Theoretical and mathematical codynamics of nonlinear tuberculosis and COVID-19 model pertaining to fractional … – Nature.com

The general population is divided into eight indistinguishable groups in this category, which are designated as susceptible people, (({textbf{S}})), latent TB patients who do not exhibit TB-associated indications and are not pathogenic ({{textbf{L}}}_{{textbf{T}}}), influential TB-infected people ({{textbf{I}}}_{{textbf{T}}}), COVID-19-infested humans who do not exhibit indications but are transmissible ({{textbf{E}}}_{{textbf{C}}}), COVID-19-diagnosed people who exhibit scientific backing indications and are pathogenic ({{textbf{I}}}_{{textbf{C}}}), both inactive TB and COVID-19-contaminated people ({{textbf{L}}}_{{textbf{T}}{textbf{C}}}), current TB and COVID-19-contaminated humans ({{textbf{I}}}_{{textbf{T}}{textbf{C}}}), and retrieved people ({{textbf{R}}}) consisting of both TB and COVID-19. The underlying computational framework for the codynamics of TB and COVID-19 is developed in this portion. Considering such, all people at moment (tau ), represented by ({textbf{N}}(tau )), are provided by

$$begin{aligned} {textbf{N}}(tau )={textbf{S}}(tau )+{{textbf{L}}}_{{textbf{T}}}(tau ) +{{textbf{I}}}_{{textbf{T}}}(tau )+{{textbf{E}}}_{{textbf{C}}}(tau )+{{textbf{I}}}_{{textbf{C}}}(tau ) +{{textbf{L}}}_{{textbf{T}}{textbf{C}}}(tau )+{{textbf{I}}}_{{textbf{T}}{textbf{C}}}(tau )+{{textbf{R}}}(tau ). end{aligned}$$

(1)

We hypothesized that acquisition increases the vulnerable community at an intensity of (nabla ). Every person in every compartment experiences an inevitable mortality rate of (beta ). Equivalent to formula (1), vulnerable individuals contract TB via interaction with current TB individuals via agent transmission (psi _{{textbf{T}}}). The acceptable interaction rate for TB transmission is indicated by (alpha _{1}) within this manifestation. It is believed that people with persistent TB are undiagnosed and cannot pass on the illness35. Likewise, those at risk contract COVID-19 at an intensity of transmission (psi _{{textbf{C}}}), which is determined as in formula (1), after effectively coming into proximity to COVID-19-infected people. The efficient interaction probability for COVID-19 infection is represented by (alpha _{2}) in this case. Furthermore, we hypothesized that people in the hidden TB segment ({{textbf{L}}}_{{textbf{T}}}) depart at an incidence of (mu ) to segment ({{textbf{I}}}_{{textbf{T}}}), at an incidence of transmission of (lambda psi _{{textbf{C}}}) to the persistent TB as well as COVID-19 contaminated group, whilst certain of them recuperate at an intervention incidence of (varpi ). Additionally, those in the TB-infected category ({{textbf{I}}}_{{textbf{T}}}) recuperate due to the illness at an incidence of (delta ), with the surviving percentage either transferring to the transmission category ({{textbf{I}}}_{{textbf{T}}{textbf{C}}}) at a pace of (varsigma _{3}) or dying at a speed of (zeta _{{textbf{T}}}) via TB-induced mortality.

The overall community in cohort ({{textbf{L}}}_{{textbf{T}}{textbf{C}}}) potentially dies at COVID-19-induced mortality pace (zeta _{{textbf{C}}}) or advances at an intensity of (rho ) to become contaminated category ({{textbf{I}}}_{{textbf{T}}{textbf{C}}}). As seen in Fig. 3, it is believed that the other people will be moved to the other cohort at a consistent multiplicity of (eta ). In other words, the general population classified as ({{textbf{L}}}_{{textbf{T}}{textbf{C}}}) migrates at an intensity of (varsigma _{2}eta ) to category ({{textbf{I}}}_{{textbf{T}}}), then at a pace of (varsigma _{1}eta ) to compartment ({{textbf{I}}}_{{textbf{C}}}) group, and finally recovers at a pace of ((1-(varsigma _{1}+varsigma _{2}))eta ). Additionally, we hypothesized that, although the codynamics-induced mortality prevalence is represented by (zeta _{{textbf{T}}{textbf{C}}}), people in compartment ({{textbf{I}}}_{{textbf{T}}{textbf{C}}}) depart for compartments ({{textbf{I}}}_{{textbf{T}}},~{{textbf{I}}}_{{textbf{C}}}) or ({textbf{R}}), correspondingly, at an intensity of (theta _{2}xi ,~theta _{1}xi ) or ((1-(theta _{1}+theta _{2}))xi .) Furthermore, at an intensity of (epsilon psi _{{textbf{T}}},phi ) or (varphi _{2},) the COVID-19 exposure people ({{textbf{E}}}_{{textbf{C}}}) can choose to depart to compartment ({{textbf{L}}}_{{textbf{T}}{textbf{C}}},~{{textbf{I}}}_{{textbf{C}}}) or ({textbf{R}}), respectively. Comparably, the number of individuals in compartment ({{textbf{I}}}_{{textbf{C}}}) is either moved to the codynamics cohort at an intensity of (nu ) or restored at a steady pace of (varphi _{3}). (zeta _{{textbf{C}}}) represents the disease-induced fatality rate within this category. Figure 3 displays the suggested systems process layout.

Flow diagram for depicting the codynamics process of TB-COVID-19 model (2).

It leads to frameworks for the subsequent nonlinear DEs determined by the procedure illustration:

$$begin{aligned} {left{ begin{array}{ll} dot{{textbf{S}}}=nabla -(psi _{{textbf{T}}}+psi _{{textbf{C}}}+beta ){textbf{S}},\ dot{{{textbf{L}}}_{{textbf{T}}}}=psi _{{textbf{T}}}{textbf{S}}-(beta +mu +lambda psi _{{textbf{C}}}+varpi ){{textbf{L}}}_{{textbf{T}}},\ dot{{{textbf{I}}}_{{textbf{T}}}}=mu {{textbf{L}}}_{{textbf{T}}}+varsigma _{2}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{2}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ){{textbf{I}}}_{{textbf{T}}},\ dot{{{textbf{E}}}_{{textbf{C}}}}=psi _{{textbf{C}}}{textbf{S}}-(beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}){{textbf{E}}}_{{textbf{C}}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0le tau le intercal _{1},\ dot{{{textbf{I}}}_{{textbf{C}}}}=varphi _{1} {{textbf{E}}}_{{textbf{C}}}+rho eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{1}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +zeta _{{textbf{C}}}+nu +varphi _{3}){{textbf{I}}}_{{textbf{C}}},\ dot{{{textbf{L}}}_{{textbf{T}}{textbf{C}}}}=lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}+epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}-(beta +zeta _{{textbf{C}}}+rho +eta ){{textbf{L}}}_{{textbf{T}}{textbf{C}}},\ dot{{{textbf{I}}}_{{textbf{T}}{textbf{C}}}}=rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+varsigma _{3}{{textbf{I}}}_{{textbf{T}}}+nu {{textbf{I}}}_{{textbf{C}}}-(beta +zeta _{{textbf{T}}{textbf{C}}}+xi ){{textbf{I}}}_{{textbf{T}}{textbf{C}}},\ dot{{{textbf{R}}}}=varpi {{textbf{L}}}_{{textbf{T}}}+varphi _{2}{{textbf{E}}}_{{textbf{C}}}+delta {{textbf{I}}}_{{textbf{T}}}+varphi _{3}{{textbf{I}}}_{{textbf{C}}}+(1-(varsigma _{1}+varsigma _{2}))eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+(1-(theta _{1}+theta _{2}))xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-beta {textbf{R}},end{array}right. } end{aligned}$$

