A fractional-order model for COVID-19 and tuberculosis co-infection using Atangana–Baleanu derivative

Abstract

This paper considers and analyzes a fractional order model for COVID-19 and tuberculosis co-infection, using the Atangana–Baleanu derivative. The existence and uniqueness of the model solutions are established by applying the fixed point theorem. It is shown that the model is locally asymptotically stable when the reproduction number is less than one. The global stability analysis of the disease free equilibrium points is also carried out. The model was simulated using data relevant to both diseases in New Delhi, India. Fitting the model to the cumulative confirmed COVID-19 cases for New Delhi from March 1, 2021 to June 26, 2021, COVID-19 and TB contact rates and some other important parameters of the model are estimated. The numerical method used combines the two-step Lagrange polynomial and the fundamental theorem of fractional calculus and has been shown to be highly accurate and efficient, user-friendly and converges quickly to the exact solution even with a large step of discretization. Simulations of the Fractional order model revealed that reducing the risk of COVID-19 infection by latently-infected TB individuals will not only bring down the burden of COVID-19, but will also reduce the co-infection of both diseases in the population. Also, the conditions for the co-existence or elimination of both diseases from the population are established.

1. Introduction

Since late 2019, the Corona Virus Disease-2019 (COVID-19) caused by Severe Acute Respiratory Syndrome-Corona Virus-2 (SARS-CoV-2) has spread globally with about 190 million cases reported worldwide, as at 31st July, 2021 [1]. According to the reports of the World Health Organization (WHO) in 2019, 10 million people were infected with tuberculosis (TB) and approximately 1.6 million died worldwide that same year [2]. Presently, the countries with highest burden of TB wordwide are: India, China and the Russia [2]. Owing to serious threats posed by COVID-19 and TB co-infections, many researchers have focused their attention on studying the epidemiology of both diseases and their co-interactions [3], [4], [5], [6], [7]. According to Chen et al. [5], TB can increase the susceptibility to COVID-19 and severity of its symptoms. Petrone et al. [4] reported that COVID-19 and TB patients have a reduced risk of developing strong immune response to SAR-COV-2. Clinical evidences have shown that co-infections of TB and COVID-19 are associated with higher morbidity and mortality [8]. In a study by Boulle et al. [3], TB is associated with higher mortality in COVID-19 patients. Also, according to Davies et al [7], patients infected with TB in the past have increased risk of mortality if they got infected with COVID-19. In a cohort study on 49 patients, Tadolini et al. [6] discovered that more than 50.0% patients having a previous history of TB, more than 27% developed COVID-19 first and 18.3% patients tested positive for both TB and COVID-19. In more than 35% patients, COVID-19 was developed during treatment for TB revealing the likelihood of the risk of transmission to care-givers. Another research carried out by Chen et al. [5] warranted that COVID-19 infection and its severity would be likely higher in patients having active and latent TB infection. In a work done by Motta et al. [9], analyses were carried out on two cohorts of patients co-infected with TB and COVID-19 and it was observed that more than 65 patients from two cohorts, about 11.6% had the co-infection of the two diseases.

Mathematical models have been developed for the dynamics of COVID-19 [10], [11], [12]. The authors in Okuonghae and Omame [10] developed and analyzed a mathematical model for COVID-19 population dynamics in Lagos, Nigeria. Simulations of the model revealed that social distancing regulation, face masks usage while in public and increased case detection for symptomatic individuals can greatly reduce the incidence of COVID-19 disease. Baleanu et al. [11] considered a fractional-order model for COVID-19 transmission with Caputo-Fabrizio derivative. They used the homotopy analysis transform method (HATM)to solve the model and also provided solution in convergent series. Rezapour et al. [12] developed an SEIR model for COVID-19 using the Caputo derivative. They obtained the approximate solution of the model using the fractional Euler method. Their model also predicted COVID-19 transmission in Iran and in the world based on real data.

Mathematical models have been formulated for the dynamics of COVID-19 and its co-infection with other diseases [13], [14], [15], [16]. Omame et al. [13] developed a model for COVID-19 and co-morbidity co-infection with optimal control. They showed that COVID-19 re-infection and modification parameter for susceptibility of co-morbid individuals induced the phenomenon of backward bifurcation in the model. Simulations of the model revealed that the strategy that prevents COVID-19 infection by comorbid susceptibles is the most cost-effective of all the control strategies for the prevention of COVID-19. The authors in Tchoumi et al. [14] considered a co-infection model for COVID-19 and Malaria with optimal control. The showed that applying both COVID-19 and malaria protective strategies could help reduce their spread in comparison to applying each preventive measures singly. More recently, Rehman et al. [15] studied a fractional order model for COVID-19, comparing the behavior of the model using different derivatives (Caputo, Caputo–Fabrizio and Atangana–Baleanu), and showed that Caputo presented better results in the form of stability as compared to the other two operators. Furthermore, in a related research, the authors in Hezam et al. [16] showed that the policy of providing resources for the distribution of chlorine water tablets, sufficient equipments for testing with adequate compliance on social distancing rules as well as quarantining infected individuals has significant impact in reducing COVID-19 ad cholera co-infections in Yemen.

Many of the research works that have been carried out on the epidemiology of diseases using integer order models, such as those in Omame et al. [13], Okuonghae and Omame [10], Omame et al. [17], Egeonu et al. [18], Omame et al. [19], Omame and Okuonghae [20], Uwakwe et al. [21] have so much limitations as they could not capture the effect of memory as a result the integer nature of the order. These limitations have created a big vacuum for other methodologies to come up, such as fractional differential operators which involve both non-local and singular kernel and uses the power law function as its kernel. It is well known that fractional order derivatives are very important in modeling as they capture the memory effect, hereditary, as well as nonlocal properties [22]. “Memory effect means the future state of the fractional operator of a given function depends on the current state and the historical behavior of the state” [22]. Fractional derivatives have been used extensively in the literature to capture the effect of memory on the system dynamics [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. In epidemiological modeling, fractional derivatives and integrals are very significant, because the effect of memory plays an important role in the spread of the disease [22]. The presence of memory effects on past events will affect the spread of the disease in the future so that the it can be controlled easily. Thus, the effects of memory on the spread of infectious diseases can be verified using fractional derivatives [22].

In this work, we shall formulate a new co-infection model for COVID-19 and TB and analyze them using Atangana–Baleanu fractional derivative with the non-singular kernel [23]. To the best of the authors’ knowledge, this research is novel and will contribute to the body of knowledge, as it seeks to assess the impact of TB control measures on COVID-19 prevention and also to determine the conditions for the co-existence and elimination of both diseases in a population. The reason for considering this special fractional operator is its feature that it avoids the singularity which is present in the Caputo fractional derivative. The Atangana–Baleanu fractional derivative is the most preferred derivative in modeling biological and physical processes, as it involves nonsingular and nonlocal kernel [23].

2. Preliminaries

In this section, we recall some basic concepts from fractional calculus and some known theorems needed in the sequel.

