COVID-19 pandemic, the number of unemployed people in the world increased from 187.3 million to 220.3 million, the largest annual increase in unemployment in this period of time, reaching the rate average of 6.5%^{28}. Many of these people who have lost their jobs have lost their only source of income, both personal and family, resulting in a drastic reduction in labor income and a consequent increase in poverty. Compared to the year 2019, 108 million more workers are now considered to be living in poverty or extreme poverty^{29}. Several studies have documented the association between unemployment and poor health status^{30,31,32}. Unemployed individuals, have their physical and mental health affected, more likely to suffer from depression, anxiety, low self-esteem, demoralization, worry and physical pain^{33}. In addition, these individuals end up having their perception of health reduced, exposing themselves more to risk of HIV infection, with risk behaviors such as exchanging sex for money, often with multiple partners and without condoms; low demand for care and health services, and consequently delay in detection and initiation of treatment for HIV/AIDS and increase considerable in mortality^{34,35}.

### Drug and alcohol abuse

Alcohol and drug abuse, whether injectable or non-injectable, are intrinsically associated with an increased risk of HIV infection. The main factor of injecting drugs is the sharing of syringes and needles, while alcohol and other types of drugs favor the increase in risky behavior due to the exchange of sex for drugs or money, sexual disinhibition^{36}. Approximately 15.6 million people inject drugs worldwide, while around 2.8 million of those are living with the HIV virus^{37}. Sharing needles and syringes has the second highest probability (of 10%) of HIV transmission per act, second only to receptive anal sex. One in 160 people becomes infected every time they share a syringe, with 10% of HIV cases in the USA being attributed to this risky behavior^{38}. In this way people who inject drugs (PWID) are exposed to the double route of contamination, sexual and intravenous. Another relevant factor is that substance abuse, alcohol consumption, as well as unemployment are factors related to reduced adherence to antiretroviral treatment^{39}. Therefore, the use of alcohol and drugs is a SDH that must be considered in the mathematical modeling of HIV/AIDS.

### Condom use

Approximately 95% of cases, worldwide, of people being infected by HIV are attributed to sexual practices without using condoms. Condoms are the best known, most accessible and effective method to prevent HIV infection and other sexually transmitted infections, such as syphilis, gonorrhea and also some types of hepatitis^{40}. Consistent condom use can reduce HIV transmission among serodiscordant individuals by up to 80%^{41}.

### Mathematical modeling

Compartmental model is a type of mathematical model that simulates the disease status of individuals within populations, which are divided into different compartments. Within each compartment, people are considered homogeneous in terms of their behavior and risk factors^{16,17}. The most generic and widely used model is the SIR model, in which individuals are classified into three types of compartments: Susceptible (S), Infected (I) and Recovered (R). Susceptible individuals are those who have never had the disease or are likely to become infected. After infection, these individuals migrate to the infected compartment and can spread the disease to susceptible individuals. The recovered ones can develop lifetime immunity or return to the Susceptible compartment, depending on the pathophysiology of the disease. In the case of HIV, the Recovered compartment does not exist, as this disease has no cure. However, a variation of the SIR model for HIV is the replacement of the Recovered compartment by the virally suppressed compartment (T), where individuals in these compartments have an undetectable viral load and their contribution to transmission is almost zero^{42}. For this study, we propose an extended model in which the population (N) is divided into Susceptible (S), HIV-positive (I), Individual with AIDS (A) and individual virally suppressed (T). Thus, we consider the total population (N), where N = S + I + A + T. One way to model the dynamics of virus transmission is given by the system of equations:

$$\frac{dS}{dt}= \kappa – \mu S – \beta \frac{I}{N}S$$

$$\frac{dI}{dt}= \beta \frac{I}{N}S{+\alpha }_{1}T-\rho I -{\gamma }_{1}I-\mu I$$

$$\frac{dA}{dt}= \rho I {+\alpha }_{2}T- {\gamma }_{2}A -{\delta }_{1}A-\mu A$$

$$\frac{dT}{dt}= {\gamma }_{1}I+{\gamma }_{2}A -{\delta }_{2}T-{\alpha }_{1}T-{\alpha }_{2}T-\mu T$$

where \(\kappa \) represents the population’s natality rate and \(\mu \) the natural mortality rate (inverse of the average life expectancy). \(\beta \) represents the effective contact rate, I/N the fraction of infected individuals. Thus, the term \(\beta \frac{I}{N}\) represents the HIV transmission rate. After a period \({\rho }^{-1}\), the infected individual becomes “full-blown aids”. We consider that the individual in class I, with the assumption that treatment occurs, after a time \({{\gamma }_{1}}^{-1}\) they reaches undetectable viral load levels, the same applies to the individual in compartment A, in this case after a time \({{\gamma }_{2}}^{-1}\). However, if this individual with an undetectable viral load ceases treatment, the viral load may increase, which may lead the individual to class I (infectious) or even to compartment A, with rates \({\alpha }_{1}\mathrm{and}\) \({\alpha }_{2}\), respectively. We assumed that individuals in compartment A have HIV-related death, \({\delta }_{1}\), as well as for individuals in compartment T, \({\delta }_{2}\) (with \({\delta }_{2}<{\delta }_{1}\)). Figure 2 shows the epidemiological scheme of the HIV/AIDS transmission model.