(2)

where (psi _{{textbf{T}}}=frac{alpha _{1}}{{textbf{N}}(tau )}big ({{textbf{I}}}_{{textbf{T}}}(tau )+{{textbf{I}}}_{{textbf{T}}{textbf{C}}}(tau )big )) and (psi _{{textbf{C}}}=frac{alpha _{2}}{{textbf{N}}(tau )}big ({{textbf{E}}}_{{textbf{C}}}(tau )+{{textbf{I}}}_{{textbf{C}}}(tau )+{{textbf{L}}}_{{textbf{T}}{textbf{C}}}(tau )+{{textbf{I}}}_{{textbf{T}}{textbf{C}}}(tau )big ),) containing positive initial conditions (ICs) ({{textbf{S}}}(0)ge 0,~{{textbf{L}}}_{{textbf{T}}}ge 0,~{{textbf{I}}}_{{textbf{T}}}ge 0,~{{textbf{E}}}_{{textbf{C}}}ge 0,~{{textbf{I}}}_{{textbf{C}}}ge 0,~{{textbf{L}}}_{{textbf{T}}{textbf{C}}}ge 0,~{{textbf{I}}}_{{textbf{T}}{textbf{C}}}ge 0,~{{textbf{R}}}ge 0.)

Table 1 provides a description of the systems characteristics.

To help readers that are acquainted with fractional calculus, we provide the related summary herein (see21,22,23 comprehensive discussion on fractional calculus).

$$begin{aligned} ,_{0}^{c}{textbf{D}}_{tau }^{omega } {mathcal {G}}(tau )=frac{1}{Gamma (1-omega )}int limits _{0}^{tau }{mathcal {G}}^{prime }({textbf{q}})(tau -{textbf{q}})^{omega }d{textbf{q}},~~omega in (0,1]. end{aligned}$$

The index kernel is involved in the Caputo fractional derivative (CFD). Whenever experimenting with a particular integral transform, such as the Laplace transform36,37, the CFD accommodates regular ICs.

$$begin{aligned} ,_{0}^{CF}{textbf{D}}_{tau }^{omega } {mathcal {G}}(tau )=frac{bar{{mathcal {M}}}(omega )}{1-omega }int limits _{0}^{tau }{mathcal {G}}^{prime }({textbf{q}})exp bigg [-frac{omega }{1-omega }(tau -{textbf{q}})bigg ]d{textbf{q}},~~omega in (0,1], end{aligned}$$

where (bar{{mathcal {M}}}(omega )) indicates the normalization function (bar{{mathcal {M}}}(0)=bar{{mathcal {M}}}(1)=1.)

The non-singular kernel of the Caputo-Fabrizio fractional derivative (CFFD) operator has drawn the attention of numerous researchers. Furthermore, representing an assortment of prevalent issues that obey the exponential decay memory is best suited to utilize the CFFD operator38. With the passage of time, developing a mathematical model using the CFFD became a remarkable field of research. In recent times, several mathematicians have been busy with the development and simulation of CFFD DEs39.

The ABC fractional derivative operator is described as follows:

$$begin{aligned} ,_{0}^{ABC}{textbf{D}}_{tau }^{omega } {mathcal {G}}(tau )=frac{ABC(omega )}{1-omega }int limits _{0}^{tau }{mathcal {G}}^{prime }({textbf{q}})E_{omega }bigg [-frac{omega }{1-omega }(tau -{textbf{q}})^{omega }bigg ]d{textbf{q}},~~omega in (0,1], end{aligned}$$

where (ABC(omega )=1-omega +frac{omega }{Gamma (omega )}) represents the normalization function.

The memory utilized in AtanganaBaleanuCaputo fractional derivative (ABCFD) can be found intuitively within the index-law analogous for an extended period as well as exponential decay in a number of scientific concerns40,41. The broad scope of the connection and the non-power-law nature of the underlying tendency are the driving forces behind the selection of this version. The impact of the kernel, considered crucial in the dynamic BaggsFreedman framework, was fully produced by the GML function42.

To far better perceive the propagation of TB and COVID-19, we indicate a dynamic mechanism (2) that includes the co-infection within the context of CFD, CFFD and ABCFD, respectively. This is because FO algorithms possess inherited properties that characterize the local/non-local and singular/non-singular dynamics of natural phenomena, presented as follows:

$$begin{aligned} {left{ begin{array}{ll} ,^{c}{textbf{D}}_{tau }^{omega }{textbf{S}}=nabla -(psi _{{textbf{T}}}+psi _{{textbf{C}}}+beta ){textbf{S}},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}}=psi _{{textbf{T}}}{textbf{S}}-(beta +mu +lambda psi _{{textbf{C}}}+varpi ){{textbf{L}}}_{{textbf{T}}},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}}=mu {{textbf{L}}}_{{textbf{T}}}+varsigma _{2}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{2}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ){{textbf{I}}}_{{textbf{T}}},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{E}}}_{{textbf{C}}}=psi _{{textbf{C}}}{textbf{S}}-(beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}){{textbf{E}}}_{{textbf{C}}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~intercal _{1}le tau le intercal _{2},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{C}}}=varphi _{1} {{textbf{E}}}_{{textbf{C}}}+rho eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{1}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +zeta _{{textbf{C}}}+nu +varphi _{3}){{textbf{I}}}_{{textbf{C}}},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}{textbf{C}}}=lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}+epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}-(beta +zeta _{{textbf{C}}}+rho +eta ){{textbf{L}}}_{{textbf{T}}{textbf{C}}},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}{textbf{C}}}=rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+varsigma _{3}{{textbf{I}}}_{{textbf{T}}}+nu {{textbf{I}}}_{{textbf{C}}}-(beta +zeta _{{textbf{T}}{textbf{C}}}+xi ){{textbf{I}}}_{{textbf{T}}{textbf{C}}},\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{R}}}=varpi {{textbf{L}}}_{{textbf{T}}}+varphi _{2}{{textbf{E}}}_{{textbf{C}}}+delta {{textbf{I}}}_{{textbf{T}}}+varphi _{3}{{textbf{I}}}_{{textbf{C}}}+(1-(varsigma _{1}+varsigma _{2}))eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+(1-(theta _{1}+theta _{2}))xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-beta {textbf{R}},end{array}right. } end{aligned}$$