Definition 2.1 [23] —

Let $f\in H^1(a_1,a_2),a_2>a_1,\varpi \in [0,1]$, then the Atangana–Baleanu Caputo integral of a function $f\left(t\right)$ of order $\varpi$ is defined by

$${}_{a_1}^{abc}{I}_t^{\varpi }=\frac{1-\varpi }{\mathcal{B}\left(\varpi \right)}f\left(t\right)+\frac{\varpi }{\mathcal{B}\left(\varpi \right)\Gamma \left(\varpi \right)}{\int }_{a_1}^{t}f\left(\tau \right){(t-\tau )}^{\varpi -1}d\tau .$$

Definition 2.2 [23] —

The laplace transform of the Atangana–Baleanu fractional derivative of order $\varpi$ in Caputo sense is given by

$$L\left\{{}_0^{ABC}{D}_t^{\varpi }f\left(t\right)\right\}=\frac{\mathcal{B}\left(\varpi \right)(s^{\varpi }\overline{f}\left(s\right)-s^{\varpi -1}f\left(0\right)))}{s^{\varpi }(1-\varpi )+\varpi },$$

where $L$ is the Laplace transform operator.

Lemma 2.1 [23] —

For$f\in H^1(a_1,a_2)$, the following Newton–Leibniz formula is satisfied,

$${}_{a_1}^{abc}{I}_t^{\varpi }\left({}_{a_1}^{abc}{\mathfrak{D}}_t^{\varpi }f\left(t\right)\right)=f\left(t\right)-f\left(a\right).$$

Theorem 2.2 [23] —

The following inequality holds for a function$f\in C[a,b]$

$${\parallel }_{a_1}^{abc}{\mathfrak{D}}_t^{\varpi }f\left(t\right)\parallel <\frac{\mathcal{B}\left(\varpi \right)}{1-\varpi }\parallel f\left(t\right)\parallel ,where\parallel f\left(t\right)\parallel =ma{x}_{a<t<b}\left|f\left(t\right)\right|.$$

Moreover, the Lipschitz condition is satisfied by ABC derivatives, that is, we have:

$${\parallel }_{a_1}^{abc}{\mathfrak{D}}_t^{\varpi }{f}_1\left(t\right){-}_{a_1}^{abc}{\mathfrak{D}}_t^{\varpi }{f}_2\left(t\right)\parallel <\delta \parallel f_1\left(t\right)-f_2\left(t\right)\parallel .$$

Theorem 2.3 [23] —

The unique solution of the differential equation with fractional order$\varpi$given by

$${}_{a_1}^{abc}{\mathfrak{D}}_t^{\varpi }f\left(t\right)=r\left(t\right),$$

is of the form

$$f\left(t\right)=\frac{1-\varpi }{\mathcal{B}\left(\varpi \right)}r\left(t\right)+\frac{\varpi }{\mathcal{B}\left(\varpi \right)\Gamma \left(\varpi \right)}{\int }_{a_1}^{t}r\left(\tau \right){(t-\tau )}^{\varpi -1}d\tau .$$

3. Model formulation

The total human population $N\left(t\right)$ at time $t$ is divided into nine compartments: unvaccinated susceptible individuals $\left(S\right(t\left)\right)$, Susceptible individuals vaccinated with the $Pfizer$ COVID-19 vaccine $\left(V\right(t\left)\right)$, infectious individuals with COVID-19 $\left(I\right(t\left)\right)$, individuals who have recovered from COVID-19 infection $\left(R\right(t\left)\right)$, infected individuals with latent TB $\left(E\right(t\left)\right)$, infected individuals with active TB $\left(A\right(t\left)\right)$, individuals treated of TB infection $\left(T\right(t\left)\right)$, individuals co-infected with latent TB and COVID-19 $\left(I_e\left(t\right)\right)$ and individuals co-infected with active TB and COVID-19 $\left(I_a\left(t\right)\right)$.

Susceptible individuals, $S$, are recruited into the population at a rate ${\Delta }_h$. This population is decreased as a result of infection with COVID-19, following effective contacts with infected individuals at the rate $\frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}$. The population also decreases due to infection with TB at the rate $\frac{{\alpha }_2(A+I_a)}{N}$. The parameter $\kappa (\kappa \ge 1)$ accounts for the increased infectivity of co-infected individuals as a result of tuberculosis. Natural mortality is assumed to be the same for all compartments, at the rate ${\beta }_h$.

The model has the following assumptions:

• i.Individuals with tuberculosis have increased susceptibility to infection with COVID-19 [5],
• ii.co-infected individuals are associated with higher mortality [8],
• iii.it is also assumed that individuals co-infected with both diseases can transmit only a single infection, and not mixed infections at the same time.

The model of fractional order $\varpi \in (0,1]$ is given by (model parameters are explained in Table 1):

$$\begin{array}{ccc}{}_0^{abc}{\mathfrak{D}}_t^{\varpi }S\hfill & =\hfill & {\Delta }_h-({\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\beta }_h+\psi +{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})S,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }V\hfill & =\hfill & \psi S-((1-\vartheta ){\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\beta }_h+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})V,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }I\hfill & =\hfill & {\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}[S+(1-\vartheta )V+T]+{\epsilon }_2{I}_a-({\varphi }_1+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}+{\beta }_h+{\delta }_1)I,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }R\hfill & =\hfill & {\varphi }_{1}I-({\beta }_h+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})R,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }E\hfill & =\hfill & (1-{\lambda }_1){\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}(S+V+R+\omega T)+{\varphi }_2{I}_e-({\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\gamma }_1+{\beta }_h)E,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }A\hfill & =\hfill & {\lambda }_1{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}(S+V+R+\omega T)+{\gamma }_{1}E+{\varphi }_3{I}_a-({\epsilon }_1+{\beta }_h+{\delta }_2+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}})A,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }T\hfill & =\hfill & {\epsilon }_{1}A-({\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+\omega {\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}+{\beta }_h)T,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }{I}_e\hfill & =\hfill & (1-{\lambda }_2){\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I+{\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}E-({\gamma }_2+{\beta }_h+{\varphi }_2+{\delta }_1)I_e,\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }{I}_a\hfill & =\hfill & {\lambda }_2{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I+{\gamma }_2{I}_e+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}A-({\epsilon }_2+{\beta }_h+{\varphi }_3+{\delta }_3)I_a.\hfill \end{array}$$ (1)

Table 1.

Model parameters and interpretation.

