Co-Dynamics of COVID-19 and Viral Hepatitis B Using a Mathematical Model of Non-Integer Order: Impact of Vaccination

Abstract

The modeling of biological processes has increasingly been based on fractional calculus. In this paper, a novel fractional-order model is used to investigate the epidemiological impact of vaccination measures on the co-dynamics of viral hepatitis B and COVID-19. To investigate the existence and stability of the new model, we use some fixed point theory results. The COVID-19 and viral hepatitis B thresholds are estimated using the model fitting. The vaccine parameters are plotted against transmission coefficients. The effect of non-integer derivatives on the solution paths for each epidemiological state and the trajectory diagram for infected classes are also examined numerically. An infection-free steady state and an infection-present equilibrium are achieved when ${\mathcal{R}}_0<1$ and ${\mathcal{R}}_0>1$, respectively. Similarly, phase portraits confirm the behaviour of the infected components, showing that, regardless of the order of the fractional derivative, the trajectories of the disease classes always converge toward infection-free steady states over time, no matter what initial conditions are assumed for the diseases. The model has been verified using real observations.

1. Introduction

The “Coronavirus Disease 2019” (COVID-19), induced by the “Severe Acute Respiratory Syndrome Coronavirus-2” (SARS-CoV-2), has infected over 674,003,861 people and led to over 6,862,846 deaths since it was first discovered in late 2019 [1]. Globally, more than 13,307,915,519 doses of the COVID-19 vaccine have been administered [1]. The hepatitis B virus (HBV) also represents a serious threat to public health worldwide, according to Wang et al. [2]. Over 290 million people worldwide have HBV infections, resulting in more than 0.9 million deaths, according to the Centers for Disease Control and Prevention (CDC) [3]. HBV is associated with chronic liver infections such as cirrhosis, liver fibrosis, and hepatocellular carcinoma. HBV causes more than 40% of cases of hepatocellular carcinoma and more than 25% of cases of liver cirrhosis [4]. The Global Burden of Disease study estimates that HBV infection accounts for more than 750,000 deaths worldwide each year [5,6]. Mokhtari et al. [7] reports that the majority of HBV cases are reported in Middle Eastern countries, Asia, and Africa. The majority of documented cases are spread from mother to child in several of these nations [7]. A fifth of Pakistan’s population has viral hepatitis B, according to Yousafzai and coauthors [8]. According to recent studies, between 2 and 11% of COVID-19 patients had a history of liver infection; about 25% experienced COVID-19-related liver issues. In undiagnosed people with viral hepatitis B, SARS-CoV-2 infection may be a risk factor for severe illness, according to Zhang et al. [9].

According to the “McGill COVID-19 vaccine tracker”, there are 242 vaccines currently being developed, of which 50 have been approved and are available in more than 200 countries [10]. Some of the vaccines authorized against COVID-19 include “BNT162b2 (Pfizer/BioNTech), AZD-1222/ChAdOx1-nCoV (Oxford/AstraZeneca), NVX-CoV2373 (Novavax), CoronaVac (Sinovac), mRNA-1273 (Moderna), rAd26-S+rAd5-S (Sputnik V), Ad26.COV2.S (Janssen)”, and so on [11]. Several vaccines that are recommended range in effectiveness from 60% to 90% [11]. Although antiviral medications have been used to treat HBV, they are insufficient to eradicate the virus completely. Vaccination is therefore essential for eradicating HBV infections completely [12]. Several countries, including China and the USA [12], have made vaccination programs for newborns and adults a top priority. A significant reduction in HBV prevalence has been reported by Zheng et al. [13]. It is still far from possible to vaccinate all adults in developing countries in Asia and sub-Saharan Africa.

In recent years, fractional derivatives have been heavily used to model real-world scenarios [14,15,16,17]. Fractional differential operators with power function kernels were initially defined in [18]. Singular kernels, however, cannot be used to simulate biological or other physical phenomena using these formulations. In order to circumvent these limitations, the authors [19,20] updated their definitions based on exponential and generalized Mittag–Leffler functions. Caputo (or “power-kernel-based”) and Caputo–Fabrizio (exponential kernel-based) derivatives have been sufficiently adopted in investigating a myriad of mathematical models. As an example, a fractional model of COVID-19 incorporating isolation and quarantine effects was considered by the authors in [21]. In comparison to the classical integer model, a version of the non-integer model behaves better with real data. Additionally, Baleanu et al. [22] investigated a tumor dynamical model using fractional-based derivatives. The model was numerically investigated using a modified predictor-corrector approach. Results showed that the non-integer model’s generalized kernel captured the real-world cases more accurately than integer-order models. According to Chu et al. [23], a fractional-order model of SARS-CoV-2 variants was investigated. In [24], the authors also investigated a hepatitis B virus model using fractional derivatives. According to Shen et al. [25], fractional calculus was applied to the analysis of a co-dynamical model for HBV and HCV.

Although a few recent research projects have examined the dynamics of COVID-19 and viral hepatitis B co-infection [26,27,28], none has taken into account the critical role that vaccination could play in controlling the co-propagation of both diseases via fractional calculus. Therefore, the goal of this research work is to design and analyze a thorough fractional-order mathematical model for the co-dynamics of the two viral illnesses with the incorporation of double dose vaccination. To the best of the authors’ knowledge, this research is original and will close any gaps left by other studies. With the help of fractional calculus tools and certain findings from fixed point theory, the constructed model will be qualitatively analyzed in order to evaluate intervention techniques for the two ailments. The major contributions of this paper are: (i) developing a fractional co-dynamical model to assess the epidemiological effect of intervention measures on viral hepatitis B and COVID-19 dynamics (see Section 2). Additionally, the existence/uniqueness results for fractional systems are established (see Section 3); (ii) Ulam–Hyers stability results are also established (see Section 4); (iii) the numerical analysis also highlights the impact of fractional derivatives and different intervention measures on the dynamics of both diseases (see Section 5).

