We studied how COVID-19 and kidney disease impact each other by examining them separately first. After understanding each individually, they’re combined to see the overall effect. The goal is to ensure the combined results are accurate and logical.
COVID-19-only model: when we exclude kidney disease infections, we can formulate a COVID-19-specific sub-model from the full disease model; we get \({I}_{k}=0,{I}_{kc}=0,{I}_{kd}=0,{I}_{kdc}=0\)
$$\begin{gathered} \frac{dS}{{dt}} = \Delta - \frac{{\phi_{1} I_{c} }}{N}S - \mu S \hfill \\ \frac{{dI_{c} }}{dt} = \frac{{\phi_{1} I_{c} }}{N}S - \tau_{1} I_{c} - \mu I_{c} \hfill \\ \frac{dR}{{dt}} = \tau_{1} I_{c} - \mu R \hfill \\ \end{gathered}$$
(2)
Table of Contents
Theorem 1
All the populations of the system with positive initial conditions are nonnegative.
Assume \({\text{S}}\left(0\right)>0,{{\text{I}}}_{{\text{C}}}(0)>0,{\text{R}}(0)>0\) are positive for time \({\text{t}}> 0\) and for all nonnegative parameters.
From the initial condition, all the state variables are nonnegative at the initial time; then, \(\mathrm{t }> 0\)
To show the solutions of the model, as it is positive, first, we take \(\frac{{\text{dS}}}{{\text{dt}}}\) from equation
$$\frac{{\text{dS}}}{{\text{dt}}}=\Delta -\frac{{\upphi }_{1}{{\text{I}}}_{{\text{c}}}}{{\text{N}}}{\text{S}}-\mathrm{\mu S}$$
$$\frac{{\text{ds}}}{{\text{dt}}}=\Delta -\left(\frac{{\upphi }_{1}{{\text{I}}}_{{\text{c}}}}{{\text{N}}}+\upmu \right){\text{s}}$$
$${\text{S}}\left({\text{t}}\right)={\text{S}}(0){\text{exp}}\left(-\underset{0}{\overset{{\text{t}}}{\int }}\left\{\frac{{\upphi }_{1}{{\text{I}}}_{{\text{c}}}}{{\text{N}}}+\upmu \right\}{\text{du}}\right)+\underset{0}{\overset{{\text{t}}}{\int }}\Delta \mathrm{ exp}(\underset{0}{\overset{{\text{x}}}{\int }}\left\{\frac{{\upphi }_{1}{{\text{I}}}_{{\text{c}}}}{{\text{N}}}+\upmu \right\}{\text{du}})\mathrm{dx }\times {\text{exp}}\left(-\underset{0}{\overset{{\text{t}}}{\int }}\left\{\frac{{\upphi }_{1}{{\text{I}}}_{{\text{c}}}}{{\text{N}}}+\upmu \right\}{\text{du}}\right)>0$$
Accordingly, all the variables are nonnegative in \([0,{\text{t}}]\), so \({\text{S}}\left(0\right)>0,\) similarly we can show \({{\text{I}}}_{{\text{C}}}(0)>0,{\text{R}}(0)>0\).
Theorem 2
The dynamical system represented by the COVID-19 submodel remains positively invariant within the closed invariant set defined by \({\rm Z}_{c} = \left\{\left( S,{I}_{c},R\right)\epsilon {R}^{3}+ : N\le \frac{\Delta }{\mu }\right\}\)
An invariant region is identified to demonstrate that the solution remains within certain bounds. This invariant region provides a constraint ensuring that the solution does not exceed these limits; we have
$$\frac{dN}{dt}=\frac{dS}{dt}+\frac{{dI}_{C}}{dt}+\frac{dR}{dt}$$
$$\frac{dN}{dt}=\Delta -\frac{{\phi }_{1}{I}_{c}}{N}S-\mu S+\frac{{\phi }_{1}{I}_{c}}{N}S-{\tau }_{1}{I}_{c}-\mu {I}_{c}+{\tau }_{1}{I}_{c}-\mu R$$
$$\frac{dN}{dt}=\Delta -\left(S+{I}_{c}+R\right)\mu$$
$$\frac{dN}{dt}=\Delta -N\mu$$
$$N\left(t\right)=N\left(0\right){e}^{-\mu t}+\frac{\Delta }{\mu }(1-{e}^{-\mu t})$$
As,\(t\to \infty\), we get \(0\le N\le \frac{\Delta }{\mu }\), the theory of differential equation27 in the region.
\({\rm Z}_{c} = \{\left( S,{I}_{c},R\right)\epsilon {R}^{3}+ : N\le \frac{\Delta }{\mu } \}\), For the autonomous system representing the COVID-19-only model, given by (2), any solution that starts in \({Z}_{c}\) will stay within \({Z}_{c}\) for all \(t\ge 0.\) Based on Cheng et al., this means that \({Z}_{c}\) acts as a stable and attractive region. Therefore, according to Naicker et al., the dynamics of model (2) are both mathematically sound and relevant to epidemiology, and it is appropriate to study its tabiliz within \({Z}_{c}.\)
Stability analysis of equilibrium states: In the only COVID-19 sub-model, the equilibrium state is reached when the following conditions are met
$$\frac{dS}{dt}=\frac{{dI}_{c}}{dt}=\frac{dR}{dt}=0$$
For the isolated COVID-19 model represented by the system (2), the state without any active disease (termed the disease-free equilibrium or DFE) is derived by setting each component of the system (2) to zero. At this DFE, neither infections nor recoveries are present.
Therefore, for the stand-alone COVID-19 model (2), the DFE is described \({\Omega }_{c}=\left(S,{I}_{C},R\right)=(\frac{\Delta }{\mu },\mathrm{0,0})\)
The sub-model’s basic reproduction number is the average number of secondary infections caused by a single COVID-19-infected person in a totally susceptible population. The system (2) calculates it using the next-generation matrix.
$${R}_{oc}=\frac{{\phi }_{1}}{({\tau }_{1}+\mu )}$$
(3)
The basic reproduction number, \({R}_{0c}\), represents the average number of people one infected individual is expected to infect over their entire infectious period within a completely susceptible population.
Theorem 3
For the kidney disease sub-model, the point of equilibrium without the disease is represented as \({\Omega }_{0c}\), remains stable as long as the basic reproduction number \({R}_{oc}\) is less than 1.