(3)

$$begin{aligned} {left{ begin{array}{ll} ,^{CF}{textbf{D}}_{tau }^{omega }{textbf{S}}=nabla -(psi _{{textbf{T}}}+psi _{{textbf{C}}}+beta ){textbf{S}},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}}=psi _{{textbf{T}}}{textbf{S}}-(beta +mu +lambda psi _{{textbf{C}}}+varpi ){{textbf{L}}}_{{textbf{T}}},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}}=mu {{textbf{L}}}_{{textbf{T}}}+varsigma _{2}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{2}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ){{textbf{I}}}_{{textbf{T}}},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{E}}}_{{textbf{C}}}=psi _{{textbf{C}}}{textbf{S}}-(beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}){{textbf{E}}}_{{textbf{C}}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~intercal _{1}le tau le intercal _{2},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{C}}}=varphi _{1} {{textbf{E}}}_{{textbf{C}}}+rho eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{1}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +zeta _{{textbf{C}}}+nu +varphi _{3}){{textbf{I}}}_{{textbf{C}}},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}{textbf{C}}}=lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}+epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}-(beta +zeta _{{textbf{C}}}+rho +eta ){{textbf{L}}}_{{textbf{T}}{textbf{C}}},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}{textbf{C}}}=rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+varsigma _{3}{{textbf{I}}}_{{textbf{T}}}+nu {{textbf{I}}}_{{textbf{C}}}-(beta +zeta _{{textbf{T}}{textbf{C}}}+xi ){{textbf{I}}}_{{textbf{T}}{textbf{C}}},\ ,^{CF}{textbf{D}}_{tau }^{omega }{{textbf{R}}}=varpi {{textbf{L}}}_{{textbf{T}}}+varphi _{2}{{textbf{E}}}_{{textbf{C}}}+delta {{textbf{I}}}_{{textbf{T}}}+varphi _{3}{{textbf{I}}}_{{textbf{C}}}+(1-(varsigma _{1}+varsigma _{2}))eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+(1-(theta _{1}+theta _{2}))xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-beta {textbf{R}},end{array}right. } end{aligned}$$

(4)

$$begin{aligned} {left{ begin{array}{ll} ,^{ABC}{textbf{D}}_{tau }^{omega }{textbf{S}}=nabla -(psi _{{textbf{T}}}+psi _{{textbf{C}}}+beta ){textbf{S}},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}}=psi _{{textbf{T}}}{textbf{S}}-(beta +mu +lambda psi _{{textbf{C}}}+varpi ){{textbf{L}}}_{{textbf{T}}},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}}=mu {{textbf{L}}}_{{textbf{T}}}+varsigma _{2}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{2}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ){{textbf{I}}}_{{textbf{T}}},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{E}}}_{{textbf{C}}}=psi _{{textbf{C}}}{textbf{S}}-(beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}){{textbf{E}}}_{{textbf{C}}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~intercal _{1}le tau le intercal _{2},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{C}}}=varphi _{1} {{textbf{E}}}_{{textbf{C}}}+rho eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{1}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +zeta _{{textbf{C}}}+nu +varphi _{3}){{textbf{I}}}_{{textbf{C}}},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}{textbf{C}}}=lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}+epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}-(beta +zeta _{{textbf{C}}}+rho +eta ){{textbf{L}}}_{{textbf{T}}{textbf{C}}},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}{textbf{C}}}=rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+varsigma _{3}{{textbf{I}}}_{{textbf{T}}}+nu {{textbf{I}}}_{{textbf{C}}}-(beta +zeta _{{textbf{T}}{textbf{C}}}+xi ){{textbf{I}}}_{{textbf{T}}{textbf{C}}},\ ,^{ABC}{textbf{D}}_{tau }^{omega }{{textbf{R}}}=varpi {{textbf{L}}}_{{textbf{T}}}+varphi _{2}{{textbf{E}}}_{{textbf{C}}}+delta {{textbf{I}}}_{{textbf{T}}}+varphi _{3}{{textbf{I}}}_{{textbf{C}}}+(1-(varsigma _{1}+varsigma _{2}))eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+(1-(theta _{1}+theta _{2}))xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-beta {textbf{R}}.end{array}right. } end{aligned}$$

(5)

The arrangement of this article is as follows: In Codynamics model and preliminaries section, explanations for fractional calculus, along with several key notions and model (2) details, are provided. Moreover, a detailed analysis of the FO co-infection systems (3) equilibrium stability is presented in Codynamics model and preliminaries section. In Stochastic configuration of codynamics of TB-COVID-19 model section, a probabilistic form of the TB and COVID-19 models (28) codynamics is proposed and a detailed description of the unique global positive solution for each positive initial requirement is presented. The dynamical characteristics of the mechanisms appropriate conditions for the presence of the distinctive stationary distribution are provided. The P.D.F enclosing a quasi-stable equilibrium of the probabilistic COVID-19 framework is presented in Stochastic COVID-19 model without TB infection section. Numerous numerical simulations in view of piecewise fractional derivative operators are presented in Numerical solutions of co-dynamics model using random perturbations section to validate the diagnostic findings we obtained in Stochastic configuration of codynamics of TB-COVID-19 model and Stochastic COVID-19 model without TB infection sections. In conclusion, we conceal our findings to conclude this study.

Since we interact with living communities, each approach ought to be constructive and centred on a workable area. We utilized the subsequent hypothesis that guarantees these.

Assume that the set ({tilde{Xi }}:=Big ({{textbf{S}}},{{textbf{L}}}_{{textbf{T}}},{{textbf{I}}}_{{textbf{T}}},{{textbf{E}}}_{{textbf{C}}},{{textbf{I}}}_{{textbf{C}}},{{textbf{L}}}_{{textbf{T}}{textbf{C}}},{{textbf{I}}}_{{textbf{T}}{textbf{C}}},{{textbf{R}}}Big )) is a positive invariant set for the suggested FO model (3).

In order to demonstrate whether the solution to a set of equations (3) is positive, then (3) yields

$$begin{aligned} {left{ begin{array}{ll} ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{S}}}big vert _{{{textbf{S}}}=0}=nabla ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{{textbf{L}}}_{{textbf{T}}}}big vert _{{{textbf{L}}}_{{textbf{T}}}}=psi _{{textbf{T}}}{{textbf{S}}}ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{{textbf{I}}}_{{textbf{T}}}}big vert _{{{textbf{I}}}_{{textbf{T}}}=0}=mu {{textbf{L}}}_{{textbf{T}}}ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{{textbf{E}}}_{{textbf{C}}}}big vert _{{{textbf{E}}}_{{textbf{C}}}=0}=psi _{{textbf{C}}}{{textbf{S}}}ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{{textbf{I}}}_{{textbf{C}}}}big vert _{{{textbf{I}}}_{{textbf{C}}}=0}=varphi _{1}{{textbf{E}}}_{{textbf{C}}}ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{{textbf{L}}}_{{textbf{T}}{textbf{C}}}}big vert _{{{textbf{L}}}_{{textbf{T}}{textbf{C}}}=0}=lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}+epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{{textbf{I}}}_{{textbf{T}}{textbf{C}}}}big vert _{{{textbf{I}}}_{{textbf{T}}{textbf{C}}}=0}=rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+varsigma _{3}{{textbf{I}}}_{{textbf{T}}}+nu {{textbf{I}}}_{{textbf{C}}}ge 0,\ ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{R}}}big vert _{{textbf{R}}=0}=varpi {{textbf{L}}}_{{textbf{T}}}+varphi _{2}{{textbf{E}}}_{{textbf{C}}}+delta {{textbf{I}}}_{{textbf{T}}}+varphi _{3}{{textbf{I}}}_{{textbf{C}}}+(1-(varsigma _{1}+varsigma _{2}))eta {L_{{textbf{T}}{textbf{C}}}}+(1-(theta _{1}+theta _{2}))xi {{{textbf{I}}}_{{textbf{T}}{textbf{C}}}}ge 0. end{array}right. } end{aligned}$$

(6)