Preliminaries

Some definitions and results needed in the paper are briefly discussed in this subsection.

Definition 1. 

Let $f\in H^1(a,b)$ with $b>a,$ and $\Im \in [0,1)$. The Atangana–Baleanu fractional derivative of f of order ℑ in Caputo sense is defined by

$${}_a^{ABC}{D}_t^{\Im }f\left(t\right)=\frac{\mathcal{F}(\Im )}{(1-\Im )}{\int }_a^t\frac{df\left(\omega \right)}{d\omega }{E}_{\Im }[-\Im \frac{{(t-\omega )}^{\Im }}{1-\Im }]d\omega ,$$

where $\mathcal{F}(\Im )=(1-\Im )+\frac{\Im }{\Gamma (\Im )}$ is a normalization function satisfying $\mathcal{F}\left(0\right)=\mathcal{F}\left(1\right)=1$ and $E_{\Im }$ (.) is the Mittag–Leffler function given by

$$E_{\Im }\left(t\right)=\sum _{k=0}^{\infty }\frac{t^k}{\Gamma (\Im k+1)},\Im >0.$$

Definition 2

([20]). The Atangana–Baleanu integral of f (in Caputo sense) of order ℑ is defined by

$${}_{a_1}^{ABC}{I}_t^{\Im }f\left(t\right)=\frac{1-\Im }{\mathcal{F}(\Im )}f\left(t\right)+\frac{\Im }{\mathcal{F}(\Im )\Gamma (\Im )}{\int }_a^{t}f\left(\omega \right){(t-\omega )}^{\Im -1}d\omega .$$

Definition 3

([20]). The Laplace transform of the Atangana–Baleanu fractional derivative of $f$ of order ℑ in Caputo sense is given by

$$\mathcal{L}\left\{{}_a^{ABC}{D}_t^{\Im }f\left(t\right)\right\}=\mathcal{F}(\Im )\frac{s^{\Im }\mathcal{L}\left\{f\left(t\right)\right\}-s^{\Im -1}f\left(a\right)}{s^{\Im }(1-\Im )+\Im },$$

where $\mathcal{L}$ is the Laplace transform operator.

Definition 4

([29]). The Laplace transform of the Mittag–Leffler function $E_{\Im }$ (.) is given by

$$\mathcal{L}\left\{E_{\Im }(-\theta t^{\Im })\right\}=\frac{s^{\Im -1}}{s^{\Im }+\theta },where\theta \in \mathbb{R}.$$

2. Model Formulation

The compartments for the formulation of the proposed model are now defined: $S\left(t\right)$: susceptibles, $V_f\left(t\right)$: those vaccinated for COVID-19, $V_g\left(t\right)$: those vaccinated for viral hepatitis B, $I_f\left(t\right)$: infected with COVID-19, $I_g\left(t\right)$: infected with viral hepatitis B, $I_{cb}\left(t\right)$: infected with the dual infections, $R_f\left(t\right)$: those who have recovered from COVID-19, $R_g\left(t\right)$: those who have recovered from viral hepatitis B, $R_{fg}\left(t\right)$: those who have recovered from the dual infections, so that at any time t, the total population is given by $N\left(t\right)=S\left(t\right)+V_f\left(t\right)+V_g\left(t\right)+I_c\left(t\right)+I_f\left(t\right)+I_{fg}\left(t\right)+R_f\left(t\right)+R_g\left(t\right)+R_{fg}\left(t\right)$. A constant recruitment rate is assumed at the rate $\Lambda$. The COVID-19 and viral hepatitis B transmission rates are defined as ${\beta }_f$ and ${\beta }_g$, respectively. The COVID-19 and viral hepatitis B primary vaccination rates are defined by $\psi$ and $\rho$. The parameters ${\theta }_c$ and ${\theta }_h$ denote the COVID-19 and viral hepatitis B booster dose vaccination rates. Immunities due to the COVID-19 and viral hepatitis B vaccines are not lifelong, and wane at the rates ${\delta }_f$ and ${\delta }_g$, respectively. Individuals who have recovered from COVID-19 are assumed to be immune to re-infection. A similar assumption is made for individuals who have recovered from viral hepatitis B. However, infection with the other disease is possible. In addition, due to the imperfect nature of the COVID-19 and viral hepatitis B vaccines, vulnerable or unvaccinated individuals have reduced rates of getting infected with either of the viral diseases, with $\sigma (0<\sigma <1)$ and $\gamma (0<\gamma <1)$ denoting the COVID-19 and viral hepatitis B vaccine inefficacies. A description of the model parameters is given in

Table 1. Model parameters and variables.