The Jacobian matrix is tabiliz to ascertain the equilibrium points’ local stability. For sub-model (2), the Jacobian matrix is formulated as \(J=\left(\begin{array}{c}\frac{\partial {f}_{1}}{\partial S} \frac{\partial {f}_{1}}{\partial {I}_{C}} \frac{\partial {f}_{1}}{R}\\ \frac{\partial {f}_{2}}{\partial S} \frac{\partial {f}_{2}}{\partial {I}_{C}} \frac{\partial {f}_{2}}{R} \\ \frac{\partial {f}_{3}}{\partial S} \frac{\partial {f}_{3}}{\partial {I}_{C}} \frac{\partial {f}_{3}}{R}\end{array}\right)\)
$$J=\left(\begin{array}{c}-\frac{{\varnothing }_{1}{I}_{c}}{N}-\mu \frac{{\varnothing }_{1}S}{N} \,\,\,\,\,\,\,0\\ \frac{{\varnothing }_{1}{I}_{c}}{N} -{\tau }_{1}-\mu \,\,\,\,\,\,\,\,\,\,0 \\ 0 \,\,\,\,\,\,\,\,\,\,{\tau }_{1} -\mu \end{array}\right)$$
The Jacobian matrix for the sub-model, when evaluated at the disease-free equilibrium point \({\Omega }_{0c}\), is expressed as
$$J({\Omega }_{0c})=\left(\begin{array}{c}-\mu \,\,\,\,\,\,\,\,\frac{{\varnothing }_{1}\Delta }{\mu N} \,\,\,\,\,\,\,0 \\ 0 \,\,\,\,\,-\left({\tau }_{1}+\mu \right) \,\,\,\,\,\,\,0\\ 0 \,\,\,\,\,\,\,\,{\tau }_{1} -\mu \end{array}\right)$$
In this context, one of the eigenvalues for \({\Omega }_{0c}\) is \(\lambda =-\mu\). The other eigenvalues can be conveniently derived from the associated submatrix.
$${J}_{1}=\left(\begin{array}{cc}-\left({\tau }_{1}+\mu \right)& 0\\ {\tau }_{1}& -\mu \end{array}\right)$$
To confirm the local stability of the disease-free equilibrium point, two conditions need to be met:
(i) The trace of \({J}_{1}\) should be less than zero. (ii) The determinant of \({J}_{1}\) should be greater than zero.
The trace is Trc \(\left({J}_{1}\right)=-({\tau }_{1}+2\mu ),\) which is less than zero.
$${\text{det}}\left({J}_{1}\right)=\left({\tau }_{1}+\mu \right)\mu >0$$
As a result, the COVID-19 sub-model’s disease-free equilibrium point is asymptotically stable.
Theorem 5. The COVID-19 submodel has an isolated endemic equilibrium point if \({R}_{0c}>1\).
The endemic equilibrium point of the COVID-19 sub-model is the solution of the system of equation in (4).
$$\Delta -\left({{\text{f}}}_{{\text{c}}}+\mu \right)S=0$$
$${f}_{c}S-\left({\tau }_{1}+\mu \right){I}_{c}=0$$
$${\tau }_{1}{I}_{c}-\mu R=0$$
To solve this system of equations,
we express it in terms of
$${f}_{c}^{*}=\frac{{\phi }_{1}{I}_{c}^{*}}{N}$$
(4)
$${S}^{*}=\frac{\Delta }{{f}_{c}^{*}+\mu }, {I}_{c}^{*}=\frac{{f}_{c}^{*}S}{({\tau }_{1}+\mu )}, {R}^{*}=\frac{{\tau }_{1}{I}_{c}*}{\mu },$$
(5)
Now,
$${I}_{c}^{*}=\frac{{f}_{c}^{*}S}{({\tau }_{1}+\mu )}$$
$${I}_{c}^{*}=\frac{\Delta {f}_{c}^{*}}{({\tau }_{1}+\mu )({f}_{c}^{*}+\mu )}$$
So, using (4)
$${f}_{c}^{*}=\frac{{\phi }_{1}{I}_{c}^{*}}{N}$$
$${f}_{c}^{*}=\frac{{\phi }_{1}\mu }{\left({\tau }_{1}+\mu \right)}-\mu$$
$${f}_{c}^{*}=\mu (\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)}-1)$$
$${f}_{c}^{*}=\mu ({R}_{0c}-1)$$
The conclusion drawn is that the infection force \({f}_{c}^{*}\) will be positive at the endemic equilibrium point \({\Omega }_{0c}\) only when \({R}_{oc}>1\). With this, we have effectively demonstrated the related theorem.
Theorem 5
Analysis of the Global Stability Analysis for the Endemic Equilibrium Point.
The endemic equilibrium point \({\Omega }_{c}\) undergoes a global stability analysis using the Lyapunov function method. To facilitate this analysis, we establish the
$$L=\frac{1}{2}((S-{S}^{*})+\left({I}_{c}-{I}_{c}^{*}\right)+{\left(R-{R}^{*}\right))}^{2}$$
(6)
The Lyapunov function L consistently maintains a positive value and only becomes zero at the endemic equilibrium point and differentiating with respect to time \(t\)
$$\begin{gathered} \frac{{{\text{dL}}}}{{{\text{dt}}}} = \left\{ {\left( {{\text{S}} - {\text{S}}^{*} } \right) + \left( {{\text{I}}_{{\text{c}}} - {\text{I}}_{{\text{c}}}^{*} } \right) + \left( {{\text{R}} - {\text{R}}^{*} } \right)} \right\}\left( {\frac{{{\text{dS}}}}{{{\text{dt}}}} + \frac{{{\text{dI}}_{{\text{c}}} }}{{{\text{dt}}}} + \frac{{{\text{dR}}}}{{{\text{dt}}}}} \right) \hfill \\ = \left\{ {\left( {{\text{S}} + {\text{I}}_{{\text{c}}} + {\text{R}}} \right) - \left( {{\text{S}}^{*} + {\text{I}}_{{\text{c}}}^{*} + {\text{R}}^{*} } \right)} \right\}\left( {\Delta - {\mu N}} \right) \hfill \\ = \frac{{\left( {{\mu N} - \Delta } \right)}}{{\upmu }}\left( {\Delta - {\mu N}} \right) \hfill \\ = - \frac{{\left( {\Delta - {\mu N}} \right)^{2} }}{{\upmu }} \hfill \\ \frac{{{\text{dL}}}}{{{\text{dt}}}} \le 0 \hfill \\ \end{gathered}$$
For \({R}_{oc}>1\), the endemic equilibrium point exists, leading to \(\frac{dL}{dt}\) is less than zero. It seems that the function L appears as a clear-cut Lyapunov function, suggesting that the endemic equilibrium point reaches asymptotic and global stability. From a biological perspective, this signifies that COVID-19 has remained prevalent in the community over a prolonged duration.
