## 2 Methods
### The SPCBIA Model  
In the SPCBIA model we now include PrEP and condom usage and explore the resulting infection projections. As before, each susceptible compartment allows for natural death or infection. There is an allowance for those using PrEP who discontinue usage and return to the general susceptible compartment for various medical reasons or life decisions. The condom usage compartment also allows for the discontinued usage of condoms, as this is the current trend in our high-risk group [17]. In the case that someone in the \(B\) compartment should cease using PrEP, it is assumed they will continue using condoms and return to that compartment. All four infected subgroups, similarly to the SIA model, allow for natural death, infection-related death, and progression to the AIDS state. Likewise, the AIDS compartment still allows for continued infection.  
Our non-normalized equations for the SPCBIA model are:  
\[S_{tot}=S+P+C+B \tag{4}\]  
\[I_{tot}=I_{S}+I_{P}+I_{C}+I_{B} \tag{5}\]  
\[\frac{dS}{dt}=G+[1-\psi(1-\chi)(1-\delta)-\delta-\chi\psi(1-\delta)]\frac{dS_{ tot}}{dt}-r\beta\left(\frac{I_{tot}+\eta A}{N}\right)S \tag{6}\]  
\[\frac{dP}{dt}=\psi(1-\chi)(1-\delta)\frac{dS_{tot}}{dt}-(1-k)r\beta\left( \frac{I_{tot}+\eta A}{N}\right)P-(\mu+\theta)P \tag{7}\]  
\[\frac{dC}{dt}=\delta\frac{dS_{tot}}{dt}-(1-L)r\beta\left(\frac{I_{tot}+\eta A }{N}\right)C-(\mu+\xi)C+\omega B \tag{8}\]  
\[\frac{dB}{dt}=\chi\psi(1-\delta)\frac{dS_{tot}}{dt}-Fr\beta\left(\frac{I_{tot }+\eta A}{N}\right)B-(\mu+\omega)B \tag{9}\]  
\[\frac{dI_{S}}{dt}=r\beta\left(\frac{I_{tot}+\eta A}{N}\right)S-(\gamma+\
u+ \mu)I_{S}+\alpha A \tag{10}\]  
\[\frac{dI_{P}}{dt}=(1-k)r\beta\left(\frac{I_{tot}+\eta A}{N}\right)P-(\gamma+ \
u+\mu)I_{P} \tag{11}\]  
\[\frac{dI_{C}}{dt}=(1-L)r\beta\left(\frac{I_{tot}+\eta A}{N}\right)C-(\gamma+ \
u+\mu)I_{C} \tag{12}\]  
\[\frac{dI_{B}}{dt}=Fr\beta\left(\frac{I_{tot}+\eta A}{N}\right)B-(\gamma+\
u+ \mu)I_{B} \tag{13}\]  
\[\frac{dA}{dt}=\gamma I_{tot}-(\alpha+\
u_{a}+\mu\ )A \tag{14}\]  
The new parameters are defined as follows: \(I_{tot}\) is the sum of the individual infected subgroups, \(\psi\) is the proportion of \(S_{tot}\) starting PrEP, \(\delta\) is the proportion of \(S_{tot}\) starting to use condoms, \(\chi\) is the proportion of \(S_{tot}\) using both PrEP and condoms, \(\theta\) is the portion of \(P\) who stop using PrEP and move back to \(S\), \(\xi\) is the proportion of \(C\) stopping condoms use and returning to \(S\), \(k\) is a modification parameter to account for PrEP effectiveness, \(L\) is a modification parameter to account for condom effectiveness, and \(F\) is a modification parameter to account for the compounded effectiveness of using both.  
Equations (4) and (5) simply represent how to calculate the total susceptible population and the total infected population, respectively. The remaining equations calculate the time evolution of the individual categories within each larger population. As an example, we discuss Equation (11) which calculates how the number of infected PrEP users changes with time. The number of \(\mathrm{I_{P}}\) cases can decrease through natural death (\(\mu\)), death through HIV (\(v\)), or can be transferred to another category (\(\gamma\)), i.e., the development of AIDS. Similarly, an increase in the number of \(\mathrm{I_{P}}\) cases is proportional to the susceptible population using PrEP, (\(P\)), with the proportionality constant including terms involving the number of partners per year (\(r\)), the probability of transmission \(\beta\), the probability of meeting an infected partner \(\left(I_{tot}+\eta A\right)/N\), and an estimation of PrEP effectiveness (\(1-k\)).  
Numerically, the solution for the coupled set of equations is carried out using Runge-Kutta integration methods with a time step of 0.001 years. For the SIA model we checked that our results were consistent with analytic equilibrium calculations.