Parameter	Description	Value	Reference
${\beta }_f$	COVID-19 transmission rate	0.0237	Fitted
${\beta }_g$	Viral hepatitis B transmission rate	0.7571	Fitted
$\Lambda$	Recruitment rate into the population	$\frac{238,181,034}{69.37\times 365}$ day ${}^{-1}$	[30]
$\mu$	Natural death rate	$\frac{1}{69.37\times 365}$ day ${}^{-1}$	[30]
${\xi }_f$	COVID-19 recovery rate	$[\frac{1}{30},\frac{1}{3}]$ day ${}^{-1}$	[26]
${\xi }_g$	Viral hepatitis B recovery rate	$\frac{1}{21}$ day ${}^{-1}$	[31]
${\xi }_{fg}$	Recovery rate for co-infected persons	$\frac{1}{21}$	[26]
${\eta }_f$	COVID-19-induced death rate	0.0020	Fitted
${\eta }_g$	Viral hepatitis B induced death rate	0.05	[31]
${\eta }_{fg}$	Co-infection death rate	0.05	[26]
$\psi$	COVID-19 primary vaccination rate	0.0015	Assumed
$\rho$	Viral hepatitis B primary vaccination rate	0.0100	Assumed
${\theta }_f$	COVID-19 booster vaccination rate	0.9998	Assumed
${\theta }_g$	Viral hepatitis B booster vaccination rate	0.0010	Assumed
${\delta }_f$	Waning immunity due to COVID-19 vaccination	0.010–0.015	[32]
${\delta }_g$	Waning immunity due to viral hepatitis B vaccination	0.010–0.015	Assumed
$\sigma$	COVID-19 vaccine inefficacy rate	(1–0.85)	[33]
$\gamma$	Viral hepatitis B vaccine inefficacy rate	(1–0.85)	[31]
${\phi }_1,{\phi }_2$	Modification parameter for vulnerability to a second infection	0.15	Assumed

. The proposed fractional model takes the form.

$$\begin{array}{cc}\hfill {}_0^{ABC}{D}_t^{\Im }S\left(t\right)& =\Lambda -\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}S\left(t\right)-\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}S\left(t\right)+{\delta }_f{V}_f\left(t\right)+{\delta }_g{V}_g\left(t\right)-(\mu +\psi +\rho )S\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{V}_f\left(t\right)& =\psi S\left(t\right)-\frac{\sigma {\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{V}_f\left(t\right)-\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{V}_f\left(t\right)-({\delta }_f+{\theta }_f+\mu )V_f\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{V}_g\left(t\right)& =\rho S\left(t\right)-\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{V}_g\left(t\right)-\frac{\gamma {\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{V}_g\left(t\right)-({\delta }_g+{\theta }_g+\mu )V_g\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{I}_f\left(t\right)& =\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}(S\left(t\right)+R_g\left(t\right)+\sigma V_f\left(t\right)+V_g\left(t\right))-({\xi }_f+{\eta }_f+\mu )I_f-{\phi }_1\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{I}_f\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{I}_g\left(t\right)& =\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}(S\left(t\right)+R_f\left(t\right)+V_f\left(t\right)+\gamma V_g\left(t\right))-({\xi }_g+{\eta }_g+\mu )I_g\left(t\right)-{\phi }_2\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{I}_g\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{I}_{fg}\left(t\right)& ={\phi }_1\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{I}_f\left(t\right)+{\phi }_2\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{I}_g\left(t\right)-({\xi }_{fg}+{\eta }_{fg}+\mu )I_{fg}\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{R}_f\left(t\right)& ={\theta }_f{V}_f\left(t\right)+{\xi }_f{I}_f\left(t\right)-\mu R_f\left(t\right)-\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{R}_f\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{R}_g\left(t\right)& ={\theta }_g{V}_g\left(t\right)+{\xi }_g{I}_g\left(t\right)-\mu R_g\left(t\right)-\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{R}_g\left(t\right),\hfill \\ \hfill {}_0^{ABC}{D}_t^{\Im }{R}_{fg}\left(t\right)& ={\xi }_{fg}{I}_{fg}\left(t\right)-\mu R_{fg}\left(t\right).\hfill \end{array}$$

(1)

2.1. Basic Reproduction Number

The disease free equilibrium (DFE) for the deterministic system is:

$$Q_0=\left(S^{\ast },V_f^{\ast },V_g^{\ast },I_f^{\ast },I_g^{\ast },I_{fg}^{\ast },R_f^{\ast },R_g^{\ast },R_{fg}^{\ast }\right),$$

with

$$\begin{array}{cc}\hfill S^{\ast }& =\frac{\Lambda ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )}{\rho ({\theta }_g+\mu )({\delta }_f+{\theta }_f+\mu )+\psi ({\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )},\hfill \\ \hfill V_f^{\ast }& =\frac{\Lambda \psi ({\delta }_g+{\theta }_g+\mu )}{\rho ({\theta }_g+\mu )({\delta }_f+{\theta }_f+\mu )+\psi ({\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )},\hfill \\ \hfill V_g^{\ast }& =\frac{\Lambda \rho ({\delta }_f+{\theta }_f+\mu )}{\rho ({\theta }_g+\mu )({\delta }_f+{\theta }_f+\mu )+\psi ({\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )},\hfill \\ \hfill R_f^{\ast }& =\frac{{\theta }_f\Lambda \psi ({\delta }_g+{\theta }_g+\mu )}{\mu [\rho ({\theta }_g+\mu )({\delta }_f+{\theta }_f+\mu )+\psi ({\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )]},\hfill \\ \hfill R_g^{\ast }& =\frac{{\theta }_g\Lambda \rho ({\delta }_f+{\theta }_f+\mu )}{\mu [\rho ({\theta }_g+\mu )({\delta }_f+{\theta }_f+\mu )+\psi ({\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )]}.\hfill \end{array}$$