Analysing the sensitivity-only COVID-19 model
We conducted a sensitivity analysis of parameters within the COVID-19 sub-model. The behavior of the model in response to modest changes in a parameter’s value is known as the parameter’s sensitivity and is tabilize by the symbol \({\phi }_{1}\). It can be expressed as
$${R}_{oc}=\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)}$$
$${S}_{{\phi }_{1}}=\frac{\partial {R}_{0c}}{\partial {\varnothing }_{1}} \frac{{\phi }_{1}}{{R}_{0c}}=\frac{1}{\left({\tau }_{1}+\mu \right)} \frac{{\phi }_{1}}{\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)}}=+1$$
$${S}_{\mu }=\frac{\partial {R}_{0c}}{\partial \mu } \frac{\mu }{{R}_{0c}}= - \frac{{\phi }_{1}}{{\left({\tau }_{1}+\mu \right)}^{2}} \frac{\mu }{\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)}}=-\frac{\mu }{(\mu +{\tau }_{1})}$$
$${S}_{{\tau }_{1} }=\frac{\partial {R}_{0c}}{\partial {\tau }_{1}} \frac{{\tau }_{1}}{{R}_{0c}}=-\frac{{\phi }_{1}}{{\left({\tau }_{1}+\mu \right)}^{2}} \frac{{\tau }_{1}}{\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)}}=-\frac{{\tau }_{1}}{\left({\tau }_{1}+\mu \right)}$$
Table 1 displays the data for the sensitivity indices related to the sole COVID-19 sub-model. This sub-model analysis reveals that the COVID-19 contact rate is \({\phi }_{1}\), play a significant role in intensifying the virus’s spread. This trend results from an upsurge in secondary infections when these parameters increase, as highlighted by (Martcheva 2015). Conversely, parameters such as \({\tau }_{1}\) and \(\mu\) have a diminishing effect, meaning an uptick in their values could reduce the infection rate. A visual depiction of the sensitivity indices for \({R}_{oc}\) is showcased in Fig. 2.
The graphical depiction of the sensitivity indices concerning the primary reproduction number \(({R}_{oc})\) parameters are shown in the COVID-19 sub-model.
Kidney disease-only model
Kidney disease-only sub-model from the co-infection model, we get \({I}_{c}=0,{I}_{kc}=0,{I}_{kdc}=0,R=0\)
$$\begin{gathered} \frac{dS}{{dt}} = \Delta - f_{k} S - \mu S \hfill \\ \frac{{dI_{k} }}{dt} = f_{k} S - \sigma_{1} I_{k} - \mu I_{k} \hfill \\ \frac{{dI_{kd} }}{dt} = \sigma_{1} I_{k} - \mu I_{kd} \hfill \\ \end{gathered}$$
(7)
Theorem 6
All the populations of the system with positive initial conditions are nonnegative.
Assume \({\text{S}}(0) > 0,{{\text{I}}}_{{\text{k}}}(0) >0,{{\text{I}}}_{{\text{k}}}(0) > 0\) are positive for time \(\mathrm{t }>0\) and all nonnegative parameters.
From the initial condition, all the state variables are nonnegative at the initial time; then, \(\mathrm{t }>0\).
To show the solutions of the model, as it is positive, first, we take \(\frac{{\text{dS}}}{{\text{dt}}}\) from equation
$$\begin{gathered} \frac{{{\text{dS}}}}{{{\text{dt}}}} = \Delta - \frac{{\phi_{2} {\text{I}}_{{\text{k}}} }}{{\text{N}}}{\text{S}} - {\mu S} \hfill \\ \frac{{{\text{dS}}}}{{{\text{dt}}}} = \Delta - \left( {\frac{{\phi_{2} {\text{I}}_{{\text{k}}} }}{{\text{N}}} + {\upmu }} \right){\text{S}} \hfill \\ {\text{S}}\left( {\text{t}} \right) = {\text{S}}\left( 0 \right)\exp \left( { - \mathop \smallint \limits_{0}^{{\text{t}}} \left\{ {\frac{{\phi_{2} {\text{I}}_{{\text{k}}} }}{{\text{N}}} + {\upmu }} \right\}{\text{du}}} \right) + \mathop \smallint \limits_{0}^{{\text{t}}} \Delta {\text{ exp}}(\mathop \smallint \limits_{0}^{{\text{x}}} \left\{ {\frac{{\phi_{2} {\text{I}}_{{\text{k}}} }}{{\text{N}}} + {\upmu }} \right\}{\text{du}}){\text{dx }} \times \exp \left( { - \mathop \smallint \limits_{0}^{{\text{t}}} \left\{ {\frac{{\phi_{2} {\text{I}}_{{\text{k}}} }}{{\text{N}}} + {\upmu }} \right\}{\text{du}}} \right) > 0 \hfill \\ \end{gathered}$$
(8)
Hence \({\text{S}}(0)>0\), similarly we can prove \({{\text{I}}}_{{\text{k}}}(0) >0,{\mathrm{ I}}_{{\text{k}}}(0) > 0\).
Theorem 7
The dynamical system (7) is positively invariant in the closed invariant set.
$${\rm Z}_{k} = \{\left( S,{I}_{k},{I}_{kd}\right)\epsilon {R}^{3}+ : N\le \frac{\Delta }{\mu } \}$$
To obtain an invariant region that shows that the solution is bounded, we have
$$\begin{gathered} N = S + I_{k} + I_{kd} \hfill \\ \frac{dN}{{dt}} = \frac{dS}{{dt}} + \frac{{dI_{k} }}{dt} + \frac{{dI_{kd} }}{dt} \hfill \\ \frac{dN}{{dt}} = \Delta - f_{k} S - \mu S + f_{k} S - \sigma_{1} I_{k} - \mu I_{k} + \sigma_{1} I_{k} - \mu I_{kd} \hfill \\ \frac{dN}{{dt}} = \Delta - \left( {S + I_{k} + I_{kd} } \right)\mu \hfill \\ \frac{dN}{{dt}} = \Delta - N\mu \hfill \\ N\left( t \right) = N\left( 0 \right)e^{ - \mu t} + \frac{\Delta }{\mu }\left( {1 - e^{ - \mu t} } \right) \hfill \\ \end{gathered}$$
As,\(t\to \infty\), we get \(0\le N\le \frac{\Delta }{\mu }\), the theory of differential equation27 in the region.
