Mathematical Analysis of the Probability of Spontaneous Mutations in HIV-1 Genome and Their Role in the Emergence of Resistance to Anti-Retroviral Therapy

$\mathbf{\text{Background:}}$ High mutability of HIV is the driving force of antiretroviral drug resistance, which represents a medical care challenge. $\mathbf{\text{Method}}\mathbf{\text{and}}\mathbf{\text{Model}}\mathbf{\text{Equation:}}$ To detect the mutability of each gene in the HIV-1 genome; a mathematical analysis of HIV-1 genome is performed, depending on a linear relation wherein the probability of spontaneous mutations emergence is directly proportional to the ratio of the gene length to the whole genome length. \begin{equation*} {P_g}{S_i} =\frac{g}{G} \end{equation*} $\mathbf{\text{Results:}}$ $\mathbf{\text{tat}}$, $\mathbf{\text{vpr}}$ and $\mathbf{\text{vpu}}$ are the least mutant genes in HIV-1 genome. Protease $\mathbf{\text{PROT}}$ gene is the least mutant gene component of the polymerases $\mathbf{\text{pol}}$. $\mathbf{\text{Conclusion:}}$ $\mathbf{\text{tat}}$, $\mathbf{\text{vpr}}$ and $\mathbf{\text{vpu}}$ are the best candidates for HIV-1 recombinant subunit vaccines or as a part of $\mathit{\text{prime}}\mathit{\text{and}}\mathit{\text{boost}}$ vaccine combinations. Also; the protease inhibitor-based regime represents a high genetic barrier for HIV to overcome.