(2)

Associated transfer matrices are:

$$\begin{array}{c}\hfill F=\left(\begin{array}{ccc}\frac{{\beta }_f(S^{\ast }+\sigma V_f^{\ast }+V_g^{\ast }+R_g^{\ast })}{N^{\ast }}& 0& 0\\ 0& \frac{{\beta }_f(S^{\ast }+V_f^{\ast }+\gamma V_g^{\ast }+R_f^{\ast })}{N^{\ast }}& 0\\ 0& 0& 0\end{array}\right),\\ \hfill V=\left(\begin{array}{ccc}{\xi }_f+{\eta }_f+\mu & 0& 0\\ 0& {\xi }_g+{\eta }_g+\mu & 0\\ 0& 0& {\xi }_{fg}+{\eta }_{fg}+\mu \end{array}\right).\end{array}$$

(3)

The basic reproduction, employing the approach in [34], is: ${\mathcal{R}}_0=\rho \left(F{V}^{-1}\right)=$$\max \{{\mathcal{R}}_{0f},{\mathcal{R}}_{0g}\}$, where ${\mathcal{R}}_{0f}$ and ${\mathcal{R}}_{0g}$ denote the deterministic reproduction numbers for viral hepatitis B and COVID-19, and are given by:

$$\begin{array}{cc}\hfill {\mathcal{R}}_{0f}& =\frac{{\beta }_f[\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu \sigma \psi ({\delta }_f+{\theta }_f+\mu )+\rho (\mu +{\theta }_g)({\delta }_f+{\theta }_f+\mu )]S^{\ast }}{\Lambda ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )({\xi }_f+{\eta }_f+\mu )},\hfill \\ \hfill {\mathcal{R}}_{0g}& =\frac{{\beta }_g[\mu ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )+\mu \gamma \rho ({\delta }_g+{\theta }_g+\mu )+\psi (\mu +{\theta }_f)({\delta }_g+{\theta }_g+\mu )]S^{\ast }}{\Lambda ({\delta }_f+{\theta }_f+\mu )({\delta }_g+{\theta }_g+\mu )({\xi }_g+{\eta }_g+\mu )}.\hfill \end{array}$$

2.2. Local Asymptotic Stability of System (1)

Theorem 1. 

System (1)’s DFE, $Q_0$, “is locally asymptotically stable” whenever ${\mathcal{R}}_0<1$, and unstable for ${\mathcal{R}}_0>1$.

Proof. 

Stability within the neighbourhood of the DFE for system (1) is analyzed with the help of its Jacobian matrix evaluated at DFE, and this is given by:

$$J=\left(\begin{array}{ccccccccc}-(\mu +\psi +\rho )& {\delta }_f& {\delta }_g& -\frac{{\beta }_f{S}^{\ast }}{N^{\ast }}& -\frac{{\beta }_g{S}^{\ast }}{N^{\ast }}& -\frac{({\beta }_f+{\beta }_g)S^{\ast }}{N^{\ast }}& 0& 0& -\frac{{\beta }_f{S}^{\ast }}{N^{\ast }}\\ \psi & -({\delta }_f+{\theta }_f+\mu )& 0& -\frac{\sigma {\beta }_f{V}_f^{\ast }}{N^{\ast }}& -\frac{{\beta }_g{V}_f^{\ast }}{N^{\ast }}& -\frac{(\sigma {\beta }_f+{\beta }_g)V_f^{\ast }}{N^{\ast }}& 0& 0& 0\\ \rho & 0& -({\delta }_g+{\theta }_g+\mu )& 0& -\frac{{\beta }_f{V}_g^{\ast }}{N^{\ast }}& -\frac{\gamma {\beta }_g{V}_f^{\ast }}{N^{\ast }}& -\frac{({\beta }_f+\gamma {\beta }_g)V_g^{\ast }}{N^{\ast }}& 0& 0\\ 0& 0& 0& {\beta }_f{A}^{\ast }-K_1& 0& {\beta }_f{A}^{\ast }& 0& 0& 0\\ 0& 0& 0& 0& {\beta }_g{B}^{\ast }-K_2& {\beta }_g{B}^{\ast }& 0& 0& 0\\ 0& 0& 0& 0& 0& -K_3& 0& 0& 0\\ 0& {\theta }_f& 0& {\xi }_f& 0& 0& -\mu & 0& 0\\ 0& 0& {\theta }_g& 0& {\xi }_g& 0& 0& -\mu & 0\\ 0& 0& 0& 0& 0& -{\xi }_{fg}& 0& 0& -\mu \end{array}\right),$$

(4)

where $A^{\ast }=\frac{(S^{\ast }+\sigma V_f^{\ast }+V_g^{\ast }+R_g^{\ast })}{N^{\ast }},B^{\ast }=\frac{(S^{\ast }+V_f^{\ast }+\gamma V_g^{\ast }+R_f^{\ast })}{N^{\ast }},K_1={\xi }_f+{\eta }_f+\mu ,K_2={\xi }_g+{\eta }_g+\mu ,K_3={\xi }_{fg}+{\eta }_{fg}+\mu$.

