Table 2.

A list of equilibrium points in the mathematical model represented by the system of ODEs shown in Eq. (7)

Fixed point	Description	Existence condition
${\widehat{V}}_1=(S^{\ast },0,0,0)$	No infections	Always present in the model
${\widehat{V}}_2=({\widehat{S}}_r,{\widehat{R}}_r,{\widehat{\mathbf{X}}}_r,0)$	Infected only with resident strain. We define this as MFE	$|\mathbf{S}{\mathbf{F}}_r-{\mathbf{D}}_r{|}_{{\widehat{V}}_2}=0$
${\widehat{V}}_3=({\widehat{S}}_m,{\widehat{R}}_m,0,{\widehat{\mathbf{X}}}_m)$	Infected only with mutant strain	$|\mathbf{S}{\mathbf{F}}_m-{\mathbf{D}}_m{|}_{{\widehat{V}}_3}=0$
${\widehat{V}}_4=(\widehat{S},\widehat{R},{\mathbf{X}}_r^{\ast },{\mathbf{X}}_m^{\ast })$	Infected with resident or mutant strain (Co-infected)	$|\mathbf{S}{\mathbf{F}}_r-{\mathbf{D}}_r{|}_{\widehat{V}4}=0$ and $|\mathbf{S}{\mathbf{F}}_m-{\mathbf{D}}_m{|}_{{\widehat{V}}_4}=0$

Understanding the properties of fixed points is essential for understanding the system’s long-term behavior. In this computation, we leverage the fundamental mathematical theorem that states that if x is a non-zero vector and $Ax=0$, then the matrix A is singular, meaning that its determinant is zero ($|A|=0$)