## 3 Results  
### Determination of Parameters  
As noted earlier, there are multiple parameters that are not well established for the epidemiological models. We have tried to reduce this problem by comparing the results of a simple model with the CDC data, initially without PrEP. Table 1 presents yearly data containing information from CDC case surveillance data related to the estimated high-risk population and infections. These data are used to calculate Figure 1, which plots the normalized percentages for the different categories with projections from our SIA model. We note that Table 1 contains estimated values for N, the total high-risk population. No value for this is given in the CDC data. To make this estimate, we took 3.9% of the U.S. population for each year minus the reported HIV and AIDS-related deaths [18]. As mentioned earlier, we use the data in Figure 1 along with our SIA model results to help determine some of the unknown (or disputed) parameters in the SIA model. We \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Year** & **Susceptibles** & **Infected** & **AIDS** & **N b** & **Susceptibles/N** & **Infected/N** & **AIDS/N** \\ \hline
2008 & 3,917,413 & 538,516 & 311,817 & 4,767,746 & 0.82207 & 0.11295 & 0.065401 \\ \hline
2009 & 3,937,002 & 556,262 & 320,702 & 4,813,966 & 0.818247 & 0.115552 & 0.066619 \\ \hline
2010 & 3,952,700 & 575,801 & 328,546 & 4,857,047 & 0.814221 & 0.11855 & 0.067643 \\ \hline
2011 & 3,973,413 & 593,774 & 335,292 & 4,902,479 & 0.810901 & 0.12117 & 0.068392 \\ \hline
2012 & 3,995,338 & 612,928 & 342,363 & 4,950,629 & 0.807443 & 0.123808 & 0.069155 \\ \hline
2013 & 4,017,589 & 630,660 & 348,925 & 4,997,174 & 0.804375 & 0.126203 & 0.069824 \\ \hline
2014 & 4,043,226 & 650,714 & 349,891 & 5,043,831 & 0.802017 & 0.129012 & 0.06937 \\ \hline
2015 & 4,066,157 & 670,052 & 353,147 & 5,089,356 & 0.799349 & 0.131658 & 0.069389 \\ \hline
2016 & 4,088,924 & 688,579 & 356,601 & 5,134,104 & 0.796817 & 0.134119 & 0.06457 \\ \hline
2017 & 4,110,186 & 706,439 & 359,520 & 5,176,145 & 0.794453 & 0.13648 & 0.06457 \\ \hline
2018 & 4,132,070 & 723,948 & 362,516 & 5,218,534 & 0.792193 & 0.138726 & 0.06467 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CDC case surveillance data with normalization a found good agreement, as shown in Figure 1, with the parameters listed in Table 2. We chose to use two average sexual partners per year and adjusted the remaining parameters to best match the CDC surveillance data. The key parameters that were determined in this way were \(\beta,\ \eta\), and \(\alpha\). We note that PrEP was introduced in 2012. This information helped determine various rates in our model involving PrEP. Figure 2 shows how several death rates and development of AIDS rate evolved during this period. It is easily seen that the rates in Figure 2 are nearly constant after 2012. Because of this, we simply used the values from 2017 for our parameters in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(N(0)\) & Initial total high-risk population. & \(4,767,746\) people 2  \\ \hline \(I(0)\) & Initial total infected. & \(538,516\) people 2  \\ \hline \(A(0)\) & Initial total HIV stage III (AIDS) population. & \(311,817\) people 2  \\ \hline \(S(0)\) & Initial susceptible population. & \(N(0)-I(0)-A(0)\) \\ \hline \(G\) & Recruitment rate. & \(67,000\) /year \\ \hline \(r\) & Average number of sexual partners per year. & 2 \\ \hline \(\beta\) & Transmissions per partner. & 0.047 \\ \hline \(\eta\) & Modification parameter for relative AIDS & 1.235 \\  & infectiousness. & 0.0045 /year \\ \hline \(\mu\) & Natural death rate. & 0.033 /year \\ \hline \(\
u\) & HIV-related death rate. & 0.015557 /year \\ \hline \(\
u_{a}\) & AIDS-related death rate. & 0.023924 /year \\ \hline \(\gamma\) & Rate of leaving high-risk population into AIDS. & 0.016960 /year \\ \hline \(\alpha\) & A into treatment and able to transmit HIV. & 0.0045 /year \\ \hline \end{tabular}
\end{table} 
Table 2: List of SIA model parameters 3.