The first seven eigenvalues are given by: ${\Psi }_1=-\mu \left(\mathrm{with}\mathrm{multiplicity}\mathrm{of}\mathrm{three}\right),{\Psi }_2=-(r-\mu +\psi +\rho ),{\Psi }_3=-({\delta }_f+{\theta }_f+\mu ),{\Psi }_4=-({\delta }_g+{\theta }_g+\mu ),{\Psi }_5=-({\xi }_{fg}+{\eta }_{fg}+\mu )$, and the solutions of these equations:

$$\left(\Psi +({\xi }_f+{\eta }_f+\mu )(1-{\mathcal{R}}_{0f})\right)=0,\left(\Psi +({\xi }_g+{\eta }_g+\mu )(1-{\mathcal{R}}_{0g})\right)=0.$$

(5)

It can be observed that the equations in (5) have roots whose real parts are negative at any time when ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0f},{\mathcal{R}}_{0g}\}<1$. Therefore, the infection-free equilibrium is “locally asymptotically stable”. □

3. Existence and Uniqueness of the Fractional Model’s Solution

Some important results from the theory of fixed point equations shall now be used to prove the existence and uniqueness of a solution to the model (1). To proceed, the model is represented in the form below:

$$\left\{\begin{array}{cc}{}_0^{ABC}{D}_t^{\Im }\Omega \left(t\right)\hfill & =\mathcal{E}(t,\Omega (t\left)\right),\hfill \\ \Omega \left(0\right)\hfill & ={\Omega }_0,\hfill \end{array}\right.$$

(6)

where the vector $\Omega \left(t\right)=\left(S\left(t\right),V_C\left(t\right),C\left(t\right),A_H\left(t\right),I_H\left(t\right),A_{HC}\left(t\right),I_{HC}\left(t\right),R\left(t\right)\right)\in {\mathbb{R}}^9$ for $t\in [0,{\mathcal{T}}_{max}]$ represents the various states of the system, and $\mathcal{E}$ is a continuous vector defined as follows:

$$\mathcal{E}=\left(\begin{array}{c}{\mathcal{E}}_1\\ {\mathcal{E}}_2\\ {\mathcal{E}}_3\\ {\mathcal{E}}_4\\ {\mathcal{E}}_5\\ {\mathcal{E}}_6\\ {\mathcal{E}}_7\\ {\mathcal{E}}_8\\ {\mathcal{E}}_9\end{array}\right)=\left(\begin{array}{c}\Lambda -\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}S\left(t\right)-\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}S\left(t\right)+{\delta }_f{V}_f\left(t\right)+{\delta }_g{V}_g\left(t\right)-(\mu +\psi +\rho )S\left(t\right),\\ \psi S\left(t\right)-\frac{\sigma {\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{V}_f\left(t\right)-\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{V}_f\left(t\right)-({\delta }_f+{\theta }_f+\mu )V_f\left(t\right),\\ \rho S\left(t\right)-\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{V}_g\left(t\right)-\frac{\gamma {\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{V}_g\left(t\right)-({\delta }_g+{\theta }_g+\mu )V_g\left(t\right),\\ \frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}(S\left(t\right)+R_g\left(t\right)+\sigma V_f\left(t\right)+V_g\left(t\right))-({\xi }_f+{\eta }_f+\mu )I_f-{\phi }_1\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{I}_f\left(t\right),\\ \frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}(S\left(t\right)+R_f\left(t\right)+V_f\left(t\right)+\gamma V_g\left(t\right))-({\xi }_g+{\eta }_g+\mu )I_g\left(t\right)-{\phi }_2\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{I}_g\left(t\right),\\ {\phi }_1\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{I}_f\left(t\right)+{\phi }_2\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{I}_g\left(t\right)-({\xi }_{fg}+{\eta }_{fg}+\mu )I_{fg}\left(t\right),\\ {\theta }_f{V}_f\left(t\right)+{\xi }_f{I}_f\left(t\right)-\mu R_f\left(t\right)-\frac{{\beta }_g{I}_g\left(t\right)}{N\left(t\right)}{R}_f\left(t\right),\\ {\theta }_g{V}_g\left(t\right)+{\xi }_g{I}_g\left(t\right)-\mu R_g\left(t\right)-\frac{{\beta }_f{I}_f\left(t\right)}{N\left(t\right)}{R}_g\left(t\right),\\ {\xi }_{fg}{I}_{fg}\left(t\right)-\mu R_{fg}\left(t\right).\end{array}\right).$$

(7)

The initial states of the model’s variables are defined by

$$\Omega \left(0\right)=\left(S\left(0\right),V_C\left(0\right),C\left(0\right),A_H\left(0\right),I_H\left(0\right),A_{HC}\left(0\right),I_{HC}\left(0\right),R\left(0\right)\right).$$

In addition, $\mathcal{E}:[0,{\mathcal{T}}_{max}]\times {\mathbb{R}}^9\to {\mathbb{R}}^9$ is said to satisfy the Lipschitz condition in the second argument, if we have:

$$\begin{array}{cc}\hfill & ∥\mathcal{E}(t,{\Omega }_1)-\mathcal{E}(t,{\Omega }_2)∥\le \mathcal{M}∥{\Omega }_1-{\Omega }_2∥,\forall t\in [0,{\mathcal{T}}_{max}],\forall {\Omega }_1,{\Omega }_2\in {\mathbb{R}}^9,\hfill \end{array}$$

(8)

where $\mathcal{M}$ > 0, ${\mathcal{T}}_{max}$ denotes final time.

We claim the results below:

Theorem 2. 