Parameter	Interpretation	Value	References
${\Delta }_h$	Recruitment rate	$\frac{30,291,000}{(70\times 365)}$	[36]
${\beta }_h$	Natural mortality rate	$\frac{1}{(70\times 365)}$	[36]
${\varphi }_2,{\varphi }_3$	COVID-19 recovery rates for individuals in the $I_e,I_a$ classes, respectively	0.02095	Assumed
$\kappa$	Modification parameter for the infectiousness of co-infected individuals	1.0	Assumed
${\delta }_1$	COVID-19 related death rate	0.015	[20]
${\delta }_2$	TB related death rate	$\frac{32}{100,000}$	[38]
${\epsilon }_1$	TB recovery rate for singly infected	2.0	[40]
$\omega$	Rate of reinfection with TB	0.2	[40]
${\lambda }_1,{\lambda }_2$	Proportion of newly infected individuals with active TB	$\frac{1}{5}$	Assumed
${\gamma }_1,{\gamma }_2$	Rate of progression to active TB	0.15	Assumed
${\alpha }_1$	COVID-19 transmission rate	2.0122	Fitted
${\alpha }_2$	TB transmission rate	2.9598	Fitted
${\Lambda }_1$	modification parameter for increased susceptibility to COVID-19 by Latent TB infected individuals	1.0147	Fitted
${\Lambda }_2$	modification parameter for increased susceptibility to COVID-19 by Latent TB infected individuals	1.0626	Fitted
${\delta }_3$	COVID-19-TB related death rate	0.5140	Fitted
$\psi$	COVID-19 vaccination rate	0.5482	Fitted
${\epsilon }_1$	TB recovery rate for co-infected	2.2186	Fitted
${\varphi }_1$	COVID-19 recovery rate for singly-infected	4.3592	Fitted

3.1. Fundamentals of the model

The boundedness and positivity of the solutions which shows that system (1) is both mathematically and biologically well-posed is presented.

3.1.1. Non-negativity of the solution

Theorem 3.1

The closed set

$$\Psi =\{(S,V,I,R,E,A,T,I_e,I_a)\in {\mathbb{R}}_{+}^9:0\le S+V+I+R+E+A+T+I_e+I_a\le \frac{{\Delta }_h}{{\beta }_h}\},$$

is positively invariant with respect to the model (1) .

Proof

Adding all the equations of the model (1), gives

$${}_0^{abc}{\mathfrak{D}}_t^{\varpi }N={\Delta }_h-{\beta }_h(S+V+I+R+E+A+T+I_e+I_a)-({\delta }_{1}I+{\delta }_{2}A+{\delta }_1{I}_e+({\delta }_1+{\delta }_2)I_a),$$

which can be re-written as:

$${}_0^{abc}{\mathfrak{D}}_t^{\varpi }N\le {\Delta }_h-{\beta }_{h}N.$$ (2)

Applying the Laplace transform on the above inequality we have

$$\begin{array}{ccc}\hfill N\left(t\right)& \le & (\frac{\mathcal{B}\left(\varpi \right)}{\mathcal{B}\left(\varpi \right)+(1-\varpi ){\beta }_h}N\left(0\right)+\frac{(1-\varpi ){\Delta }_h}{\mathcal{B}\left(\varpi \right)+(1-\varpi ){\beta }_h})E_{\varpi ,1}(-\frac{\varpi {\beta }_h}{\mathcal{B}\left(\varpi \right)+(1-\varpi ){\beta }_h}{t}^{\varpi })\hfill \\ & & +\frac{\varpi {\Delta }_h}{\mathcal{B}\left(\varpi \right)+(1-\varpi ){\beta }_h}{E}_{\varpi ,\varpi +1}(-\frac{\varpi {\beta }_h}{\mathcal{B}\left(\varpi \right)+(1-\varpi ){\beta }_h}{t}^{\varpi }).\hfill \end{array}$$

 □

The Mittag–Leffler function $E_{m,n}$ is asymptotic in nature [23]. Thus, we have that $N\left(t\right)\le \frac{{\Delta }_h}{{\beta }_h}$ as $t\to \infty$. As a result, the system (1) has the solution in $\Psi$. Thus, the given system is positively invariant.

3.2. Basic reproduction number of the co-infection model (1)

The model (1) has a DFE given by

$$\begin{array}{ccc}{\mathcal{J}}_0\hfill & =\hfill & (S^{*},V^{*},I^{*},R^{*},E^{*},A^{*},T^{*},I_e^{*},I_a^{*})\hfill \\ \hfill & =\hfill & (\frac{{\Delta }_h}{{\beta }_h+\psi },\frac{\psi {\Delta }_h}{{\beta }_h({\beta }_h+\psi )},0,0,0,0,0,0,0).\hfill \end{array}$$ (3)

The basic reproduction number of the model (1), using the approach in van den and Watmough [33], is given by ${\mathcal{R}}_0=\text{max}\{{\mathcal{R}}_{0t},{\mathcal{R}}_{0c}\}$ where ${\mathcal{R}}_{0t}$ and ${\mathcal{R}}_{0c}$ are, respectively, given by

$${\mathcal{R}}_{0c}=\frac{{\alpha }_1[{\beta }_h+(1-\vartheta )]}{{\mathcal{L}}_1({\beta }_h+\psi )},$$

and

$${\mathcal{R}}_{0t}=\frac{{\alpha }_2[{\gamma }_1(1-{\lambda }_1)+{\mathcal{L}}_2{\lambda }_1]}{{\mathcal{L}}_2{\mathcal{L}}_3},$$

where,

$${\mathcal{L}}_1={\varphi }_1+{\beta }_h+{\delta }_1,{\mathcal{L}}_2={\gamma }_1+{\beta }_h,{\mathcal{L}}_3={\epsilon }_1+{\beta }_h+{\delta }_2.$$

3.3. Local asymptotic stability of the disease free equilibrium (DFE) of the co-infection model

Theorem 3.2

The DFE, ${\mathcal{J}}_0$ , of the model (1) is locally asymptotically stable (LAS) if ${\mathcal{R}}_0<1$ , and unstable if ${\mathcal{R}}_0>1$ .

Proof

The local stability of the model (1) is analysed by the Jacobian matrix of the system (1) evaluated at the COVID-19-free equilibrium, ${\mathcal{J}}_0$, given by:

$$\left(\begin{array}{ccccccccc}-({\beta }_h+\psi )& 0& {\displaystyle \frac{-{\alpha }_1{S}^{*}}{N^{*}}}& 0& 0& {\displaystyle \frac{-{\alpha }_2{S}^{*}}{N^{*}}}& 0& {\displaystyle \frac{-{\alpha }_{1}kS}{N^{*}}}& {\displaystyle \frac{-({\alpha }_{1}k+{\alpha }_2)S^{*}}{N^{*}}}\\ \psi & -{\beta }_h& {\displaystyle \frac{-{\alpha }_1(1-\vartheta )V^{*}}{N^{*}}}& 0& 0& {\displaystyle \frac{-{\alpha }_2{V}^{*}}{N^{*}}}& 0& {\displaystyle \frac{-{\alpha }_1(1-\vartheta )kV}{N^{*}}}& {\displaystyle \frac{-{\alpha }_1(1-\vartheta )k{V}^{*}-{\alpha }_2{V}^{*}}{N^{*}}}\\ 0& 0& {\displaystyle \frac{{\alpha }_1(S^{*}+(1-\vartheta )V^{*})}{N^{*}}}-{\mathcal{L}}_1& 0& 0& 0& 0& {\displaystyle \frac{{\alpha }_{1}k(S^{*}+(1-\vartheta )V^{*})}{N^{*}}}& {\displaystyle \frac{{\alpha }_1(S^{*}+(1-\vartheta )V^{*})}{N^{*}}}+{\epsilon }_2\\ 0& 0& {\varphi }_1& -{\beta }_h& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -{\mathcal{L}}_2& (1-{\lambda }_1){\alpha }_2& 0& 0& (1-{\lambda }_1){\alpha }_2\\ 0& 0& 0& 0& {\gamma }_1& {\lambda }_1{\alpha }_2-{\mathcal{L}}_3& 0& 0& {\lambda }_1{\alpha }_2\\ 0& 0& 0& 0& 0& {\epsilon }_1& -{\beta }_h& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& -{\mathcal{L}}_4& 0\\ 0& 0& 0& 0& 0& 0& 0& {\gamma }_2& -{\mathcal{L}}_5\end{array}\right),$$

where,

$${\mathcal{L}}_1={\varphi }_1+{\beta }_h+{\delta }_1,{\mathcal{L}}_2={\gamma }_1+{\beta }_h,{\mathcal{L}}_3={\epsilon }_1+{\beta }_h+{\delta }_2,{\mathcal{L}}_4={\gamma }_2+{\beta }_h+{\varphi }_2+{\delta }_1,{\mathcal{L}}_5={\epsilon }_2+{\beta }_h+{\varphi }_3+{\delta }_3.$$

The characteristic equation of the above matrix is given by

$${(\chi +{\beta }_h)}^3(\chi +{\mathcal{L}}_4)(\chi +{\mathcal{L}}_5)(\chi +{\beta }_h+\psi )(\chi +{\mathcal{L}}_1(1-{\mathcal{R}}_{0c}))({\chi }^2+({\mathcal{L}}_2+{\mathcal{L}}_3-{\alpha }_2{\lambda }_1)\chi +{\mathcal{L}}_2{\mathcal{L}}_3(1-{\mathcal{R}}_{0t}))=0.$$ (4)

For the roots of $(\chi +{\mathcal{L}}_4)=0$, $(\chi +{\mathcal{L}}_5)=0$, $(\chi +{\beta }_h+\psi )=0$ and $\chi +{\beta }_h=0$, the arguments are

$$\text{arg}\left({\chi }_k\right)>\frac{\pi }{a}>k\frac{2\pi }{a}>\frac{\pi }{M}>\frac{\pi }{2M},\text{where},k=0,1,2,3,...,a-1$$

Applying the Routh–Hurwitz criterion, the equations

$$(\chi +{\mathcal{L}}_1(1-{\mathcal{R}}_{0c}))=0,$$

and

$$({\chi }^2+({\mathcal{L}}_2+{\mathcal{L}}_3-{\alpha }_2{\lambda }_1)\chi +{\mathcal{L}}_2{\mathcal{L}}_3(1-{\mathcal{R}}_{0t}))=0$$

will have roots with negative real parts if and only if ${\mathcal{R}}_{0c}<1$ and ${\mathcal{R}}_{0t}<1$, respectively. Thus, the DFE, ${\mathcal{J}}_0$ is locally asymptotically stable if ${\mathcal{R}}_0=\text{max}\{{\mathcal{R}}_{0t},{\mathcal{R}}_{0c}\}<1$. □

The epidemiological implication of Theorem 3.2 is that both diseases can be eliminated from the population when ${\mathcal{R}}_0<1$ and if the initial sizes of the population of the model are in the region of attraction of the DFE.

3.4. Global asymptotic stability(GAS) of the disease-free equilibrium(DFE) of the model

The approach in Castillo-Chavez et al. [34] shall be applied to investigate the global asymptotic stability (GAS) of the DFE of the model. We list two conditions that if met, also guarantee the GAS of the DFE. System (1) is re-written as follows:

$$\begin{array}{ccc}{}_0^{abc}{\mathfrak{D}}_t^{\varpi }\mathcal{V}\hfill & =\hfill & P(\mathcal{V},\mathcal{K}),\hfill \\ {}_0^{abc}{\mathfrak{D}}_t^{\varpi }\mathcal{K}\hfill & =\hfill & \Psi (\mathcal{V},\mathcal{K}),\Psi (\mathcal{V},0)=0,\hfill \end{array}$$ (5)

where $\mathcal{V}=(S,V,R,T)\in {\mathbb{R}}^4$ denotes the number of uninfected components and $\mathcal{K}=(I,E,A,I_e,I_a)\in {\mathbb{R}}^5$ denotes the number of infected components. ${\mathcal{U}}_0=({\mathcal{V}}^{*},0)$ denotes the DFE of this system. The conditions below must be met so as to guarantee the local asymptotic stability:

$\left({\Sigma }_1\right)$: For $\frac{d\mathcal{V}}{dt}=P(\mathcal{V},0),{\mathcal{V}}^{*}$is globally asymptotically stable (GAS),

$\left({\Sigma }_2\right)$: $\Psi (\mathcal{V},\mathcal{K})=\mathcal{W}\mathcal{K}-\widehat{\Psi }(\mathcal{V},\mathcal{K})\mathcal{V},Q(\mathcal{V},\mathcal{K})\ge 0$ for $(\mathcal{V},\mathcal{K})\in \Omega ,$ where $\mathcal{W}=D_{\mathcal{K}}\Psi ({\mathcal{V}}^{*},0)$ is an M-matrix and $\Omega$ is the invariant domain. If system (1) fulfills two conditions above, then we have the following result:

Theorem 3.3

The fixed point${\mathcal{U}}_0=({\mathcal{V}}^{*},0)$is a globally asymptotic stable (GAS) equilibrium of(1)provided that${\mathcal{R}}_0<1$(LAS) and that assumptions$\left({\Sigma }_1\right)$and$\left({\Sigma }_2\right)$are met

Proof

$${}_0^{abc}{\mathfrak{D}}_t^{\varpi }\mathcal{V}=\mathcal{P}(\mathcal{V},\mathcal{K})=\left(\begin{array}{c}{\Delta }_h-({\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\beta }_h+\psi +{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})S\\ \psi S-((1-\vartheta ){\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\beta }_h+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})V\\ {\varphi }_{1}I-({\beta }_h+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})R\\ {\epsilon }_{1}A-({\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+\omega {\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}+{\beta }_h)T\end{array}\right),$$ (6)

$$\mathcal{P}(\mathcal{V},0)=\left(\begin{array}{c}{\Delta }_h-{\beta }_{h}S\\ \psi S-{\beta }_{h}V\\ 0\\ 0\end{array}\right),$$ (7)