Therefore, the outcomes related to the FO model (3) are positive. Finally, the variation in the entire community is described by

$$begin{aligned} ,_{0}^{c}{textbf{D}}_{tau }^{omega }{tilde{Xi }}{} & {} le nabla +zeta _{{textbf{T}}}{{textbf{I}}}_{{textbf{T}}}-zeta _{{textbf{C}}}({{textbf{I}}}_{{textbf{C}}}+{{textbf{L}}}_{{textbf{T}}{textbf{C}}})-zeta _{{textbf{T}}{textbf{C}}}{{textbf{I}}}_{{textbf{T}}{textbf{C}}}-beta {textbf{N}}nonumber \{} & {} le nabla -beta {{textbf{N}}}. end{aligned}$$

Addressing the variant previously mentioned, we get

$$begin{aligned} {tilde{Xi }}(tau )le bigg ({tilde{Xi }}(0)-frac{nabla }{beta }bigg )E_{omega }bigg (-beta tau ^{omega }bigg )+frac{nabla }{beta }. end{aligned}$$

Consequently, we derive the GML functions asymptotic operation43 as

$$begin{aligned} {tilde{Xi }}(tau )le frac{nabla }{beta }. end{aligned}$$

Adopting the same procedure for other systems of equations in the model (3), which indicates that the closed set ({tilde{Xi }}) is a positive invariant domain for the FO system (3).(square )

Assuming that every requirement is non-negative throughout time (tau ), we exhibit that the outcomes remain non-negative and bounded in the proposed region, (Pi ). Well look at the co-infection model (3) ({tilde{Xi }}:=Big ({{textbf{S}}},{{textbf{L}}}_{{textbf{T}}},{{textbf{I}}}_{{textbf{T}}},{{textbf{E}}}_{{textbf{C}}},{{textbf{I}}}_{{textbf{C}}},{{textbf{L}}}_{{textbf{T}}{textbf{C}}},{{textbf{I}}}_{{textbf{T}}{textbf{C}}},{{textbf{R}}}Big )) spreads in the domain, which is described as (Pi :=Big {{tilde{Xi }}in Re _{+}^{8}:0le {textbf{N}}le frac{nabla }{beta }Big }.)

According to the afflicted categories in co-infection model (3), disease-free equilibrium (DFE) and endemic equilibrium (EE) are the biologically significant steady states of FO model (3). We establish the fractional derivative to get the immune-to-infection steady state as ( ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{S}}},~,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}},~,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}},,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{E}}}_{{textbf{C}}},~,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{C}}}, ,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}{textbf{C}}},~,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}{textbf{C}}},~,_{0}^{c}{textbf{D}}_{tau }^{omega }{{textbf{R}}},) to zero of the FO model (3) have no infection, and get

$$begin{aligned} {mathcal {E}}_{0}=Big (frac{nabla }{beta },0,0,0,0,0,0,0Big ). end{aligned}$$

The dominating eigenvalue of the matrix ({textbf{F}}{textbf{G}}^{-1}) correlates with the basic reproductive quantity ({mathbb {R}}_{0}^{CT}) of structure (3), in accordance with the next generation matrix approach44. Thus, we find

$$begin{aligned}{mathcal {F}}= begin{pmatrix} psi _{{textbf{T}}}{{textbf{S}}}\ 0\ psi _{{textbf{C}}}{textbf{S}}\ 0\ 0\ 0 end{pmatrix},~~~Phi =begin{pmatrix} (beta +mu +lambda psi _{{textbf{C}}}+varpi ){{textbf{L}}}_{{textbf{T}}}\ -theta _{2}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-varsigma _{2}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}-mu {{textbf{L}}}_{{textbf{T}}}+(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ){{textbf{I}}}_{{textbf{T}}}\ (beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}){{textbf{E}}}_{{textbf{C}}}\ -varphi _{1}{{textbf{E}}}_{{textbf{C}}}-varsigma _{1}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}-theta _{1}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}+(beta +nu +zeta _{{textbf{C}}}+varphi _{3}){{textbf{I}}}_{{textbf{C}}}\ -lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}-epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}+(beta +zeta _{{textbf{C}}}+rho +eta ){{textbf{L}}}_{{textbf{T}}{textbf{C}}}\ -rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}-varsigma _{3}{{textbf{I}}}_{{textbf{T}}}-nu {{textbf{I}}}_{{textbf{C}}}+(beta +zeta _{{textbf{T}}{textbf{C}}}+xi ) end{pmatrix}. end{aligned}$$

The next generation matrix at DEF can then be obtained by using the Jacobian of ({textbf{F}}) and ({textbf{G}}) examined at ({mathcal {E}}_{0}) as

$$begin{aligned} {textbf{F}}{textbf{G}}^{-1}=begin{pmatrix} frac{mu {mathcal {K}}_{1}}{(beta +mu +varpi ){mathcal {K}}_{7}}&{}frac{{mathcal {K}}_{1}}{{mathcal {K}}_{7}}&{}frac{varphi _{1}{mathcal {K}}_{3}}{(beta +omega +varphi _{2}){mathcal {K}}_{7}}&{}frac{{mathcal {K}}_{3}}{{mathcal {K}}_{7}}&{}frac{{mathcal {K}}_{5}}{(beta +zeta _{{textbf{C}}}+rho +eta ){mathcal {K}}_{7}}&{}frac{-alpha _{1}(beta +nu +zeta _{{textbf{C}}}+varphi _{3})(beta +varsigma _{3}+zeta _{{textbf{T}}}delta +theta _{2}xi )}{{mathcal {K}}_{7}}\ 0&{}0&{}0&{}0&{}0&{}0\ frac{mu {mathcal {K}}_{2}}{(beta +mu +varpi ){mathcal {K}}_{7}}&{}frac{{mathcal {K}}_{2}}{{mathcal {K}}_{7}}&{}frac{varphi _{1}{mathcal {K}}_{4}}{(beta +omega +varphi _{2}){mathcal {K}}_{7}}&{}frac{-alpha _{2}(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}}+nu )-theta _{2}varsigma _{3}xi }{{mathcal {K}}_{7}}&{}frac{{mathcal {K}}_{6}}{(beta +varphi _{1}+varphi _{2}){mathcal {K}}_{7}}&{}frac{-alpha _{2}(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +zeta _{{textbf{C}}}+rho +eta )}{{mathcal {K}}_{7}}\ 0&{}0&{}0&{}0&{}0&{}0\ 0&{}0&{}0&{}0&{}0&{}0\ 0&{}0&{}0&{}0&{}0&{}0 end{pmatrix}, end{aligned}$$