The initial value problem (IVP) (6) will have a unique solution in $C([0,{\mathcal{T}}_{max}],{\mathbb{R}}^9)$ as long as:

$$\begin{array}{ccc}\hfill \left(\frac{(1-\Im )\mathcal{M}}{\mathcal{H}(\Im )}+\frac{\Im \mathcal{M}}{\mathcal{H}(\Im )\Gamma (\Im +1)}{\mathcal{T}}_{max}^{\Im }\right)& <& 1.\hfill \end{array}$$

(9)

Proof. 

Upon the application of an AB integral to both sides of (6), we obtain

$$\begin{array}{ccc}\hfill Z\left(t\right)& =& Z_0+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{E}(t,Z\left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{E}(\omega ,Z\left(\omega \right))d\omega .\hfill \end{array}$$

(10)

Let $\mathcal{W}=(0,{\mathcal{T}}_{max})$.

Let us define the operator ${\rm Y}:C(\mathcal{W},{\mathbb{R}}^9)\to \mathcal{C}(\mathcal{W},{\mathbb{R}}^9)$ by:

$${\rm Y}\left[Z\right]\left(t\right)=\mathcal{Q}\left(t\right)$$

(11)

where

$$\mathcal{Q}\left(t\right)=Z_0+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{E}(t,\mathcal{Q}\left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{E}(\omega ,\mathcal{Q}\left(\omega \right))d\omega .$$

The sup norm on $C(\mathcal{W},{\mathbb{R}}^9){,\left|\right|.\left|\right|}_{\mathcal{W}}$ is defined by:

$${\left|\right|\mathcal{Q}\left|\right|}_{\mathcal{W}}{=}_{t\in \mathcal{W}}^{sup}\left|\right|\mathcal{Q}\left(t\right)\left|\right|,\forall \mathcal{Q}\in C(\mathcal{W},{\mathbb{R}}^9).$$

It is clear that $C(\mathcal{W},{\mathbb{R}}^9)$ together with ${\left|\right|.\left|\right|}_{\mathcal{W}}$ is a “Banach space”.

Let $\zeta$ be the fixed point of the operator ${\rm Y}:\mathcal{C}(\mathcal{W},{\mathbb{R}}^9)\to \mathcal{C}(\mathcal{W},{\mathbb{R}}^9)$. Then, $\zeta$ becomes a solution of (6), and this holds:

$${\rm Y}\left[\zeta \right]\left(t\right)=\zeta \left(t\right),$$

where

$$\zeta \left(t\right)=Z_0+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{E}(t,\zeta \left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{E}(\omega ,\zeta \left(\omega \right))d\omega .$$

Applying Equation (11), we have that

$$\begin{array}{cc}\hfill {∥{\rm Y}\left[\mathcal{Q}\right]\left(t\right)-{\rm Y}\left[\zeta \right]\left(t\right)∥}_{\mathcal{W}}& =∥Z_0+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{E}(t,\mathcal{Q}\left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{E}(\omega ,\mathcal{Q}\left(\omega \right))d\omega \hfill \\ \hfill & -\left[Z_0+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{E}(t,\zeta \left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{E}(\omega ,\zeta \left(\omega \right))d\omega \right]∥\hfill \\ \hfill & \le ∥\frac{1-\Im }{\mathcal{H}(\Im )}(\Im (t,\mathcal{Q}\left(t\right))-\mathcal{E}(t,\zeta \left(t\right)))\hfill \\ \hfill & +\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}(\Im (\omega ,\mathcal{Q}\left(\omega \right))-\mathcal{E}(\omega ,\zeta \left(\omega \right)))d\omega ∥.\hfill \end{array}$$

(12)

Because the operator $\mathcal{E}$ fulfills the Lipschitz criteria, we obtain

$$\begin{array}{cc}\hfill & \le \frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{M}∥\mathcal{Q}\left(t\right)-\zeta \left(t\right)∥+\frac{\Im \mathcal{M}}{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}∥\mathcal{Q}\left(t\right)-\zeta \left(t\right)∥d\omega \hfill \\ \hfill & \le \frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{M}{}_{t\in \mathcal{W}}^{sup}∥\mathcal{Q}\left(t\right)-\zeta \left(t\right)∥+\frac{\Im \mathcal{M}}{\mathcal{H}(\Im )\Gamma (\Im )}\left({\int }_0^t{\left(t-\omega \right)}^{\Im -1}\right)d\omega {}_{t\in \mathcal{W}}^{sup}∥\mathcal{Q}\left(t\right)-\zeta \left(t\right)∥\hfill \\ \hfill & \le \left(\frac{\left(1-\Im \right)\mathcal{M}}{\mathcal{H}(\Im )}+\frac{\Im \mathcal{M}{\mathcal{T}}_{max}^{\Im }}{\mathcal{H}(\Im )\Gamma (\Im +1)}\right)∥\mathcal{Q}-\zeta ∥,\mathrm{where}{\mathcal{T}}_{max}^{\Im }={\int }_0^{{\mathcal{T}}_{max}}{\left(t-\omega \right)}^{\Im -1}d\omega .\hfill \end{array}$$

(13)

Hence, given that the criteria in (9) are satisfied,

$$∥{\rm Y}\left[\mathcal{Q}\right]-{\rm Y}\left[\zeta \right]∥\le \left(\frac{\left(1-\Im \right)\mathcal{M}}{\mathcal{H}(\Im )}+\frac{\Im \mathcal{M}{\mathcal{T}}_{max}^{\Im }}{\mathcal{H}(\Im )\Gamma (\Im +1)}\right)∥\mathcal{Q}-\zeta ∥,$$

and the earlier defined operator ${\rm Y}$ is therefore a contraction. Thus, ${\rm Y}$ has a “unique fixed point”, which automatically becomes the solution of the IVP (6) and equivalently solves the fractional system (1). □

4. Ulam–Hyers (UH) Stability

The stability of the fractional system in the framework of generalized UH stability [35] shall now be discussed.