 □

where $\mathcal{V}$ denotes the number of non-infectious compartments and $\mathcal{K}$ denotes the number of infectious compartments

$$\Psi (\mathcal{V},\mathcal{K})=\left(\begin{array}{c}{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}[S+(1-\vartheta )V+T]+{\epsilon }_2{I}_a-({\varphi }_1+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}+{\beta }_h+{\delta }_1)I\\ (1-{\lambda }_1){\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}(S+V+R+\omega T)-({\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\gamma }_1+{\beta }_h)E\\ {\lambda }_1{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}(S+V+R+\omega T)+{\gamma }_{1}E-({\epsilon }_1+{\beta }_h+{\delta }_2+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}})A\\ (1-{\lambda }_2){\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I+{\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}E-({\gamma }_2+{\beta }_h+{\varphi }_2+{\delta }_1)I_e\\ {\lambda }_2{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I+{\gamma }_2{I}_e+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}A-({\epsilon }_2+{\beta }_h+{\varphi }_3+{\delta }_3)I_a\end{array}\right),$$

$$\mathcal{W}=D_{\mathcal{K}}\Psi ({\mathcal{V}}^{*},0)=\left(\begin{array}{ccccc}{\displaystyle \frac{{\alpha }_1[S^{*}+(1-\vartheta )V^{*}]}{N^{*}}}-{\mathcal{L}}_1& 0& 0& {\displaystyle \frac{{\alpha }_1\kappa [S^{*}+(1-\vartheta )V^{*}]}{N^{*}}}& {\displaystyle \frac{{\alpha }_1\kappa [S^{*}+(1-\vartheta )V^{*}]}{N^{*}}}+{\epsilon }_2\\ 0& -{\mathcal{L}}_2& (1-{\lambda }_1){\alpha }_2& 0& (1-{\lambda }_1){\alpha }_2\\ 0& {\gamma }_1& {\lambda }_1{\alpha }_2-{\mathcal{L}}_3& 0& {\lambda }_1{\alpha }_2\\ 0& 0& 0& -{\mathcal{L}}_4& 0\\ 0& 0& 0& {\gamma }_2& -{\mathcal{L}}_5\end{array}\right),$$

with

$${\mathcal{L}}_1={\varphi }_1+{\beta }_h+{\delta }_{},{\mathcal{L}}_2={\gamma }_1+{\beta }_h{\mathcal{L}}_3={\epsilon }_1+{\beta }_h+{\delta }_2,{\mathcal{L}}_4={\gamma }_2+{\beta }_h+{\varphi }_2+{\delta }_1,{\mathcal{L}}_5={\epsilon }_2+{\beta }_h+{\varphi }_3+{\delta }_3,$$

so that

$$\mathcal{W}\mathcal{K}=\left(\begin{array}{c}{\alpha }_1{\displaystyle \frac{[S^{*}+(1-\vartheta )V^{*}]}{N^{*}}}(I+\kappa (I_a+I_e\left)\right)-{\mathcal{L}}_{1}I+{\epsilon }_2{I}_a\\ -{\mathcal{L}}_{2}E+(1-{\lambda }_1){\alpha }_2(A+I_a)\\ {\gamma }_{1}E-{\mathcal{L}}_{3}A+{\lambda }_1{\alpha }_2(A+I_a)\\ -{\mathcal{L}}_4{I}_e\\ {\gamma }_2{I}_e-{\mathcal{L}}_5{I}_a\end{array}\right),$$

$$\widehat{\Psi }(\mathcal{V},\mathcal{K})=\mathcal{W}\mathcal{K}-\Psi (\mathcal{V},\mathcal{K})=\left(\begin{array}{c}{\alpha }_1[I+\kappa (I_a+I_e)]({\displaystyle \frac{S^{*}+(1-\vartheta )V^{*}}{N^{*}}}-{\displaystyle \frac{S+(1-\vartheta )V+T}{N}})+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I\\ (1-{\lambda }_1){\alpha }_2(A+I_a)(1-{\displaystyle \frac{S+V+R+\omega T}{N}})+{\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}E\\ {\lambda }_1{\alpha }_2(A+I_a)(1-{\displaystyle \frac{S+V+R+\omega T}{N}})+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}A\\ -(1-{\lambda }_2){\alpha }_2(A+I_a)I-{\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}E\\ -{\lambda }_2{\alpha }_2(A+I_a)I-{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}A\end{array}\right).$$

It is clear from the above, that, $\widehat{\Psi }(\mathcal{V},\mathcal{K})\ngeqq 0$. Hence the DFE may not be globally asymptotically stable.

3.5. Existence and uniqueness of the solution

In this section, we shall apply some basic results from fixed point theory to the model (1), in order to establish existence and uniqueness of solution. The model (1) is re-written in the following form:

$$\{\begin{array}{cc}{}_0^{ABC}{D}_t^{\varpi }Q\left(t\right)\hfill & =\mathcal{Q}(t,Q(t\left)\right),\hfill \\ Q\left(0\right)\hfill & =Q_0,\hfill \end{array}$$ (8)

where the vector $Q=(S,V,I,R,E,A,T,I_e,I_a)$ represents the compartments of the model and $\mathcal{Q}$ denotes a continuous vector defined as follows:

$$\mathcal{Q}=\left(\begin{array}{c}{\mathcal{Q}}_1\\ {\mathcal{Q}}_2\\ {\mathcal{Q}}_3\\ {\mathcal{Q}}_4\\ {\mathcal{Q}}_5\\ {\mathcal{Q}}_6\\ {\mathcal{Q}}_7\\ {\mathcal{Q}}_8\\ {\mathcal{Q}}_9\end{array}\right)=\left(\begin{array}{c}{\Delta }_h-({\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\beta }_h+\psi +{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})S\\ \psi S-((1-\vartheta ){\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\beta }_h+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})V\\ {\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}[S+(1-\vartheta )V+T]+{\epsilon }_2{I}_a-({\varphi }_1+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}+{\beta }_h+{\delta }_1)I\\ {\varphi }_{1}I-({\beta }_h+{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}})R\\ (1-{\lambda }_1){\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}(S+V+R+\omega T)-({\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+{\gamma }_1+{\beta }_h)E\\ {\lambda }_1{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}(S+V+R+\omega T)+{\gamma }_{1}E-({\epsilon }_1+{\beta }_h+{\delta }_2+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}})A\\ {\epsilon }_{1}A-({\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}+\omega {\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}+{\beta }_h)T\\ (1-{\lambda }_2){\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I+{\Lambda }_1{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}E-({\gamma }_2+{\beta }_h+{\varphi }_2+{\delta }_1)I_e\\ {\lambda }_2{\displaystyle \frac{{\alpha }_2(A+I_a)}{N}}I+{\gamma }_2{I}_e+{\Lambda }_2{\displaystyle \frac{{\alpha }_1[I+\kappa (I_a+I_e)]}{N}}A-({\epsilon }_2+{\beta }_h+{\varphi }_3+{\delta }_3)I_a\end{array}\right).$$