\({\rm Z}_{k} = \{\left( S,{I}_{k},{I}_{kd}\right)\epsilon {R}^{3}+ : N\le \frac{\Delta }{\mu } \}\) For the autonomous system representing the Kidney disease-only model, given by (7), any solution that starts in \({Z}_{k}\) will stay within \({Z}_{k}\) for all \(t\ge 0\)
Kidney disease sub-model with disease-free equilibrium (DFE)
By equating Eq. (10) to zero \(\frac{dS}{dt}=\frac{{dI}_{k}}{dt}=\frac{d{I}_{kd}}{dt}=0\)
The disease-free equilibrium (DFE) of the COVID-19-only model system (7) is obtained by setting each of the systems of model system (10) to zero. Also, at the DFE, there are no infections. Thus, the DFE of the COVID-19-only model (10) is given by \({\Omega }_{0k}=\left( S,{I}_{k},{I}_{kd}\right)=(\frac{\Delta }{\mu },\mathrm{0,0})\)
Basic reproduction number \({R}_{0k}\)
Employing the next-generation matrix method outlined in (Yang 2014), we derive the related next-generation matrix as
$$\begin{gathered} F = \left[ {\begin{array}{*{20}c} {\frac{{\phi_{2} \left( {I_{k} + \theta I_{kd} } \right)}}{N}S} \\ 0 \\ \end{array} } \right] \hfill \\ V = \left[ {\begin{array}{*{20}c} {\left( {\sigma_{1} + \mu } \right)I_{k} } \\ { - \sigma_{1} I_{k} + \mu I_{kd} } \\ \end{array} } \right] \hfill \\ \end{gathered}$$
Consequently, the terms for new infections, F and the subsequent transfer components, V are provided as follows:
$$\begin{gathered} F = \left[ {\begin{array}{*{20}c} {\phi_{2} } & {\phi_{2} \theta } \\ 0 & 0 \\ \end{array} } \right] \hfill \\ V = \left[ {\begin{array}{*{20}c} {\left( {\sigma_{1} + \mu } \right)} & 0 \\ { - \sigma_{1} } & \mu \\ \end{array} } \right] \hfill \\ {\text{So}},\,\,V^{ - 1} = \frac{1}{{\left( {\sigma_{1} + \mu } \right)\mu }}\left[ {\begin{array}{*{20}c} \mu & 0 \\ {\sigma_{1} } & {\left( {\sigma_{1} + \mu } \right)} \\ \end{array} } \right] \hfill \\ \end{gathered}$$
The next-generation matrix \(F{V}^{-1}\)’s leading eigenvalue, which is also known as the spectral radius, represents the fundamental reproductive number and is defined as:
$${R}_{ok}=\frac{{\phi }_{2}(\mu +\theta {\sigma }_{1})}{\left({\sigma }_{1}+\mu \right)\mu }$$
(9)
\({R}_{ok}\) represents the anticipated count of secondary infections produced by a single infected person throughout their entire infectious phase within a wholly susceptible community.
Theorem 8
The DFE is locally asymptotically stable if \({R}_{Ok}< 1\) and unstable if \({R}_{Ok}>1\)
We use the Jacobian matrix to ascertain the local stability of equilibrium points. For sub-model (7), the Jacobian matrix is given as
$$J=\left(\begin{array}{c}\frac{\partial {f}_{1}}{\partial S} \frac{\partial {f}_{1}}{\partial {I}_{k}} \frac{\partial {f}_{1}}{{I}_{kd}}\\ \frac{\partial {f}_{2}}{\partial S} \frac{\partial {f}_{2}}{\partial {I}_{k}} \frac{\partial {f}_{2}}{{I}_{kd}} \\ \frac{\partial {f}_{3}}{\partial S} \frac{\partial {f}_{3}}{\partial {I}_{k}} \frac{\partial {f}_{3}}{{I}_{kd} }\end{array}\right)$$
$$J = \left( {\begin{array}{*{20}c} { - \frac{{\emptyset_{2} \left( {I_{k} + \theta I_{kd} } \right)}}{N} - \mu - \frac{{\emptyset_{2} S}}{N} - \frac{{\emptyset_{2} \theta S}}{N}} \\ { \frac{{\emptyset_{2} \left( {I_{k} + \theta I_{kd} } \right)}}{N} \frac{{\emptyset_{2} S}}{N} {-}\sigma_{1} - \mu \frac{{\emptyset_{2} \theta S}}{N} } \\ { 0 \,\,\,\,\sigma_{1} - \mu } \\ \end{array} } \right)$$
At the disease-free equilibrium point \({\Omega }_{0k} ,\) the Jacobian matrix of the sub-model is given
$$J({\Omega }_{0k})=\left(\begin{array}{c}-\mu -\frac{{\varnothing }_{2}\Delta }{N\mu } -\frac{{\varnothing }_{2}\theta\Delta }{N\mu }\\ 0 \frac{{\varnothing }_{2}\Delta }{N\mu }-\left({\sigma }_{1}+\mu \right) \frac{{\varnothing }_{2}\theta\Delta }{N\mu }\\ 0 {\sigma }_{1} -\mu \end{array}\right)$$
For \(J({\Omega }_{0k})\) the eigenvalues is λ = − μ, and the other eigenvalues can be swiftly obtained using the submatrix
$${J}_{2}=\left(\begin{array}{cc}{\phi }_{2}-\left({\sigma }_{1}+\mu \right)& {\upphi }_{2}\theta \\ {\sigma }_{1}& -\mu \end{array}\right)$$
We must show that \({J}_{2}^{\prime}s\) trace is negative, and its determinant is positive to determine the local stability of the disease-free equilibrium point.
Trc \(\left({J}_{2}\right)={\phi }_{2}-({\tau }_{2}+2\mu ),\) which is less than zero. \(if {\phi }_{2}<({\tau }_{2}+2\mu )\)
det \(\left({J}_{2}\right)=-{\phi }_{2}\left(\mu +\theta {\sigma }_{1}\right)+\mu (\mu +{\sigma }_{1})\)
This value is greater than zero if \(\frac{{\phi }_{2}\left({\theta \sigma }_{1}+\mu \right)}{\mu (\mu +{\sigma }_{1})}<1\) that is \({\text{det}}\left({J}_{2}\right)>0\) if \({R}_{0k}<1\) and \({\text{det}}\left({J}_{2}\right)<0\) if \({R}_{0k}>1\)
For the kidney disease sub-model, the disease-free equilibrium point is stable when \({R}_{ok} < 1\) and unstable when \({R}_{ok} > 1\).
Theorem 9
Only when \({R}_{ok}>1\) does the endemic equilibrium point exist?
