A Mathematical Model Incorporating Pre-Exposure Treatment for HIV 

We modified the Kermack-McKendrick compartmental model. There is no recovered state (R) for HIV, but we will include two states for both Pre-Exposure and Post-Exposure treatment methods (PrEP and ART respectively). The state variables of our system are S, P, I, A which correspond to the Susceptible, on PrEP, Infected, and on ART states. The infected states of our model are those individuals in the I state and those in the A state. The individuals in the I state are those infected and not taking the antiretroviral therapy treatment, while the individuals in the A state are those infected and taking the treatment. The total population of the system is defined as N [4].

Susceptible State

The first equation corresponds to the rate of change in the number of people in the susceptible population. Individuals can enter the susceptible state through recruitment or exiting PrEP treatment and thus becoming susceptible again. In our case, we are focusing on people in the United States between the ages of 20-49. Therefore, recruitment means becoming 20 years old. Meanwhile, people can leave the susceptible state by becoming infected from someone who is infected or on ART. However, the chance of becoming infected from someone who is on ART is much lower than getting the disease from someone who is infected and not on ART. In addition, people can also leave the susceptible state by entering PrEP or dying.

Pre-Exposure Prophylaxis State

The second equation relates to the rate of change of people on PrEP. People can enter the PrEP state by entering treatment through being prescribed the drug and taking it. People can exit the PrEP state by dying, dropping out of care by no longer taking the drug, becoming infected from someone who is infected, or becoming infected by someone who is on ART. It should be noted that the chance of becoming infected while on PrEP is extremely low, and in reality these terms related to infection will most likely approach zero.

Infected State

The third equation relates to the rate of change in the infected population. People can enter the infected state by exiting ART through no longer taking their pills or by getting the disease. There are three ways the disease can be transmitted in this model. Either a susceptible person is infected by someone in the infected state, an individual on PrEP is infected by someone in the infected state, or a susceptible person is infected by someone in the ART state. It should be noted that out of all of these, the first way is by far the most likely. Meanwhile, people can exit the infected state by dying or starting Antiretroviral Therapy.

Antiretroviral Therapy State

The fourth equation corresponds to the rate of change of people on ART. A person can enter this state by beginning treatment and taking their prescribed pills. A person can exit the ART state by dying or dropping out of treatment through no longer taking their medication.

Total Population Equation

The last equation simply states that the total population (N) is equivalent to the summation of all the other people in the four aforementioned states.