The initial condition of the variables of the model is denoted by

$Q\left(0\right)=(S\left(0\right),V\left(0\right),I\left(0\right),R\left(0\right),E\left(0\right),A\left(0\right),T\left(0\right),I_e\left(0\right),I_a\left(0\right))$. In addition, $\mathcal{Q}$ is said to satisfy the Lipschitz condition in the second argument, if we have:

$$\parallel \mathcal{Q}(t,Q_1\left(t\right))-\mathcal{Q}(t,Q_2\left(t\right))\parallel \le \mathcal{M}\parallel Q_1\left(t\right)-Q_2\left(t\right)\parallel .$$ (9)

The existence of a unique solution is established in the following theorem:

Theorem 3.4

There exists a unique solution in$C^1([0,\mathcal{T}],D)$to the initial value problem for$t\in [0.\mathcal{T}]$provided that(9)and

$$(\frac{(1-\varpi )\mathcal{M}}{\mathcal{B}\varpi )}+\frac{\varpi \mathcal{M}}{\mathcal{B}\varpi \left)\Gamma \right(\varpi +1)}{\mathcal{T}}_{\max }^{\varpi })<1,$$ (10)

are satisfied.

Proof

Applying the Atangana–Baleanu fractional integral on the both sides of (8), we have

$$Q\left(t\right)=Q_0+\frac{1-\varpi }{\mathcal{B}\varpi )}\mathcal{Q}(t,Q\left(t\right))+\frac{\varpi }{\mathcal{B}\varpi \left)\Gamma \right(\varpi )}{\int }_0^t{(t-\tau )}^{\varpi -1}\mathcal{Q}(\tau ,Q\left(\tau \right))d\tau .$$ (11)

Let $\mathcal{W}=(0,\mathcal{M})$ Now define the operator $\mathcal{G}:C(\mathcal{W},{\mathbb{R}}^9)\to \mathcal{C}(\mathcal{W},{\mathbb{R}}^9)$ by:

$$\mathcal{G}\left[Q\left(t\right)\right]=Q_0+\frac{1-\varpi }{\mathcal{B}\varpi )}\mathcal{Q}(t,Q\left(t\right))+\frac{\varpi }{\mathcal{B}\varpi \left)\Gamma \right(\varpi )}{\int }_0^t{(t-\tau )}^{\varpi -1}\mathcal{Q}(\tau ,Q\left(\tau \right))d\tau ,$$ (12)

Eq. (11) becomes

$$Q\left(t\right)=\mathcal{G}\left[Q\right(t\left)\right].$$ (13)

The supremum norm on ${\mathcal{W},\parallel .\parallel }_{\mathcal{W}}$ is given by:

$${\parallel Q\left(t\right)\parallel }_{\mathcal{W}}{=}_{t\in \mathcal{W}}^{sup}\parallel Q\left(t\right)\parallel ,Q\left(t\right)\in C.$$

Clearly, $C(\mathcal{W},{\mathbb{R}}^9)$ equipped with ${\parallel .\parallel }_{\mathcal{W}}$ is a Banach space. Also, the following inequality holds:

$$\parallel {\int }_0^t\mathcal{A}(t,\tau )Q\left(\tau \right)d\tau \parallel \le {\mathcal{M}\parallel \mathcal{A}(t,\tau )\parallel }_{\mathcal{W}}{\parallel Q\left(t\right)\parallel }_{\mathcal{W}},$$ (14)

 □

with $Q\left(t\right)\in C(\mathcal{W},{\mathbb{R}}^9)$, $\mathcal{A}(t,\tau )\in C({\mathcal{W}}^2,\mathbb{R})$, in such a way that

$${\left|\mathcal{A}\right(t,\tau \left)\right|}_{\mathcal{W}}{=}_{t,\tau \in W}^{sup}\left|\mathcal{A}(t,\tau )\right|.$$

Applying Eq. (3.5), we have that

$$\begin{array}{cc}{\parallel \mathcal{G}\left[Q_1\left(t\right)\right]-\mathcal{G}\left[Q_2\left(t\right)\right]\parallel }_{\mathcal{W}}\hfill & \le \parallel {\displaystyle \frac{1-\varpi }{\mathcal{B}\varpi )}}(\mathcal{Q}(t,Q_1\left(t\right))-\mathcal{Q}(t,Q_2\left(t\right)))\hfill \\ & +{\displaystyle \frac{\varpi }{\mathcal{B}\varpi \left)\Gamma \right(\varpi )}}{\int }_0^t{(t-\tau )}^{\varpi -1}(\mathcal{Q}(\tau ,Q_1\left(\tau \right))-\mathcal{Q}(\tau ,Q_2\left(\tau \right)))d\tau \parallel ,\hfill \\ & \le {\displaystyle \frac{1-\varpi }{\mathcal{B}\varpi )}}\mathcal{M}\parallel Q_1\left(t\right)-Q_2\left(t\right)\parallel \hfill \\ & +{\displaystyle \frac{\varpi \mathcal{M}}{\mathcal{B}\varpi \left)\Gamma \right(\varpi )}}{\int }_0^t{(t-\tau )}^{\varpi -1}\parallel Q_1\left(t\right)-Q_2\left(t\right)\parallel d\tau ,\hfill \\ & \le {\displaystyle \frac{1-\varpi }{\mathcal{B}\varpi )}}\mathcal{M}\underset{Q\in [0,\mathcal{T}]}{\sup }\parallel Q_1\left(t\right)-Q_2\left(t\right)\parallel \hfill \\ & +{\displaystyle \frac{\varpi \mathcal{M}}{\mathcal{B}\varpi \left)\Gamma \right(\varpi )}}\left({\int }_0^t{(t-\tau )}^{\varpi -1}\right)d\tau \underset{Q\in [0,\mathcal{T}]}{\sup }\parallel Q_1\left(\tau \right)-Q_2\left(\tau \right)\parallel ,\hfill \\ & \le ({\displaystyle \frac{(1-\varpi )\mathcal{M}}{\mathcal{B}\varpi )}}+{\displaystyle \frac{\varpi \mathcal{M}{\mathcal{T}}_{\max }^{\varpi }}{\mathcal{B}\varpi \left)\Gamma \right(\varpi +1)}})\parallel Q_1\left(t\right)-Q_2\left(t\right)\parallel .\hfill \end{array}$$ (15)

Thus if the condition (10) holds then $\parallel \mathcal{G}\left[Q_1\left(t\right)\right]-\mathcal{G}\left[Q_2\left(t\right)\right]\parallel <\parallel Q_1\left(t\right)-Q_2\left(t\right)\parallel$ and the operator $\mathcal{G}$ becomes a contraction. Therefore $\mathcal{G}$ has a unique fixed point which is a solution to the initial value problem (8) and hence a solution to the system (1).