By resolving the above system of equations, we were also able to determine the endemic (disease present) equilibrium point of the renal disease sub-model:
$$\begin{gathered} \Delta - f_{k} S - \mu S = 0 \hfill \\ f_{k} S - \sigma_{1} I_{k} - \mu I_{k} = 0 \hfill \\ \sigma_{1} I_{k} - \mu I_{kd} = 0 \hfill \\ \end{gathered}$$
Here \({f}_{k}=\frac{{\phi }_{2}[{I}_{k}+\theta {{\text{I}}}_{{\text{kd}}}]}{N}\)
Solving the equation
\({S}^{*}=\frac{\Delta }{{f}_{k}^{*}+\mu }\), \({I}_{k}^{*}=\frac{{f}_{k}^{*}{S}^{*}}{\mu +{\sigma }_{1}},{I}_{kd}^{*}=\frac{{\sigma }_{1}{f}_{k}^{*}{S}^{*}}{\mu (\mu +{\sigma }_{1})}\) applying this value we get,
$$\begin{gathered} f_{k}^{*} = \frac{{\phi_{2} \left[ {I_{k}^{*} + \theta {\text{I}}_{{{\text{kd}}}}^{*} } \right]}}{N} = \frac{{\phi_{2} f_{k}^{*} S^{*} }}{N}\left( {\frac{1}{{\mu + \sigma_{1} }} + \frac{{\theta \sigma_{1} }}{{\mu \left( {\mu + \sigma_{1} } \right)}}} \right) \hfill \\ f_{k}^{*} = \frac{{\mu \phi_{2} }}{{\left( {\mu + \sigma_{1} } \right)}}\left[ {1 + \frac{{\theta \sigma_{1} }}{\mu }} \right] - \mu \hfill \\ f_{k}^{*} = \mu \frac{{\phi_{2} \left( {\mu + \theta \sigma_{1} } \right)}}{{\mu \left( {\mu + \sigma_{1} } \right)}} - \mu \hfill \\ f_{k}^{*} = \mu \left( {R_{0k} - 1} \right) \hfill \\ \end{gathered}$$
Hence the endemic equilibrium point exists when \({R}_{ok} > 1\)
Global stability of DFE
Theorem 10
The disease-free equilibrium point of the Kidney disease-sub model (7) is globally asymptotically stable. If \({R}_{ok}<1\)
Proof Considering the Lyapunov function
$$T=\mu {I}_{k}+{\phi }_{2}\theta {I}_{kd}$$
(10)
Differentiating with respect to time
$$\begin{gathered} \frac{dT}{{dt}} = \mu \frac{{dI_{k} }}{dt} + \phi_{2} \theta \frac{{dI_{kd} }}{dt} \hfill \\ = \mu \phi_{2} \left( {\frac{{I_{k} + \theta I_{kd} }}{N}} \right)S - \mu \left( {\sigma_{1} + \mu } \right)I_{k} + \phi_{2} \theta \sigma_{1} I_{k} - \phi_{2} \theta \mu I_{kd} \hfill \\ \frac{dT}{{dt}} \le \mu \phi_{2} \left( {I_{k} + \theta I_{kd} } \right) - \mu \left( {\sigma_{1} + \mu } \right)I_{k} + \phi_{2} \theta \sigma_{1} I_{k} - \phi_{2} \theta \mu I_{kd} \hfill \\ \le \mu \phi_{2} I_{k} - \mu \left( {\sigma_{1} + \mu } \right)I_{k} + \phi_{2} \theta \sigma_{1} I_{k} \hfill \\ \le \phi_{2} (\theta \sigma_{1} + \mu )I_{k} - \mu \left( {\sigma_{1} + \mu } \right)I_{k} \hfill \\ \le R_{ok} \mu \left( {\sigma_{1} + \mu } \right)I_{k} {-}\mu \left( {\sigma_{1} + \mu } \right)I_{k} \hfill \\ \le (R_{ok} - 1)\mu \left( {\sigma_{1} + \mu } \right)I_{k} \hfill \\ \le 0,\,\,{\text{for}}\,\,R_{ok} \le 1 \hfill \\ \end{gathered}$$
since all the model parameters are positive, so that \(\frac{dT}{dt}\le 0\) for \({R}_{ok}\le 1\), with \(\frac{dT}{dt}=0\) when \({I}_{k}={I}_{kd}=0\). Using \(\left({I}_{k},{I}_{kd}\right)=(\mathrm{0,0})\) into the Kidney disease only sub- model (7) represents that \(S\to \frac{\Delta }{\mu }\) as \(t\to \infty\). Hence \(T\) is a Lyapunov function on \({\Omega }_{0k}\) and the largest compact invariant set in \(\{\left( S,{I}_{k},{I}_{kd}\right)\in {\Omega }_{k}:\frac{dT}{dt}=0\}\) is \({\Omega }_{0k}\). So every solution of (7), with an initial condition in \({\Omega }_{k}\) approaches \({\Omega }_{0k}\), as \(t\to \infty\) whenever \({R}_{ok}\le 1\).
Theorem 11
In the kidney disease-only model, the equilibrium point indicating the existence of the disease is globally stable when \({R}_{0k}\) \(> 1\).
Denote the endemic equilibrium is denoted by \({E}_{k}=({S}^{*},{I}_{k}^{*},{I}_{kd}^{*}\)), At the steady state, the force of infection \({f}_{k}\) is represented as:
$${f}_{k}^{*}=\frac{{\phi }_{2}({I}_{k}^{*}+\theta {I}_{kd}^{*})}{{S}^{*}+{I}_{k}^{*}+{I}_{kd}^{*}}$$
(11)
In the sub-model (7), we obtain by setting the right-hand sides equal to zero
$${S}^{*}=\frac{\Delta }{({f}_{k}^{*}+\mu )}$$
$${I}_{k}^{*}=\frac{\Delta {f}_{k}^{*}}{\left({f}_{k}^{*}+\mu \right)\mu }$$
$${I}_{kd}^{*}=\frac{\Delta {\sigma }_{1}{f}_{k}^{*}}{\left({f}_{k}^{*}+\mu \right)\mu }$$
Using (10),
$$\left(\mu +{\sigma }_{1}\right){f}_{k}^{*}+\mu \left(\mu +{\sigma }_{1}\right)-{\phi }_{2}\left({\theta \sigma }_{1}+\mu \right)=0$$
(12)
The linear Eq. (12) has a unique positive solution given by
$${f}_{k}^{*}=\frac{{\phi }_{2}\left({\theta \sigma }_{1}+\mu \right)-\mu \left(\mu +{\sigma }_{1}\right)}{(\mu +{\sigma }_{1})}$$
$$\left(\mu +{\sigma }_{1}\right){f}_{k}^{*}=\mu \left(\mu +{\sigma }_{1}\right)({R}_{ok}-1)$$
$${f}_{k}^{*}=\mu ({R}_{ok}-1)$$
This has biological significance when \({R}_{ok}>1\). It is mentioned that \({R}_{ok}<1\) implies that \({\phi }_{2}\left(\theta {\sigma }_{1}+\mu \right)-\mu \left(\mu +{\sigma }_{1}\right)<0.\) When this occurs, the force of infection \({f}_{k}\) is negative, suggesting that the disease’s equilibrium point shifts to global stability.