### Education

One way to include the education factor would be to incorporate a new compartment, denoted here by (R-removed), representing individuals who through educational campaigns change their sexual behavior so as not to be susceptible to HIV transmission, as shown by^{42}. Parameter \({\theta }_{1}\) represents this behavior. However, these individuals can change their behaviors again and return to infection susceptibility at a rate \({\theta }_{2}\). The dynamics of this class is given by the equation,

$$\frac{dR}{dt}={ \theta }_{1}S-{ \theta }_{2}R-\mu R$$

Bearing in mind that as we incorporated a new compartment into the model, the total population is now given by N = S + I + A + T + R and the susceptible dynamics is modified by including the term \({-\theta }_{1}S\).

### Condom use

To incorporate the effect of condom use, we can include this factor through the parameters \(\varepsilon \) and \(\nu \), which represent the condom efficacy and compliance, respectively. So, the product \(c=\varepsilon \nu \) represents the level of protection against HIV through condom use^{43}. We modified the effective contact rate, which goes as follows:

where \(1-c\) measures the failure to prevent transmission through condom.

### Drug and alcohol abuse

One way to incorporate the drug abuse as SDH in a mathematical model is to restrict the study to this subpopulation, that is, consider a compartmental model in which all compartments are related to drug users^{44}. For instance, Burattini et al. (1998)^{45} showed the impact of crack-cocaine use on the prevalence of HIV/AIDS among drug users. Alcohol abuse can be incorporated in the model decreasing (or increasing) the parameter values, which can be influenced by alcohol use. For instance, the level of protection against HIV through condom use (\(c\)) as

with \({\phi }_{c}<1\), representing the alcohol abuse effect. In the case of absence of alcohol abuse effect in the mathematical model, we assume \({\phi }_{c}=1.\)

### Poverty and unemployment

The inclusion of factors such as poverty and unemployment can be applied to several parameters in the model. Galanis and Hanieh (2021)^{19} explore the use of these factors by modifying the transmission rate by a linear approximation of

$$\beta {(x}_{1},{x}_{2}) = {\beta }_{0}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2}$$

where \({x}_{1}\) and \({x}_{2}\) are the poverty and unemployment rates, respectively.

We can apply this modification to parameters such as \({\alpha }_{1}\), \({\alpha }_{2}\), \({\gamma }_{1}\) and \({\gamma }_{2}\) , as they can also be influenced by social determinants. Thus, the model with the inclusion of the social determinants mentioned here has the following form:

$$\frac{dS}{dt}= \kappa – \mu S – (1-c)\beta {(x}_{1},{x}_{2}) \frac{I}{N}S{ -\theta }_{1}S+{ \theta }_{2}R$$

$$\frac{dI}{dt}=(1-c)\beta {(x}_{1},{x}_{2}) \frac{I}{N}S{+\alpha }_{1}T-\rho I -{\gamma }_{1}I-\mu I$$

$$\frac{dA}{dt}= \rho I {+\alpha }_{2}T- {\gamma }_{2}A -{\delta }_{1}A-\mu A$$

$$\frac{dT}{dt}= {\gamma }_{1}I+{\gamma }_{2}A -{\delta }_{2}T-{\alpha }_{1}T-{\alpha }_{2}T-\mu T$$

$$\frac{dR}{dt}={ \theta }_{1}S-{ \theta }_{2}R-\mu R$$

### Numerical simulation

Here, we present an application of the model described above. We fitted the model with data of new cases and deaths of AIDS collected from the Brazilian Health Ministry^{46}. We estimated the natural mortality rate and birth rate in order to the model approximate the total population of Brazil between 2003 and 2030. Table 1 shows the values of parameters used in the simulations.

Assuming the range for each parameter shown in Table 1, we conducted a sensitivity analysis using a statistical variance-based method^{53} to evaluate the model parameter effects in the dynamics of the all variables over time. Since we considered AIDS data for the calibration process, we present in Fig. 3 the sensitivity analysis for the A compartment over 2003 to 2019. The most influent parameter, for the AIDS compartment, is the HIV transmission rate.

Hence, we assumed the correlation of poverty for only this parameter. The poverty rate was calculated using three National Household Surveys (PNAD)—the regular PNAD, Continuous-PNAD, and PNAD-COVID for 2001–2011, 2012–2019, and 2020, respectively. After 2020, we assumed two different scenarios of poverty rate, scenarios of growth and poverty reduction between 2020 and 2030 (Fig. 4). We forecast the HIV/AIDS incidence and mortality rates (per 100,000 individuals), according to the level of poverty (Fig. 4).

Figure 5 shows that depending on the level of poverty in the country, in the near future, we can expect of an increase (decrease) on these health outcomes. This result shows the importance of governmental policies to poverty mitigation in order to reduce the incidence and mortality of HIV/AIDS.