4. Numerical scheme for the solution of the model

In this section, we derive the numerical method for the stated fractional differential system (1). We shall adopt the scheme given in Toufik and Atangana [35] in order to approximate the Atangana–Baleanu fractional Integral. The numerical method used combines the two-step Lagrange polynomial and the fundamental theorem of fractional calculus. This method has been shown to be highly accurate and efficient, user-friendly and converges quickly to the exact solution even with a large step of discretization [35].

Applying the fundamental theorem of fractional calculus on (8), we have

$$Q\left(t\right)=Q_0+{\displaystyle \frac{1-\varpi }{\mathcal{B}\left(\varpi \right)}}\mathcal{Q}(t,Q\left(t\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)\Gamma \left(\varpi \right)}}{\int }_0^t{(t-\tau )}^{\varpi -1}\mathcal{Q}(\tau ,Q\left(\tau \right))d\tau .$$

At $t=t_{p+1}=(p+1)h$, where $h=t_{p+1}-t_p$ is the time space, the above equation discretizes to

$$Q\left(t_{p+1}\right)=Q\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}\mathcal{Q}(t_p,Q)+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)\Gamma \left(\varpi \right)}}\sum _{r=0}^p{\int }_{t_r}^{t_{r+1}}{(t_{r+1}-\tau )}^{\varpi -1}\mathcal{Q}(\tau ,Q)d\tau .$$ (16)

Applying Lagrange two-points interpolation polynomial into (16), the numerical scheme for the general fractional system reduces to

$$\begin{array}{ccc}\hfill Q\left(t_{p+1}\right)& =& Q\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}\mathcal{Q}(t_p,Q)+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }\mathcal{Q}(t_r,Q)}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }F(t_{r-1},Q)[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$ (17)

Adopting the numerical scheme (17) into the fractional system (1) yields the following numerical solution;

$$\begin{array}{ccc}\hfill S\left(t_{p+1}\right)& =& S\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_1(t_p,S\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_1(t_r,S\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_1(t_{r-1},S\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill V\left(t_{p+1}\right)& =& V\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_2(t_p,V\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_2(t_r,V\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_2(t_{r-1},V\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I\left(t_{p+1}\right)& =& I\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_3(t_p,I\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_3(t_r,I\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_3(t_{r-1},I\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill R\left(t_{p+1}\right)& =& R\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_4(t_p,R\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_4(t_r,R\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_4(t_{r-1},R\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill E\left(t_{p+1}\right)& =& E\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_5(t_p,E\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_5(t_r,E\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_5(t_{r-1},E\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$ (18)

$$\begin{array}{ccc}\hfill A\left(t_{p+1}\right)& =& A\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_6(t_p,A\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_6(t_r,A\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_6(t_{r-1},A\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill T\left(t_{p+1}\right)& =& T\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_7(t_p,T\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_7(t_r,T\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_7(t_{r-1},T\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_e\left(t_{p+1}\right)& =& I_e\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_8(t_p,I_e\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_8(t_r,I_e\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_8(t_{r-1},I_e\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_a\left(t_{p+1}\right)& =& I_a\left(t_0\right)+{\displaystyle \frac{(1-\varpi )}{\mathcal{B}\left(\varpi \right)}}{\mathcal{Q}}_8(t_p,I_a\left(t_p\right))+{\displaystyle \frac{\varpi }{\mathcal{B}\left(\varpi \right)}}\sum _{r=0}^p\{{\displaystyle \frac{h^{\varpi }{\mathcal{Q}}_8(t_r,I_a\left(t_r\right))}{\Gamma (\varpi +2)}}[{(p-r+1)}^{\varpi }(p-r+\varpi +2)\hfill \\ \hfill & & -{(p-r)}^{\varpi }(p-r+2\varpi +2)]-h^{\varpi }{\mathcal{Q}}_8(t_{r-1},I_a\left(t_{r-1}\right))[{(p-r+1)}^{\varpi +1}-{(p-r)}^{\varpi }(p-r+\varpi +1)]\}.\hfill \end{array}$$

5. Numerical simulations

5.1. Baseline values and initial conditions

The relative infectiousness of co-infected individuals when compared to singly infected individuals with COVID-19 is still unknown; we have therefore assumed $\kappa =1$. COVID-19 induced death rate was set to be 0.015, following the work in Omame and Okuonghae [20], so that ${\delta }_1=0.015$. The total population of New Delhi is estimated to be 30,291,000 [36]. Hence, we set $S\left(0\right)=28,500,000$. The total number vaccinated with the Covishield and Covaxine vaccines as at 1st March, 2021 in New Delhi is 367,699 [37]. Therefore, we set $V\left(0\right)=367,699$. The total number of recovered individuals as at that day is 627,044 [37]. Hence, $R\left(0\right)=627,044$. Total number of confirmed COVID-19 cases in New Delhi on 1s March, 2021 is 639,289. Hence the total number of active cases as at that same day (which is the difference between the confirmed and recovered cases as well as deaths) was set at $I\left(0\right)=12,245$. Based on the available records for TB prevalence in India and New Delhi [38], the other initial conditions are set as follows: $E\left(0\right)=500,000$, $A\left(0\right)=5000$, $T\left(0\right)=0$, $I_e\left(0\right)=1000$, $I_a\left(0\right)=50$.

5.2. Model fitting

For the model fitting, we used the genetic algorithm (GA) [39], which provides the starting values (for the parameters being estimated) for use in the fmincon function in the Optimization Toolbox of MATLAB. Using the data available for the daily cumulative number of confirmed cases in New Delhi, India [37], we estimated the COVID-19 as well as TB contact rates ${\alpha }_1=$ and ${\alpha }_2$, respectively. Other parameters estimated from the fitting are given in Table 1. The model (1) is fitted to the cumulative confirmed daily COVID-19 cases for New Delhi, India, from March 1, 2021 to June 26, 2021. This is shown in Fig. 1 . It is observed from the figure that the model fits well to the data.

Fig. 1.

Fitting the model to cumulative number of confirmed COVID-19 cases in New Delhi, India.

5.3. Impact of modification parameter accounting for susceptibility of TB infected individuals to COVID-19 infection

Simulation of the fractional order model (1) with $\varpi =0.8$ for total number of individuals infected with COVID-19 ($I$), at different values of the modification paramter accounting for susceptibility of latent TB-infected individuals to COVID-19, ${\Lambda }_1$, is depicted in Fig. 2 . It is observed from this figure, that reducing the risk of COVID-19 infection by latent TB-infected individuals could greatly reduce the COVID-19 cases in the population. The simulation of the total number of individuals co-infected with COVID-19 and latent TB for $\varpi =0.8$, depicted in Fig. 3 shows that averting the risk of COVID-19 infection by individuals infected with latent TB could also bring down the burden of COVID-19 and latent TB co-infections. Also, simular trend is observed in Fig. 4 , revealing that reducing the rate of COVID-19 acquisition by latent TB-infected individuals could also help curb the co-infections of COVID-19 and active TB.

Fig. 2.

Simulations of the total number of individuals infected with COVID-19, at different values of ${\Lambda }_1$. Here, $\varpi =0.8$. All other parameters are as in Table 1.