Analysis of sensitivity for the kidney disease model
Equation (7) specifies the renal sub-model and the examination of sensitivity for its basic reproduction number uses Yang’s (2014) tabilize forward sensitivity index for that basic reproduction number.
$${R}_{ok}=\frac{{\phi }_{2}(\mu +\theta {\sigma }_{1})}{\mu \left(\mu +{\sigma }_{1}\right)}$$
$${S}_{{\phi }_{2}}=\frac{\partial {R}_{0k}}{\partial {\varnothing }_{2}} \frac{{\phi }_{2}}{{R}_{0c}}=\frac{(\mu +\theta {\sigma }_{1})}{\mu \left(\mu +{\sigma }_{1}\right)} \frac{{\phi }_{2}}{\frac{{\phi }_{2}(\mu +\theta {\sigma }_{1})}{\mu \left(\mu +{\sigma }_{1}\right)}}=+1$$
$${S}_{{\sigma }_{1} }=\frac{\partial {R}_{0k}}{\partial {\upsigma }_{1}} \frac{{\sigma }_{1}}{{R}_{0k}}=\frac{{\phi }_{2}(\theta -1)}{{\left({\sigma }_{1}+\mu \right)}^{2}} \frac{{\sigma }_{1}}{\frac{{\phi }_{2}(\mu +\theta {\sigma }_{1})}{\mu \left(\mu +{\sigma }_{1}\right)}}=\frac{{\sigma }_{1}\mu (\theta -1)}{\left({\sigma }_{1}+\mu \right)(\mu +\theta {\sigma }_{1})}$$
$${S}_{\uptheta }=\frac{\partial {R}_{0k}}{\partial\uptheta } \frac{\uptheta }{{R}_{0k}}=\frac{{{\phi }_{2}\sigma }_{1}}{\mu \left(\mu +{\sigma }_{1}\right)}\frac{\uptheta }{\frac{{\phi }_{2}(\mu +\theta {\sigma }_{1})}{\mu \left(\mu +{\sigma }_{1}\right)}}=\frac{\uptheta {\upsigma }_{1}}{(\mu +\theta {\sigma }_{1})}$$
$${S}_{\upmu }=\frac{\partial {R}_{0k}}{\partial\upmu } \frac{\upmu }{{R}_{0k}}=-\frac{{\upphi }_{2}\left({\mu }^{2}+2\mu {\sigma }_{1}+\theta {\sigma }_{1}^{2}\right)}{{\mu }^{2}{\left(\mu +{\sigma }_{1}\right)}^{2}}\frac{\upmu }{\frac{{\phi }_{2}\left(\mu +\theta {\sigma }_{1}\right)}{\mu \left(\mu +{\sigma }_{1}\right)}}=-\frac{\left({\mu }^{2}+2\mu {\sigma }_{1}+\theta {\sigma }_{1}^{2}\right)}{(\mu +\theta {\sigma }_{1})}$$
Based on the sensitivity indices presented in Table 2, several observations can be made regarding the factors influencing the spread of kidney disease: 1. The contact rate specific to kidney disease is represented by \({\phi }_{2}\) exhibit a pronounced positive correlation with the disease’s propagation. This implies that as these rates increase, the disease spreads more aggressively. 2. The parameter adjusting for the enhanced transmission of kidney disease among co-infected individuals and those in the end-stage of the disease, denoted as \(\theta\), and Progression rates \({\sigma }_{1}\) also positively influences the spread of the disease. This suggests a higher transfer rate among the co-infected exacerbates the spread of the disease. 3. Conversely, certain parameters, namely μ, mitigate the spread of kidney disease. Specifically, elevating the values of this parameter leads to a reduction in the number of individuals afflicted with kidney disease.
COVID-19 and kidney disease full model
By analyzing the equations’ right-hand sides, we could derive the equilibrium locations for the entire model (1).
$$\begin{gathered} \Delta - f_{k} S - {\text{f}}_{{\text{c}}} S - \mu S = 0 \hfill \\ f_{k} S - \sigma_{1} I_{k} - \alpha_{1} f_{c} I_{k} + {\text{f}}_{{\text{k}}} R + \tau_{2} I_{kc} - \mu I_{k} = 0 \hfill \\ \phi R + f_{c} S - {\text{f}}_{{\text{k}}} I_{c} - \tau_{1} I_{c} - \mu I_{c} = 0 \hfill \\ \alpha_{1} f_{c} I_{k} + {\text{f}}_{{\text{k}}} I_{c} - \sigma_{2} I_{kc} - \tau_{2} I_{kc} - \mu I_{kc} = 0 \hfill \\ \sigma_{1} I_{k} + \tau_{3} I_{kdc} - \alpha_{2} f_{c} I_{kd} - \mu I_{kd} = 0 \hfill \\ \sigma_{2} I_{kc} + \alpha_{2} f_{c} I_{kd} - \tau_{3} I_{kdc} - \mu I_{kdc} = 0 \hfill \\ \tau_{1} I_{c} - f_{k} R - \phi R - \mu R = 0 \hfill \\ \end{gathered}$$
(13)
where the forces of infection \({f}_{k}\) and \({f}_{c}\) are identical to those in Eqs. (5) and (10). The whole model’s disease-free equilibrium point \(({\Omega }_{0ck})\) is then calculated as
$${\Omega }_{0ck} =(\frac{\Delta }{\mu },\mathrm{0,0},\mathrm{0,0},\mathrm{0,0})$$
(14)