where

$$begin{aligned} {mathcal {K}}_{kappa }= {left{ begin{array}{ll} -alpha _{1}big ((beta +nu +zeta _{{textbf{C}}}+varphi _{3})(beta +xi +zeta _{{textbf{T}}{textbf{C}}})+(beta +nu +zeta _{{textbf{C}}}+varphi _{3})varsigma _{3}-theta _{1}nu xi big ),~~~~kappa =1,\ -alpha _{2}varsigma _{3}(theta _{1}xi +beta +nu +zeta _{{textbf{C}}}+varphi _{3}),~~~~kappa =2,\ -alpha _{1}nu (theta _{2}xi +beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ),~~~~kappa =3,\ -alpha _{2}(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )big ((beta +nu +varphi _{3}+zeta _{{textbf{C}}})(beta +xi +zeta _{{textbf{T}}{textbf{C}}})+varphi _{1}(beta +xi +zeta _{{textbf{T}}{textbf{C}}})+nu varphi _{1}-theta _{1}nu xi big )\ qquad -theta _{2}varsigma _{3}xi (varphi _{1}+beta +nu +zeta _{{textbf{C}}}+varphi _{3}),~~~~kappa =4,\ nu rho xi (theta _{2}varsigma _{1}-theta _{1}varsigma _{2})+(beta +nu +zeta _{{textbf{C}}}+varphi _{3})big (-alpha _{1}rho (nu xi +beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )+eta varsigma _{2}(beta +xi +varsigma _{3}+zeta _{{textbf{T}}{textbf{C}}})big )\ qquad +varsigma _{1}nu eta (beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ),~~~~kappa =5,\ -alpha _{2}big ((beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +nu +zeta _{C{1}}+varphi _{3})(beta +xi +zeta _{{textbf{T}}{textbf{C}}}+rho )+theta _{1}xi (rho -nu )(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )\ qquad +(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )eta varsigma _{1}(beta +xi +zeta _{{textbf{T}}{textbf{C}}}+nu )+varsigma _{2}eta varsigma _{3}(beta +nu +zeta _{{textbf{C}}}+varphi _{3}+theta _{1}xi )\ qquad -theta _{2}varsigma _{3}xi (varsigma _{1}eta +beta +nu +zeta _{{textbf{C}}}+varphi _{3})big ),~~~kappa =6,\ theta _{1}nu xi (beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )+(beta +zeta _{{textbf{C}}}+rho +eta )big (theta _{2}varsigma _{3}xi -(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta =xi +zeta _{{textbf{T}}{textbf{C}}})big ),~~~kappa =7. end{array}right. } end{aligned}$$

The fundamental reproducing quantity of the pairing system is shown by the highest spectral radius of the subsequent generations matrix. It is evident that the matrix ({textbf{F}}{textbf{G}}^{-1}) has four eigenvalues that are equivalent to zero. The truncated matrix yields the additional eigenvalues as

$$begin{pmatrix}frac{mu {mathcal {K}}_{1}}{(beta +mu +varpi ){mathcal {K}}}&frac{varphi _{1}{mathcal {K}}_{3}}{(beta +varphi _{1}+varphi _{2}){mathcal {K}}} frac{mu {mathcal {K}}_{2}}{(beta +mu +varpi ){mathcal {K}}}&frac{{mathcal {K}}_{4}}{(beta +varphi _{1}+varphi _{2}){mathcal {K}}}end{pmatrix}.$$

Consequently, by calculating the eigenvalues of ({textbf{F}}{textbf{G}}^{-1}), it is possible to simply determine that

$$begin{aligned} tilde{delta _{1}}{} & {} =frac{(beta +mu +varpi ){textbf{Q}}_{4}+mu (beta +varphi _{1}+varphi _{2}){textbf{Q}}_{1}-nabla _{1}^{2}}{2(beta +mu +varpi )(beta +varphi _{1}+varphi _{2})big ( theta _{1}nu xi (beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )+(beta +zeta _{{textbf{C}}}+rho +eta )big (theta _{2}varsigma _{3}xi -(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}})big )big )},nonumber \ tilde{delta _{2}}{} & {} =frac{(beta +mu +varpi ){textbf{Q}}_{4}+mu (beta +varphi _{1}+varphi _{2}){textbf{Q}}_{1}+nabla _{1}^{2}}{2(beta +mu +varpi )(beta +varphi _{1}+varphi _{2})big ( theta _{1}nu xi (beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )+(beta +zeta _{{textbf{C}}}+rho +eta )big (theta _{2}varsigma _{3}xi -(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}})big )big )}, end{aligned}$$

where

$$begin{aligned} nabla _{1}=sqrt{mu ^{2}{mathcal {K}}_{1}^{2}(beta +varphi _{1}+varphi _{2})^{2}-2mu (beta +mu +varpi )(beta +varphi _{1}+varphi _{2}){textbf{Q}}_{1}{textbf{Q}}_{2}+4varphi _{1}mu (beta +mu +varpi )(beta +varphi _{1}+varphi _{2}){mathcal {K}}_{2}^{2}+(beta +mu +varpi )^{2}{mathcal {K}}_{4}^{2}}. end{aligned}$$

Therefore, the co-dynamics structures (3) fundamental reproductive quantity ({mathbb {R}}_{0}) is provided by ({mathbb {R}}_{0}^{CT}=max {{mathbb {R}}_{0}^{C},{mathbb {R}}_{0}^{T}}.)

Here, we shall then demonstrate how transmission persists in the FO mechanism. It explains how widespread the virus is within the framework. From the viewpoint of biology, the virus continues in the bloodstream if the infectious proportion is elevated for a sufficiently long time (tau ).

However, the linearization technique is used to examine the local stabilization of the codynamics algorithms DFE state. At the DFE state ({mathcal {E}}_{0},) the Jacobean matrix of system (3) is displayed as

$$begin{aligned} {mathcal {J}}_{{mathcal {E}}_{0}}=begin{pmatrix} -beta &{}0&{}-alpha _{1}&{}-alpha _{2}&{}-alpha _{2}&{}-alpha _{2}&{}-(alpha _{1}+alpha _{2})&{}0\ 0&{}-(beta +mu +varpi )&{}alpha _{1}&{}0&{}0&{}0&{}alpha _{1}&{}0\ 0&{}mu &{}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )&{}0&{}0&{}varsigma _{2}eta &{}theta _{2}xi &{}0\ 0&{}0&{}0&{}alpha _{2}-beta -varphi _{1}-varphi _{2}&{}alpha _{2}&{}alpha _{2}&{}alpha _{2}&{}0\ 0&{}0&{}0&{}varphi _{1}&{}-beta -nu -zeta _{{textbf{C}}}-varphi _{3}&{}varsigma _{1}eta &{}theta _{1}xi &{}0\ 0&{}0&{}0&{}0&{}0&{}-(beta +zeta _{{textbf{C}}}+rho +eta )&{}0&{}0\ 0&{}0&{}varsigma _{3}&{}0&{}nu &{}rho &{}-(beta +xi +zeta _{{textbf{T}}{textbf{C}}})&{}0\ 0&{}varpi &{}delta &{}varphi _{2}&{}varphi _{3}&{}(1-(varsigma _{1}+varsigma _{2}))eta &{}(1-(theta _{1}+theta _{2}))eta &{}-beta . end{pmatrix} end{aligned}$$

(7)

The analysis of ({mathcal {E}}_{0})s localized temporal equilibrium relies upon the eigenvalues interpretation. Here, (tilde{delta _{1,2}}=-beta ) and (tilde{delta _{3}}=-(beta +rho +eta +zeta _{{textbf{C}}})) are obtained by broadening the following polynomial (vert {mathcal {J}}_{{mathcal {E}}_{0}}-delta {mathcal {I}}vert =0). Moreover, we get the additional (delta )s based on the simplified matrixs (vert {mathcal {J}}_{{mathcal {E}}_{0}}-delta {mathcal {I}}vert =0) described as

$$begin{aligned}{} & {} {mathcal {J}}-delta {mathcal {I}}_{5}nonumber \ {}{} & {} =begin{pmatrix} mu &{}-(beta +varsigma _{3}+delta +zeta _{{textbf{T}}}+{tilde{delta }})&{}0&{}0&{}theta _{2}xi \ 0&{}varsigma _{3}&{}0&{}nu &{}-({tilde{delta }}+beta +xi +zeta _{{textbf{C}}})\ 0&{}0&{}varphi _{1}&{}-({tilde{delta }}+beta +nu +varphi _{3}+zeta _{{textbf{C}}})&{}theta _{1}xi \ 0&{}0&{}0&{}alpha _{2}varphi _{1}+({tilde{delta }}+beta +nu +varphi _{3}+zeta _{{textbf{C}}})(alpha _{2}-{tilde{delta }}-beta -xi -varphi _{1})/varphi _{1}&{}Im _{1}\ 0&{}0&{}0&{}0&{}Im _{2} end{pmatrix}, end{aligned}$$

