A mathematical model for the transmission of co-infection with COVID-19 and kidney disease | Scientific Reports – Nature.com

We studied how COVID-19 and kidney disease impact each other by examining them separately first. After understanding each individually, theyre 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)

All the populations of the system with positive initial conditions are nonnegative.

Assume ({text{S}}left(0right)>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(0right)>0,) similarly we can show ({{text{I}}}_{{text{C}}}(0)>0,{text{R}}(0)>0).

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},Rright)epsilon {R}^{3}+ : Nle 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

$$frac{dN}{dt}=frac{dS}{dt}+frac{{dI}_{C}}{dt}+frac{dR}{dt}$$

$$frac{dN}{dt}=Delta -frac{{phi }_{1}{I}_{c}}{N}S-mu S+frac{{phi }_{1}{I}_{c}}{N}S-{tau }_{1}{I}_{c}-mu {I}_{c}+{tau }_{1}{I}_{c}-mu R$$

$$frac{dN}{dt}=Delta -left(S+{I}_{c}+Rright)mu$$

$$frac{dN}{dt}=Delta -Nmu$$

$$Nleft(tright)=Nleft(0right){e}^{-mu t}+frac{Delta }{mu }(1-{e}^{-mu t})$$

As,(tto infty), we get (0le Nle frac{Delta }{mu }), the theory of differential equation27 in the region.

({rm Z}_{c} = {left( S,{I}_{c},Rright)epsilon {R}^{3}+ : Nle 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 (tge 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},Rright)=(frac{Delta }{mu },mathrm{0,0}))

The sub-models 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.

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}+2mu ),) 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-models 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.

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.

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 parameters value is known as the parameters 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 viruss 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 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)

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).

The dynamical system (7) is positively invariant in the closed invariant set.

$${rm Z}_{k} = {left( S,{I}_{k},{I}_{kd}right)epsilon {R}^{3}+ : Nle frac{Delta }{mu } }$$

To obtain an invariant region that shows that the solution is bounded, we have

$$begin{gathered} N = S + I_{k} + I_{kd} hfill \ frac{dN}{{dt}} = frac{dS}{{dt}} + frac{{dI_{k} }}{dt} + frac{{dI_{kd} }}{dt} hfill \ frac{dN}{{dt}} = Delta - f_{k} S - mu S + f_{k} S - sigma_{1} I_{k} - mu I_{k} + sigma_{1} I_{k} - mu I_{kd} hfill \ frac{dN}{{dt}} = Delta - left( {S + I_{k} + I_{kd} } right)mu hfill \ frac{dN}{{dt}} = Delta - Nmu hfill \ Nleft( t right) = Nleft( 0 right)e^{ - mu t} + frac{Delta }{mu }left( {1 - e^{ - mu t} } right) hfill \ end{gathered}$$

As,(tto infty), we get (0le Nle frac{Delta }{mu }), the theory of differential equation27 in the region.

({rm Z}_{k} = {left( S,{I}_{k},{I}_{kd}right)epsilon {R}^{3}+ : Nle 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 (tge 0)

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}))

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)

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A mathematical model for the transmission of co-infection with COVID-19 and kidney disease | Scientific Reports - Nature.com