We have now calculated the basic reproduction number \({R}_{0}\) of the complete model using the next-generation matrix. Using the notation of the diseased states \(({I}_{c} , {I}_{k} ,{I}_{kd}, {I}_{kc}, {I}_{kdc} )\), Given the vector differential equations form \(\frac{dX}{dt}=F\left(x\right)-V(x)\), where \(V(x) = {V}^{-}(x)-{V}^{+}(x). F (x)\) is the rate at which new infections arise in compartments, \({V}^{+} (x)\) is the rate at which people are transferred into the compartment, and \({V}^{-} \left(x\right)\) is the rate at which people are transferred out of the compartments \({I}_{k},{I}_{c},{I}_{kc},{I}_{kd},{I}_{kdc}\)
$$F\left(x\right)=\left(\begin{array}{c}{f}_{k}S+{{\text{f}}}_{{\text{k}}}R\\ {f}_{c}S\\ {{\text{f}}}_{{\text{k}}}{I}_{c}\\ 0\\ 0\end{array}\right)and V(x)=\left(\begin{array}{c}{(\sigma }_{1}+{\alpha }_{1}{f}_{c}+\mu ){I}_{k}-{\tau }_{2}{I}_{kc}\\ ({{\text{f}}}_{{\text{k}}}+{\tau }_{1}+\mu {)I}_{c}\\ ({\sigma }_{2}+{\tau }_{2}+\mu ){I}_{kc}{-\alpha }_{1}{f}_{c}{I}_{k}\\ \left({\alpha }_{2}{f}_{c}+\mu \right){I}_{kd}-{\sigma }_{1}{I}_{k}-{\tau }_{3}{I}_{kdc}\\ ({\tau }_{3}+\mu {)I}_{kdc}-{\sigma }_{2}{I}_{kc}-{\alpha }_{2}{f}_{c} {I}_{kd}\end{array}\right)$$
At\(,{E}_{0}\)
$$F=\left(\begin{array}{ccccc}{\phi }_{2} & 0& \theta & \theta & \theta \\ 0& {\phi }_{1}& \gamma & 0& \gamma \\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$$
$$V=\left(\begin{array}{ccccc}{(\sigma }_{1}+\mu )& 0& -{\tau }_{2}& 0& 0\\ 0& {\tau }_{1}+\mu & 0& 0& 0\\ 0& 0& ({\sigma }_{2}+{\tau }_{2}+\mu )& 0& 0\\ -{\sigma }_{1}& 0& 0& \mu & -{\tau }_{3}\\ 0& 0& -{\sigma }_{2}& 0& ({\tau }_{3}+\mu )\end{array}\right)$$
$${V}^{-1}=\left(\begin{array}{ccccc}\frac{1}{{(\sigma }_{1}+\mu )}& 0& -{\tau }_{2}& 0& 0\\ 0& \frac{1}{{\tau }_{1}+\mu }& 0& 0& 0\\ 0& 0& \frac{1}{{\sigma }_{2}+{\tau }_{2}+\mu }& 0& 0\\ -\frac{{\sigma }_{1}}{{(\sigma }_{1}+\mu )}& 0& 0& \frac{1}{\mu }& -{\tau }_{3}\\ 0& 0& -\frac{{\sigma }_{2}}{({\sigma }_{2}+{\tau }_{2}+\mu ){(\tau }_{3}+\mu )}& 0& \frac{1}{{\tau }_{3}+\mu }\end{array}\right)$$
$${FV}^{-1}=\left(\begin{array}{ccccc}\frac{{\phi }_{2}(\theta {\sigma }_{1}+\mu )}{\mu {(\sigma }_{1}+\mu )} & 0& \frac{\theta }{\left({\sigma }_{2}+{\tau }_{2}+\mu \right)}& \frac{\theta \Delta }{\mu }& \frac{\theta \Delta }{\mu }\\ 0& \frac{{\phi }_{1}\Delta }{\mu ({\tau }_{1}+\mu )}& \gamma \frac{\Delta }{\mu ({\sigma }_{2}+{\tau }_{2}+\mu )}& 0& \gamma \frac{\Delta }{\mu }\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right)$$
To determine the basic reproduction number \({R}_{ck}\) of the system, the eigenvalues can be employed, specifically by examining the spectral radius of the matrix \({FV}^{-1}.\) The eigenvalues can be determined by assessing the equation:
$$\begin{gathered} \det \left[ {FV^{ - 1} - \lambda I} \right] = 0 \hfill \\ |FV^{ - 1} - \lambda I| = \left| {\begin{array}{*{20}c} {\frac{{\phi_{2} \left( {\theta \sigma_{1} + \mu } \right)}}{{\mu (\sigma_{1} + \mu )}} - \lambda } & 0 & {\frac{\theta }{{\left( {\sigma_{2} + \tau_{2} + \mu } \right)}}} & \theta & \theta \\ 0 & {\frac{{\phi_{1} }}{{\left( {\tau_{1} + \mu } \right)}} - \lambda } & {\frac{\gamma }{{\left( {\sigma_{2} + \tau_{2} + \mu } \right)}}} & 0 & \gamma \\ 0 & 0 & { - \lambda } & 0 & 0 \\ 0 & 0 & 0 & { - \lambda } & 0 \\ 0 & 0 & 0 & 0 & { - \lambda } \\ \end{array} } \right| = 0 \hfill \\ \end{gathered}$$
Here eigenvalues are \({\lambda }_{1}=\frac{{\phi }_{2}(\theta {\sigma }_{1}+\mu )}{\mu {(\sigma }_{1}+\mu )},{\lambda }_{2}=\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)},{\lambda }_{3}=0,{\lambda }_{4}=0,{\lambda }_{5}=0\)
Thus, it can be concluded that the COVID-19 and kidney disease co-infection model has a reproduction number given by \({R}_{ck}=\{{R}_{oc},{R}_{ok}\}\);
where \({R}_{0k}=\frac{{\phi }_{2}(\theta {\sigma }_{1}+\mu )}{\mu {(\sigma }_{1}+\mu )}\) and \({R}_{0c}=\frac{{\phi }_{1}}{\left({\tau }_{1}+\mu \right)}\)
Stability of \({\Omega }_{0ck}\) for the full co-infection model
Theorem 12
When \({R}_{ck}>1\), model (1) has \({(\Omega }_{0ck})\) that is locally asymptotically stable.