where (Im _{1}=alpha _{2}varphi _{1}+({tilde{delta }}+beta +nu +varphi _{3}+zeta _{{textbf{C}}})(alpha _{2}-{tilde{delta }}-beta -xi -varphi _{1})/varphi _{1},Im _{2}=alpha _{1}(varsigma _{3}+({tilde{delta }}+beta +xi +zeta _{{textbf{C}}}))/varsigma _{3}+big (beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +{tilde{delta }}/mu big )big (theta _{2}xi varsigma _{3}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +{tilde{delta }})(beta +xi +zeta _{{textbf{T}}{textbf{C}}}+{tilde{delta }})/varsigma _{3})big ).) After simple computations, the characteristic polynomial of the above matrix is presented as

$$begin{aligned} {textbf{U}}({tilde{delta }}){} & {} =-mu varsigma _{3}varphi _{1} frac{alpha _{2}varphi _{1}+({tilde{delta }}+beta +nu +varphi _{3}+zeta _{{textbf{C}}})(alpha _{2}-{tilde{delta }}-beta -xi -varphi _{1})}{varphi _{1}}Bigg {frac{alpha _{1}(varsigma _{3}+({tilde{delta }}+beta +xi +zeta _{{textbf{C}}}))}{varsigma _{3}}\{} & {} quad +frac{(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +{tilde{delta }}}{mu }frac{theta _{2}xi varsigma _{3}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +{tilde{delta }})(beta +xi +zeta _{{textbf{T}}{textbf{C}}}+{tilde{delta }})}{varsigma _{3}}Bigg }. end{aligned}$$

(8)

In other words, the outcomes to the ({textbf{U}}({tilde{delta }})) are the eigenvalues:

$$begin{aligned} {textbf{U}}({tilde{delta }})={tilde{delta }}^{5}+{textbf{d}}_{1}{tilde{delta }}^{4}+{textbf{d}}_{2}{tilde{delta }}^{3}+{textbf{d}}_{3}{tilde{delta }}^{2}+{textbf{d}}_{4}{tilde{delta }}+{textbf{d}}_{5}=0, end{aligned}$$

(9)

where

$$begin{aligned} {textbf{d}}_{1}{} & {} =alpha _{2}-beta -varphi _{1}-varphi _{2},nonumber \ {textbf{d}}_{2}{} & {} =alpha _{2}varphi _{1}+(beta +nu +zeta _{{textbf{C}}}+varphi _{3})(alpha _{2}-beta -varphi _{1}-varphi _{2})-(alpha _{2}-2beta -varphi _{1}-varphi _{2}-nu -zeta _{{textbf{C}}}-varphi _{3})nonumber \{} & {} quad times (beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +xi +zeta _{{textbf{T}}{textbf{C}}}-mu -varpi )+big (mu alpha _{1}+theta _{2}xi varsigma _{3}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}})nonumber \{} & {} quad -(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +mu +varpi )-(beta +mu +varpi )(beta +xi +zeta _{{textbf{T}}{textbf{C}}})big ),nonumber \ {textbf{d}}_{3}{} & {} =mu alpha _{1}(varsigma _{3}+beta +xi +zeta _{{textbf{T}}{textbf{C}}})+(beta +mu +varpi )big (theta _{2}xi varsigma _{3}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}})big )nonumber \{} & {} quad -big (alpha _{2}varphi _{1}+(beta +nu +varphi _{3}+zeta _{{textbf{C}}})(alpha _{2}-beta -varphi _{1}-varphi _{2})big )(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +varphi _{1}+varphi _{2}-xi -zeta _{{textbf{T}}{textbf{C}}})nonumber \ {}{} & {} quad -(alpha _{2}-2beta -varphi _{1}-varphi _{2}-nu -zeta _{{textbf{C}}}-varphi _{3})big (mu alpha _{1}+theta _{2}xi varsigma _{3}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}})nonumber \ {}{} & {} quad -(beta +mu +varpi )(2beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +xi +zeta _{{textbf{T}}{textbf{C}}})big ),nonumber \{textbf{d}}_{4}{} & {} = -big (alpha _{2}varphi _{1}+(beta +nu +zeta _{{textbf{C}}}+varphi _{3})(alpha _{2}-beta -varphi _{1}-varphi _{2})big )big (mu alpha _{1}+theta _{2}xi varsigma _{3}-(beta +mu +varpi )(2beta +varsigma _{3}+zeta _{{textbf{T}}}+delta +xi +zeta _{{textbf{T}}{textbf{C}}})big )nonumber \ {}{} & {} quad -(alpha _{2}-2beta -varphi _{1}-varphi _{2}-nu -zeta _{{textbf{C}}}-varphi _{3})big (mu alpha _{1}(varsigma _{3}+beta +xi +zeta _{{textbf{T}}{textbf{C}}})+(beta +mu +varpi )nonumber \ {}{} & {} quad times (theta _{2}xi varsigma _{3}-(beta varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}}))big ),nonumber \{textbf{d}}_{5}{} & {} =big (alpha _{2}varphi _{1}+(beta +nu +varphi _{3}+zeta _{{textbf{C}}})(alpha _{2}-beta -varphi _{1}-varphi _{2})big )big (mu alpha _{1}(varsigma _{3}+beta +xi +zeta _{{textbf{T}}{textbf{C}}})nonumber \ {}{} & {} quad +(beta +mu +varpi )(theta _{2}xi varsigma _{3}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta )(beta +xi +zeta _{{textbf{T}}{textbf{C}}}))big ). end{aligned}$$

(10)

Therefore, if the subsequent requirements apply, the roots of expression (10) exhibit negative real portions according to the RouthHurwitz stability specifications as

$$begin{aligned} {left{ begin{array}{ll} {textbf{d}}_{jmath }>0,~~forall ~jmath =1,...,5,~{textbf{d}}_{1}{textbf{d}}_{2}{textbf{d}}_{3}>{textbf{d}}_{3}^{2}+{textbf{d}}_{1}^{2}{textbf{d}}_{4},\ ({textbf{d}}_{1}{textbf{d}}_{4}-{textbf{d}}_{5})({textbf{d}}_{1}{textbf{d}}_{2}{textbf{d}}_{3}-{textbf{d}}_{3}^{2}-{textbf{d}}_{1}^{2}{textbf{d}}_{4})>{textbf{d}}_{5}({textbf{d}}_{1}{textbf{d}}_{2}-{textbf{d}}_{3})^{2}+{textbf{d}}_{1}{textbf{d}}_{5}^{2}. end{array}right. } end{aligned}$$

(11)

Evolution of the basic reproduction number ({mathbb {R}}_{0}^{CT}) with the aid of ({mathbb {R}}_{0}^{C}) and ({mathbb {R}}_{0}^{T}).

Figure 4 is illustrated by depicting in 3D evolution of the threshold parameter ({mathbb {R}}_{0}^{CT}) of model (3) as a function of ({mathbb {R}}_{0}^{C}) and ({mathbb {R}}_{0}^{T}.)

The forthcoming result is established thanks to Theorem 2 in44.

The DFE point of the FO codynamics model (3) is locally asymptotically stable if the prerequisite specified in formula (12) is satisfied.