Suppose $\mathbb{E}=\mathcal{C}(\mathcal{I},{\mathbb{R}}^9)$ represent “the space of continuous functions” from $\mathcal{I}$ to ${\mathbb{R}}^9$ coupled with this norm: $\parallel \Omega \parallel {=}_{t\in \mathcal{I}}^{sup}\left|\Omega \left(t\right)\right|$, where $\mathcal{I}=[0,b]$.

Definition 5. 

The system (1) in the transformed form:

$$\left\{\begin{array}{cc}{}_0^{ABC}{D}_t^{\Im }\Omega \left(t\right)\hfill & =\mathcal{K}(t,\Omega (t\left)\right),\hfill \\ \Omega \left(0\right)\hfill & ={\Omega }_0,\hfill \end{array}\right.$$

(14)

is stable (in the sense of Ulam–Hyers) if ∃ $h>0$ such that, for $ϵ>0$ and $\overline{\Omega }\in \mathbb{E}$ and given the inequality

$${\parallel }^{ABC}{D}^{\Im }\overline{\Omega }\left(t\right)-\mathcal{K}(t,\overline{\Omega }\left(t\right)\parallel \le ϵ,t\in \mathcal{I},ϵ=max{\left({ϵ}_i\right)}^T,i=1,2,\dots 10,$$

(15)

∃ unique solution $\Omega \in \mathbb{E}$ of system (14) such that

$$\parallel \overline{\Omega }\left(t\right)-\Omega \left(t\right)\parallel \le hϵ,t\in \mathcal{I},h=max{\left(h_j\right)}^T,j=1,2,\dots 10.$$

Definition 6. 

System (14) is “generalized UH stable” if ∃ a function $\vartheta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\vartheta \left(0\right)=0$, which is continuous, such that, for any other solution $\overline{\Omega }\in \mathbb{E}$ which solves system (15), ∃ unique solution $\Omega \in \mathbb{E}$ whereby this inequality below is satisfied:

$$\parallel \overline{\Omega }\left(t\right)-\Omega \left(t\right)\parallel \le \vartheta \left(ϵ\right),t\in \mathcal{I},\vartheta =max{\left({\vartheta }_j\right)}^T,j=1,2,\dots 10.$$

Remark 1. 

“A function $\overline{\Omega }\in \mathbb{E}$ satisfies the inequality (15) if and only if there exists a function $h\in \mathbb{E}$, having the following properties:”

i. 

$∥\varphi \left(t\right)∥\le ϵ,t\in \mathcal{I}$.

ii. 

${}^{ABC}{D}^{\Im }\overline{\Omega }\left(t\right)=\mathcal{K}(t,\overline{\Omega }\left(t\right)+\varphi \left(t\right)$,$t\in \mathcal{I}$.

Lemma 1. 

For $\overline{\Omega }\in \mathbb{E}$ in (15), the inequality below is satisfied:

$$\left|\overline{\Omega }\left(t\right)-\left({\overline{\Omega }}_0\left(t\right)+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{K}(t,\overline{\Omega }\left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{K}(\omega ,\overline{\Omega }\left(\omega \right))d\omega \right)\right|\le \Phi ϵ$$

(16)

For the proof, see Appendix A.

Theorem 3. 

Let $\Omega \in \mathbb{E}$ and $\mathcal{K}:\mathcal{I}\times {\mathbb{R}}^9\to {\mathbb{R}}^9$. If the “Lipschitz condition” is met with Lipschitz constant ${\mathcal{E}}_{\mathcal{K}}>0$ and $1-\Phi {\mathcal{E}}_{\mathcal{K}}>0$, where $\Phi =\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}+\frac{b^{\Im }}{\mathcal{H}(\Im )\Gamma (\Im )}$, then the model (14) is generalized UH stable.

For the proof, see Appendix B.

5. Numerical Schemes and Simulations

The numerical scheme adopted in [36] shall be applied to the system (6). On the application of fractional calculus’ fundamental theorem, we have

$$\Omega \left(t\right)={\Omega }_0+\frac{1-\Im }{\mathcal{H}(\Im )}\mathcal{G}(t,\Omega \left(t\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^t{\left(t-\omega \right)}^{\Im -1}\mathcal{G}(\omega ,\Omega \left(\omega \right))d\omega .$$

At any point $t=t_{\varsigma +1}=(\varsigma +1)h$, where $h=t_{\varsigma +1}-t_{\varsigma }$ is the step-size with $\varsigma =0,1,2\dots ,$ the discretization of the above equation is given by

$$\begin{array}{cc}\hfill \Omega \left(t_{\varsigma +1}\right)& =\Omega \left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}\mathcal{G}(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}{\int }_0^{t_{\varsigma +1}}{\left(t_{\varsigma +1}-\omega \right)}^{\Im -1}\mathcal{G}(\omega ,\Omega \left(\omega \right))d\omega \hfill \\ \hfill \Omega \left(t_{\varsigma +1}\right)& =\Omega \left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}\mathcal{G}(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}\sum _{\varrho =0}^{\varsigma }{\int }_{t_{\varrho }}^{t_{\varrho +1}}{\left(t_{\varsigma +1}-\omega \right)}^{\Im -1}\mathcal{G}(\omega ,\Omega \left(\omega \right))d\omega .\hfill \\ \hfill \end{array}$$