Fig. 3.

Simulations of the total number of individuals co-infected with COVID-19 and latent TB, at different values of ${\Lambda }_1$. Here, $\varpi =0.8$. All other parameters are as in Table 1.

Fig. 4.

Simulations of the total number of individuals co-infected with COVID-19 and active TB, at different values of ${\Lambda }_1$. Here, $\varpi =0.8$. All other parameters are as in Table 1.

5.4. Impact of TB re-infection on COVID-19 dynamics

Simulation of the fractional order model (1) with $\varpi =0.8$ for total number of individuals infected with COVID-19 ($I$), at different values of TB re-infection rate, $\omega$, is depicted in Fig. 5 . It is observed from this figure, that reducing the risk of TB re-infection could greatly reduce the COVID-19 cases in the population. The simulation of the total number of individuals co-infected with COVID-19 and active TB for $\varpi =0.8$, depicted in Fig. 6 shows that averting the risk of TB re-infection by individuals who have recovered from a previous TB infection could curb the burden of COVID-19 and active TB co-infections.

Fig. 5.

Simulations of the total number of individuals infected with COVID-19, at different values of $\omega$. Here, $\varpi =0.8$. All other parameters are as in Table 1.

Fig. 6.

Simulations of the total number of individuals co-infected with COVID-19 and active TB, at different values of $\omega$. Here, $\varpi =0.8$. All other parameters are as in Table 1.

5.5. Conditions for co-existence or elimination of both diseases

Simulations of the populations of infected individuals at different initial conditions when ${\mathcal{R}}_{0c}=1.4914\approxeq {\mathcal{R}}_{0t}=1.4794>1$, and at ${\varphi }_1=1.2592$ and $\varpi =0.7$ are depicted in Fig. 7 . This Figure reveals that, for lower COVID-19 treatment rate (1.2592 per day) and high contact rates for COVID-19 (${\alpha }_1=2.0122$) and TB (${\alpha }_2=2.9598$), both diseases will persist in the population, with COVID-19 dominating TB. Simulations of the populations of infected individuals at different initial conditions when ${\mathcal{R}}_{0t}=2.9789>{\mathcal{R}}_{0c}=0.0027$, and at ${\alpha }_1=0.0122,{\alpha }_2=5.9598$ and ${\varphi }_1=4.2592$ and $\varpi =0.7$ is depicted in Fig. 8 . It is revealed here, that for higher COVID-19 treatment rate (${\varphi }_1=4.2592$ per day), lower COVID-19 contact rate (${\alpha }_1=0.0122$) and higher TB contact rate (${\alpha }_2=5.9598$), TB will persist in the population dominating and driving COVID-19 to extinction over time. Simulations of the populations of infected individuals at different initial conditions when ${\mathcal{R}}_{0c}=0.0027<1$ and${\mathcal{R}}_{0t}=0.0530<1$, and at ${\varphi }_1=4.2592$ and $\varpi =0.7$ are depicted in Fig. 9 . This Figure reveals that for low COVID-19 and TB contact rates as well as for high COVID-19 treatment rate, both diseases will die out of the population over time.

Fig. 7.

Simulations of the fractional order model (1) showing the number of infected individuals at various initial conditions. Here, ${\alpha }_1=2.0122$, ${\alpha }_2=2.9598,{\varphi }_1=4.2592,\varpi =0.7,(\text{so}\text{that}{\mathcal{R}}_{0c}=1.4914\approxeq {\mathcal{R}}_{0t}=1.4794)$. All other parameters as in Table 1

Fig. 8.

Simulations of the fractional order model (1) showing the number of infected individuals at various initial conditions. Here, ${\alpha }_1=0.0122$, ${\alpha }_2=5.9598,{\varphi }_1=4.2592,\varpi =0.7,(\text{so}\text{that}{\mathcal{R}}_{0t}=2.9789>{\mathcal{R}}_{0c}=0.0027)$. All other parameters as in Table 1

Fig. 9.

Simulations of the fractional order model (1) showing the number of infected individuals at various initial conditions. Here, ${\alpha }_1=0.0122$, ${\alpha }_2=0.10598,{\varphi }_1=4.2592,\varpi =0.7,(\text{so}\text{that}{\mathcal{R}}_{0c}=0.0027<1,{\mathcal{R}}_{0t}=0.0530<1)$. All other parameters as in Table 1

6. Conclusion

In this work, we have analyzed a fractional order model for COVID-19 and tuberculosis co-infection, using the Atangana–Baleanu derivative. The existence and uniqueness of the model solutions were established applying the fixed point theorem. The model was shown to be locally asymptotically stable when the reproduction number is less than unity. The global stability analysis at the disease free equilibrium point was also carried out. The model was simulated using data relevant to both diseases in New Delhi, India. Fitting the model to the cumulative confirmed COVID-19 cases for New Delhi from March 1, 2021 to June 26, 2021, we estimated the COVID-19 and TB contact rates, respectively, as well as other important parameters of the model. The numerical method used for the approximate solution of the model combines the two-step Lagrange polynomial and the fundamental theorem of fractional calculus. This method has been shown to be highly accurate and efficient, user-friendly and converges quickly to the exact solution even with a large step of discretization [35]. In addition, simulations of the Fractional order model reveal that reducing the risk of COVID-19 infection by latently-infected TB individuals will not only bring down the burden of COVID-19, but will also reduce the co-infection of both diseases in the population. Also, we establish conditions for the co-existence or elimination of both diseases from the population. In particular,

• i.Simulations of the populations of infected individuals at different initial conditions when ${\mathcal{R}}_{0c}=1.4914\approxeq {\mathcal{R}}_{0t}=1.4794>1$, and at ${\varphi }_1=1.2592$ and $\varpi =0.7$, depicted in Fig. 7 reveals that, for lower COVID-19 treatment rate (1.2592 per day) and high contact rates for COVID-19 (${\alpha }_1=2.0122$) and TB (${\alpha }_2=2.9598$), both diseases will persist in the population, with COVID-19 dominating TB.
• ii.Simulations of the populations of infected individuals at different initial conditions when ${\mathcal{R}}_{0t}=2.9789>{\mathcal{R}}_{0c}=0.0027$, and at ${\alpha }_1=0.0122,{\alpha }_2=5.9598$ and ${\varphi }_1=4.2592$ and $\varpi =0.7$, depicted in Fig. 8 showed that for higher COVID-19 treatment rate (${\varphi }_1=4.2592$ per day), lower COVID-19 contact rate (${\alpha }_1=0.0122$) and higher TB contact rate (${\alpha }_2=5.9598$), TB will persist in the population dominating and driving COVID-19 to extinction over time.

CRediT authorship contribution statement

A. Omame: Conceptualization, Data curation, Writing – review & editing, Writing – original draft. M. Abbas: Supervision, Writing – review & editing, Writing – original draft. C.P. Onyenegecha: Conceptualization, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.