This section shows that there is only one solution for the system (3). Now, we demonstrate that the frameworks solution is distinctive. Initially, we construct framework (3) in the form of:

$$begin{aligned} {left{ begin{array}{ll} ,^{c}{textbf{D}}_{tau }^{omega }{textbf{S}}={mathcal {Q}}_{1}big (tau ,{textbf{S}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}}={mathcal {Q}}_{2}big (tau ,{{textbf{L}}}_{{textbf{T}}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}}={mathcal {Q}}_{3}big (tau ,{{textbf{I}}}_{{textbf{T}}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{E}}}_{{textbf{C}}}={mathcal {Q}}_{4}big (tau ,{{textbf{E}}}_{{textbf{C}}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{C}}}={mathcal {Q}}_{5}big (tau ,{{textbf{I}}}_{{textbf{C}}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{L}}}_{{textbf{T}}{textbf{C}}}={mathcal {Q}}_{6}big (tau ,{{textbf{L}}}_{{textbf{T}}{textbf{C}}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{I}}}_{{textbf{T}}{textbf{C}}}={mathcal {Q}}_{7}big (tau ,{{textbf{I}}}_{{textbf{T}}{textbf{C}}}(tau )big ),\ ,^{c}{textbf{D}}_{tau }^{omega }{{textbf{R}}}={mathcal {Q}}_{8}big (tau ,{{textbf{R}}}(tau )big ),\ end{array}right. } end{aligned}$$

(12)

where

$$begin{aligned} {left{ begin{array}{ll} {mathcal {Q}}_{1}big (tau ,{textbf{S}}(tau )big )=nabla -(psi _{{textbf{T}}}+psi _{{textbf{C}}}+beta ){textbf{S}},\ {mathcal {Q}}_{2}big (tau ,{{textbf{L}}}_{{textbf{T}}}(tau )big )=psi _{{textbf{T}}}{textbf{S}}-(beta +mu +lambda psi _{{textbf{C}}}+varpi ){{textbf{L}}}_{{textbf{T}}},\ {mathcal {Q}}_{3}big (tau ,{{textbf{I}}}_{{textbf{T}}}(tau )big )=mu {{textbf{L}}}_{{textbf{T}}}+varsigma _{2}eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{2}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ){{textbf{I}}}_{{textbf{T}}},\ {mathcal {Q}}_{4}big (tau ,{{textbf{E}}}_{{textbf{C}}}(tau )big )=psi _{{textbf{C}}}{textbf{S}}-(beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}){{textbf{E}}}_{{textbf{C}}},\ {mathcal {Q}}_{5}big (tau ,{{textbf{I}}}_{{textbf{C}}}(tau )big )=varphi _{1} {{textbf{E}}}_{{textbf{C}}}+rho eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+theta _{1}xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-(beta +zeta _{{textbf{C}}}+nu +varphi _{3}){{textbf{I}}}_{{textbf{C}}},\ {mathcal {Q}}_{6}big (tau ,{{textbf{L}}}_{{textbf{T}}{textbf{C}}}(tau )big )=lambda psi _{{textbf{C}}}{{textbf{L}}}_{{textbf{T}}}+epsilon psi _{{textbf{T}}}{{textbf{E}}}_{{textbf{C}}}-(beta +zeta _{{textbf{C}}}+rho +eta ){{textbf{L}}}_{{textbf{T}}{textbf{C}}},\ {mathcal {Q}}_{7}big (tau ,{{textbf{I}}}_{{textbf{T}}{textbf{C}}}(tau )big )=rho {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+varsigma _{3}{{textbf{I}}}_{{textbf{T}}}+nu {{textbf{I}}}_{{textbf{C}}}-(beta +zeta _{{textbf{T}}{textbf{C}}}+xi ){{textbf{I}}}_{{textbf{T}}{textbf{C}}},\ {mathcal {Q}}_{8}big (tau ,{{textbf{R}}}(tau )big )=varpi {{textbf{L}}}_{{textbf{T}}}+varphi _{2}{{textbf{E}}}_{{textbf{C}}}+delta {{textbf{I}}}_{{textbf{T}}}+varphi _{3}{{textbf{I}}}_{{textbf{C}}}+(1-(varsigma _{1}+varsigma _{2}))eta {{textbf{L}}}_{{textbf{T}}{textbf{C}}}+(1-(theta _{1}+theta _{2}))xi {{textbf{I}}}_{{textbf{T}}{textbf{C}}}-beta {textbf{R}}.end{array}right. } end{aligned}$$

(13)

Integral transform applied to both sides of equations (14) yields

$$begin{aligned} {left{ begin{array}{ll} {textbf{S}}(tau )-{textbf{S}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{1}big ({mathfrak {p}},{textbf{S}}big )d{mathfrak {p}},\ {{textbf{L}}}_{{textbf{T}}}(tau )-{{textbf{L}}}_{{textbf{T}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{2}big ({mathfrak {p}},{{textbf{L}}}_{{textbf{T}}}big )d{mathfrak {p}},\ {{textbf{I}}}_{{textbf{T}}}(tau )-{{textbf{I}}}_{{textbf{T}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{3}big ({mathfrak {p}},{{textbf{I}}}_{{textbf{T}}}big )d{mathfrak {p}},\ {{textbf{E}}}_{{textbf{C}}}(tau )-{{textbf{E}}}_{{textbf{C}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{4}big ({mathfrak {p}},{{textbf{E}}}_{{textbf{C}}}big )d{mathfrak {p}},\ {{textbf{I}}}_{{textbf{C}}}(tau )-{{textbf{I}}}_{{textbf{C}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{5}big ({mathfrak {p}},{{textbf{E}}}_{{textbf{C}}}big )d{mathfrak {p}},\ {{textbf{L}}}_{{textbf{T}}{textbf{C}}}(tau )-{{textbf{L}}}_{{textbf{T}}{textbf{C}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{6}big ({mathfrak {p}},{{textbf{L}}}_{{textbf{T}}{textbf{C}}}big )d{mathfrak {p}},\ {{textbf{I}}}_{{textbf{T}}{textbf{C}}}(tau )-{{textbf{I}}}_{{textbf{T}}{textbf{C}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{7}big ({mathfrak {p}},{{textbf{I}}}_{{textbf{T}}{textbf{C}}}big )d{mathfrak {p}},\ {{textbf{R}}}(tau )-{{textbf{R}}}(0)=frac{1}{Gamma (omega )}int limits _{0}^{tau }(tau -{mathfrak {p}})^{omega -1}{mathcal {Q}}_{8}big ({mathfrak {p}},{{textbf{R}}}big )d{mathfrak {p}}.\ end{array}right. } end{aligned}$$

(14)

The kernels ({mathcal {Q}}_{iota },~(iota =1,...,8)) satisfies the Lipschitz condition and contraction, as demonstrated.

({mathcal {Q}}_{1}) satisfies the Lipschitz condition and contraction if the following condition holds: (0le alpha _{1}(sigma _{3}+sigma _{7})+alpha _{2}(sigma _{4}+sigma _{5}+sigma _{2}+sigma _{5})+beta <1.)