The eigenvalues of each equilibrium were used to examine its local stability (Fudolig and Howard, 2020). The eigenvalues are found in the Jacobian matrix, which each equilibrium has replaced. The model (1)’s Jacobian matrix can be described as
$$\Delta -{f}_{k}S-{{\text{f}}}_{{\text{c}}}S-\mu S=0$$
$${f}_{k}S-{\sigma }_{1}{I}_{k}-{\alpha }_{1}{f}_{c}{I}_{k}+{{\text{f}}}_{{\text{k}}}R+{\tau }_{2}{I}_{kc}-\mu {I}_{k}=0$$
$${f}_{c}S-{{\text{f}}}_{{\text{k}}}{I}_{c}-{\tau }_{1}{I}_{c}-\mu {I}_{c}=0$$
$${\alpha }_{1}{f}_{c}{I}_{k}+{{\text{f}}}_{{\text{k}}}{I}_{c}-{\sigma }_{2}{I}_{kc}-{\tau }_{2}{I}_{kc}-\mu {I}_{kc}=0$$
$${\sigma }_{1}{I}_{k}+{\tau }_{3}{I}_{kdc}-{\alpha }_{2}{f}_{c} {I}_{kd}-\mu {I}_{kd}=0$$
$${\sigma }_{2}{I}_{kc}+{\alpha }_{2}{f}_{c} {I}_{kd}-{\tau }_{3}{I}_{kdc}-\mu {I}_{kdc}=0$$
$${\tau }_{1}{I}_{c}-{f}_{k}R-\mu R=0$$
$$J=\left(\begin{array}{ccccccc}-\mu & -{\phi }_{2}& {-\phi }_{1}& -({\phi }_{2}\theta +{\phi }_{1}\gamma )& -{\phi }_{2}\theta & -({\phi }_{2}\theta +{\phi }_{1}\gamma )& 0\\ 0& {\phi }_{2}-({\sigma }_{1}+\mu ) & 0& {{\phi }_{2}\theta +\tau }_{2}& {\phi }_{2}\theta & {\phi }_{2}\theta & 0\\ 0& 0& {\phi }_{1}-({\tau }_{1}+\mu )& {\phi }_{1}\gamma & 0& 0& 0\\ 0& 0& 0& -({\sigma }_{2}+{\tau }_{2}+\mu )& 0& 0& 0\\ 0& {\sigma }_{1}& 0& 0& -\mu & {\tau }_{3}& 0\\ 0& 0& 0& {\sigma }_{2}& 0& -{(\tau }_{3}+\mu )& 0\\ 0& 0& {\tau }_{1}& 0& 0& 0& -\mu \end{array}\right)$$
At the disease-free equilibrium, we obtained the following characteristic polynomial:
$${Q}_{0} (\lambda ) =({\lambda }_{1}+\mu )({\lambda }_{2}+\mu )({\lambda }_{3}+\mu )({\lambda }_{4}+{\sigma }_{2}+{\tau }_{2}+\mu )({\lambda }_{5}-{\phi }_{2}+{\sigma }_{1}+\mu )({\lambda }_{6}-{\phi }_{1}+{\tau }_{1}+\mu )({\lambda }_{7}+{\tau }_{3}+\mu )$$
(15)
We get \({\lambda }_{1}=-\mu ,{\lambda }_{2}=-\mu ,{\lambda }_{3}=-\mu ,{\lambda }_{4}=-({\sigma }_{2}+{\tau }_{2}+\mu )\)
And \({\lambda }_{5}=-{\phi }_{2}+{\sigma }_{1}+\mu <0\) and \({\lambda }_{6}=-{\phi }_{1}+{\tau }_{1}+\mu <0\)
\({\phi }_{2}<{\sigma }_{1}+\mu\), \({\phi }_{1}<{\tau }_{1}+\mu\)
\(\frac{\mu }{(\theta {\sigma }_{1}+\mu }\frac{{\phi }_{2}(\theta {\sigma }_{1}+\mu )}{\mu ({\sigma }_{1}+\mu )}<1\) and \(\frac{{\phi }_{1}}{{\tau }_{1}+\mu }<1\)
\(\frac{\mu }{(\theta {\sigma }_{1}+\mu )}{R}_{0k} <1\) and \(\frac{{\phi }_{1}}{{(\tau }_{1}+\mu )}<1\)
So, \({R}_{ok}<1\) and \({R}_{oc}<1\)
So, the co-infection full model (1), \({ \Omega }_{0ck}\) reaches local asymptotic stability as a disease-free equilibrium point.
Global stability analysis of co-infection full model
From the full model \(\frac{dX}{dt}=F(X,Z)\), \(\frac{dZ}{dt}=T\left(X,Z\right), T\left(X,0\right)=0,\)
Here \(X=(S,R)\) and \(Z=({I}_{k},{I}_{c},{I}_{kc},{I}_{kd},{I}_{kdc})\). In this case, representation \(X,\) which belongs to \({R}^{2}\) signifies the compartments of healthy individuals, while \(Z\), a part of \({R}^{5}\), stands for the infected population compartments. The disease-free equilibrium state is denoted by \({U}_{0}=({X}_{0},0)\), where \({X}_{0}=(\frac{\Delta }{\mu },0)\)
The following assumptions \(({H}_{1})\) and \(({H}_{2})\) ensure that \({\Omega }_{0ck}\) for \({R}_{ck}\) is globally asymptotically stable. \(({H}_{1})\) For \(\frac{dX}{dt} = F(X, 0),\) the equilibrium point \({U}_{0}\) is globally stable;
(\(H_{2} ) G\left( {X,Z} \right) = AZ {-} T\left( {X,Z} \right), \hat{G}\left( {X,Z} \right) \ge 0\) for \((X,Z) \in \Omega\), The feasible area of the constructed model is denoted by \(\Omega\), and A = \({D}_{Z} T({U}_{0},0)\) is a Metzler matrix. From our co-infection mathematical model Eq. (1), we have \(\frac{dX}{dt}=F\left(X,Z\right)=\left[\begin{array}{c}\Delta -{f}_{k}S-{{\text{f}}}_{{\text{c}}}S-\mu S\\ {\tau }_{1}{I}_{c}-{f}_{k}R-\mu R\end{array}\right]\)
So, \(T\left(X,0\right)=\left[\begin{array}{c}\Delta -\mu S\\ 0\end{array}\right]\) and
$$\begin{gathered} \frac{dZ}{{dt}} = T\left( {X,Z} \right) = \left[ {\begin{array}{*{20}c} {f_{k} S - \sigma_{1} I_{k} - \alpha_{1} f_{c} I_{k} + f_{k} R + \tau_{2} I_{kc} - \mu I_{k} } \\ {f_{c} S - f_{k} I_{c} - \tau_{1} I_{c} - \mu I_{c} } \\ { \alpha_{1} f_{c} I_{k} + f_{k} I_{c} - \sigma_{2} I_{kc} - \tau_{2} I_{kc} - \mu I_{kc} } \\ { \sigma_{1} I_{k} + \tau_{3} I_{kdc} - \alpha_{2} f_{c} I_{kd} - \mu I_{kd} } \\ {\sigma_{2} I_{kc} + \alpha_{2} f_{c} I_{kd} - \tau_{3} I_{kdc} - \mu I_{kdc} } \\ \end{array} } \right] \hfill \\ \hat{T}\left( {X,Z} \right) = AZ - T\left( {X,Z} \right) \hfill \\ So,\,\,\hat{T}\left( {X,Z} \right) = \left[ {\begin{array}{*{20}c} {\hat{T}_{1} \left( {X,Z} \right)} \\ {\hat{T}_{2} \left( {X,Z} \right)} \\ {\hat{T}_{3} \left( {X,Z} \right)} \\ {\hat{T}_{4} \left( {X,Z} \right)} \\ {\widehat{{T_{5} }}\left( {X,Z} \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - f_{k} S - f_{k} R + \alpha_{1} f_{c} I_{k} } \\ { - f_{c} S + f_{k} I_{c} } \\ { {-}\alpha_{1} f_{c} I_{k} - f_{k} I_{c} } \\ {\alpha_{2} f_{c} I_{kd} } \\ { - \alpha_{2} f_{c} I_{kd} } \\ \end{array} } \right] \hfill \\ \end{gathered}$$
Thus,\({-\alpha }_{1}{f}_{c}{I}_{k}-{{\text{f}}}_{{\text{k}}}{I}_{c}<0\) and \(-{\alpha }_{2}{f}_{c} {I}_{kd}<0\). From this, condition \({H}_{2}\) is not met. Consequently,\({U}_{0}\) and subsequently, the disease-free equilibrium point \({\Omega }_{ck}\), cannot achieve global asymptotic stability.