For ({textbf{S}}) and (mathbf {S_{1}},) we have

$$begin{aligned} big Vert {mathcal {Q}}_{1}big (tau ,{{textbf{S}}}big )-{mathcal {Q}}_{1}big (tau ,{{textbf{S}}}_{1}big )big Vert{} & {} =big Vert big (alpha _{1}({textbf{I}}_{{textbf{T}}}+{textbf{I}}_{textbf{TC}})+alpha _{2}({textbf{E}}_{{textbf{C}}}+{textbf{I}}_{{textbf{C}}}+{textbf{I}}_{textbf{TC}}+{textbf{L}}_{{textbf{T}}})+beta big )big ({{textbf{S}}}(tau )-{{textbf{S}}}_{1}(tau )big )big Vert nonumber \ {}{} & {} le big (alpha _{1}big (big Vert {textbf{I}}_{{textbf{T}}}big Vert +big Vert {textbf{I}}_{textbf{TC}})big Vert big )+alpha _{2}big (big Vert {textbf{E}}_{{textbf{C}}}big Vert +big Vert {textbf{I}}_{{textbf{C}}}big Vert +big Vert {textbf{I}}_{textbf{TC}}big Vert +big Vert {textbf{L}}_{{textbf{T}}}big Vert big )+beta big )big Vert {{textbf{S}}}(tau )-{{textbf{S}}}_{1}(tau )big Vert . end{aligned}$$

Suppose ({mathcal {V}}_{1}=alpha _{1}(sigma _{3}+sigma _{7})+alpha _{2}(sigma _{4}+sigma _{5}+sigma _{2}+sigma _{5})+beta ), where ({textbf{I}}_{{textbf{T}}}le sigma _{3},~{textbf{I}}_{textbf{TC}}le sigma _{7},~{textbf{E}}_{{textbf{C}}}le sigma _{4}~{textbf{I}}_{{textbf{C}}}le sigma _{5},~{textbf{I}}_{textbf{TC}}le sigma _{7},~{textbf{L}}_{{textbf{T}}}le sigma _{2}) are a bounded functions. So, we have

$$begin{aligned} big Vert {mathcal {Q}}_{1}big (tau ,{{textbf{S}}}big )-{mathcal {Q}}_{1}big (tau ,{{textbf{S}}}_{1}big )big Vert le {mathcal {V}}_{1}big Vert {{textbf{S}}}(tau )-{{textbf{S}}}_{1}(tau )big Vert . end{aligned}$$

(15)

After obtaining the Lipschitz criterion for ({mathcal {Q}}_{1}), hence, ({mathcal {Q}}_{1}) is a contraction if (0le alpha _{1}(sigma _{3}+sigma _{7})+alpha _{2}(sigma _{4}+sigma _{5}+sigma _{2}+sigma _{5})+beta <1).

In the same manner, ({mathcal {Q}}_{jmath }~(jmath =2,..,7)) satisfy the Lipschitz condition as follows:

$$begin{aligned}{} & {} big Vert {mathcal {Q}}_{2}big (tau ,{{textbf{L}}}_{{textbf{T}}}big )-{mathcal {Q}}_{2}big (tau ,{{{textbf{L}}}_{{textbf{T}}}}_{1}big )big Vert le {mathcal {V}}_{2}big Vert {{textbf{L}}}_{{textbf{T}}}(tau )-{{{textbf{L}}}_{{textbf{T}}}}_{1}(tau )big Vert ,nonumber \ {}{} & {} big Vert {mathcal {Q}}_{3}big (tau ,{{textbf{I}}}_{{textbf{T}}}big )-{mathcal {Q}}_{3}big (tau ,{{{textbf{I}}}_{{textbf{T}}}}_{1}big )big Vert le {mathcal {V}}_{3}big Vert {{textbf{I}}}_{{textbf{T}}}(tau )-{{{textbf{I}}}_{{textbf{T}}}}_{1}(tau )big Vert ,nonumber \ {}{} & {} big Vert {mathcal {Q}}_{4}big (tau ,{{textbf{E}}}_{{textbf{C}}}big )-{mathcal {Q}}_{4}big (tau ,{{{textbf{E}}}_{{textbf{C}}}}_{1}big )big Vert le {mathcal {V}}_{4}big Vert {{textbf{E}}}_{{textbf{C}}}(tau )-{{{textbf{E}}}_{{textbf{C}}}}_{1}(tau )big Vert ,nonumber \ {}{} & {} big Vert {mathcal {Q}}_{5}big (tau ,{{textbf{I}}}_{{textbf{C}}}big )-{mathcal {Q}}_{5}big (tau ,{{{textbf{I}}}_{{textbf{C}}}}_{1}big )big Vert le {mathcal {V}}_{5}big Vert {{textbf{L}}}_{{textbf{T}}}(tau )-{{{textbf{I}}}_{{textbf{C}}}}_{1}(tau )big Vert ,nonumber \ {}{} & {} big Vert {mathcal {Q}}_{6}big (tau ,{{textbf{L}}}_{textbf{TC}}big )-{mathcal {Q}}_{6}big (tau ,{{{textbf{L}}}_{textbf{TC}}}_{1}big )big Vert le {mathcal {V}}_{6}big Vert {{textbf{S}}}(tau )-{{{textbf{L}}}_{textbf{TC}}}_{1}(tau )big Vert ,nonumber \ {}{} & {} big Vert {mathcal {Q}}_{7}big (tau ,{{textbf{I}}}_{textbf{TC}}big )-{mathcal {Q}}_{T}big (tau ,{{{textbf{I}}}_{textbf{TC}}}_{1}big )big Vert le {mathcal {V}}_{7}big Vert {{textbf{S}}}(tau )-{{textbf{S}}}_{1}(tau )big Vert ,nonumber \ {}{} & {} big Vert {mathcal {Q}}_{8}big (tau ,{{textbf{r}}}big )-{mathcal {Q}}_{8}big (tau ,{{textbf{R}}}_{1}big )big Vert le {mathcal {V}}_{8}big Vert {{textbf{R}}}(tau )-{{textbf{R}}}_{1}(tau )big Vert ,end{aligned}$$

where ({mathcal {V}}_{2}=psi _{{textbf{T}}}sigma _{1}-(beta +mu +lambda psi _{{textbf{C}}}+varpi ),~{mathcal {V}}_{3}=mu sigma _{2}+varsigma _{2}eta sigma _{6}+theta _{2}xi sigma _{7}-(beta +varsigma _{3}+zeta _{{textbf{T}}}+delta ),~{mathcal {V}}_{4}=psi _{{textbf{C}}}sigma _{1}-(beta +epsilon psi _{{textbf{T}}}+varphi _{1}+varphi _{2}),~{mathcal {V}}_{5}=varphi _{1}sigma _{4}+rho eta sigma _{6}+theta _{1}xi sigma _{7}-(beta +zeta _{{textbf{C}}}+nu +varphi _{3}),~{mathcal {V}}_{6}=lambda psi _{{textbf{C}}}sigma _{2}+epsilon psi _{{textbf{T}}}sigma _{4}-(beta +zeta _{{textbf{C}}}+rho +xi ),~{mathcal {V}}_{7}=rho sigma _{6}+varsigma _{3}sigma _{3}+nu sigma _{5}-(beta +zeta _{textbf{TC}}+xi ),~{mathcal {V}}_{8}=varpi sigma _{2}+varphi _{2}sigma _{4}+delta sigma _{3}+varphi _{3}sigma _{5}+(1-(varsigma _{1}+varsigma _{2}))eta sigma _{6}+(1-(theta _{1}+theta _{2}))xi sigma _{7}-beta .)

For (jmath =2,...,8,) we find (0le {mathcal {V}}_{jmath }<1,) then ({mathcal {V}}_{jmath }) are contractions. Assume the following recursive pattern, as suggested by system (15):

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