(17)

A function $\mathcal{G}(\omega ,\Omega (\omega \left)\right)$ in $[t_{\varrho },t_{\varrho +1}]$ can be approximated with the help of the Lagrange two-points interpolation as follows:

$$\begin{array}{cc}\hfill \mathcal{G}(\omega ,\Omega (\omega \left)\right)& \approxeq \frac{\omega -t_{\varrho -1}}{t_{\varrho }-t_{\varrho -1}}\mathcal{G}(t_{\varrho },\Omega \left(t_{\varrho }\right))-\frac{\omega -t_{\varrho }}{t_{\varrho }-t_{\varrho -1}}\mathcal{G}(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right))\hfill \\ \hfill & =\frac{\mathcal{G}(t_{\varrho },\Omega \left(t_{\varrho }\right))}{h}(\omega -t_{\varrho -1})-\frac{\mathcal{G}(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right))}{h}(\omega -t_{\varrho })\hfill \\ \hfill \end{array}$$

(18)

Substituting (18) into (17), we obtain

$$\begin{array}{cc}\hfill \Omega \left(t_{\varsigma +1}\right)& =\Omega \left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}\mathcal{G}(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )\Gamma (\Im )}\sum _{\varrho =0}^{\varsigma }(\frac{\mathcal{G}(t_{\varrho },\Omega \left(t_{\varrho }\right))}{h}{\int }_{t_{\varrho }}^{t_{\varrho +1}}(\omega -t_{\varrho -1}){\left(t_{\varsigma +1}-\omega \right)}^{\Im -1}d\omega \hfill \\ \hfill & -\frac{\mathcal{G}(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right))}{h}{\int }_{t_{\varrho }}^{t_{\varrho +1}}(\omega -t_{\varrho }){\left(t_{\varsigma +1}-\omega \right)}^{\Im -1}d\omega ).\hfill \\ \hfill \end{array}$$

(19)

On integrating, we have

$$\begin{array}{ccc}\hfill \Omega \left(t_{\varsigma +1}\right)& =& \Omega \left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}\mathcal{G}(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }\mathcal{G}(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }\mathcal{G}\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\}.\hfill \end{array}$$

(20)

Adopting the numerical scheme (20) into the fractional system (6) yields the following numerical solution:

$$\begin{array}{ccc}\hfill S\left(t_{\varsigma +1}\right)& =& S\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_1(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_1(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_1\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill V_f\left(t_{\varsigma +1}\right)& =& V_f\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_2(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_2(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_2\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill V_g\left(t_{\varsigma +1}\right)& =& V_g\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_3(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_3(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_3\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_f\left(t_{\varsigma +1}\right)& =& I_f\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_4(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_4(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_3\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_g\left(t_{\varsigma +1}\right)& =& I_g\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_5(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_5(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_5\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_{fg}\left(t_{\varsigma +1}\right)& =& I_{fg}\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_6(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_6(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_6\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill R_f\left(t_{\varsigma +1}\right)& =& R_f\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_7(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_7(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_7\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill R_g\left(t_{\varsigma +1}\right)& =& R_g\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_8(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_8(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_8\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill R_{fg}\left(t_{\varsigma +1}\right)& =& R_{fg}\left(0\right)+\frac{\left(1-\Im \right)}{\mathcal{H}(\Im )}{\mathcal{G}}_9(t_{\varsigma },\Omega \left(t_{\varsigma }\right))+\frac{\Im }{\mathcal{H}(\Im )}\sum _{\varrho =0}^{\varsigma }\left\{\frac{h^{\Im }{\mathcal{G}}_9(t_{\varrho },\Omega \left(t_{\varrho }\right))}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im }\left(\varsigma -\varrho +\Im +2\right)\right.\right.\hfill \\ & -& \left.\left.{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +2\Im +2\right)\right]-\frac{h^{\Im }{\mathcal{G}}_9\left(t_{\varrho -1},\Omega \left(t_{\varrho -1}\right)\right)}{\Gamma (\Im +2)}\left[{\left(\varsigma -\varrho +1\right)}^{\Im +1}-{\left(\varsigma -\varrho \right)}^{\Im }\left(\varsigma -\varrho +\Im +1\right)\right]\right\}.\hfill \end{array}$$

The stability and error analysis, as well as the accuracy of the scheme, have been presented in [36].

5.1. Numerical Simulations

The system (1) shall be the subject of numerical experiments in this section. Demographic information from Pakistan will be used for the numerical assessments. The initial states for the various components of the fractional-order system are assumed as follows: $S\left(0\right)$ = 175,000,000, $V_f\left(0\right)$ = 15,000,000, $V_g\left(0\right)$ = 15,000,000, $I_f\left(0\right)$ = 1,296,527, $I_g\left(0\right)$= 250,895, $I_{fg}\left(0\right)$ = 5000, $R_f\left(0\right)$ = 0, $R_g\left(0\right)$ = 0, $R_{fg}\left(0\right)$ = 0. The designed model was fitted to the data sets for daily confirmed COVID-19 cases recorded for Pakistan [37] from 1 January 2022 and 10 April 2022 (presented in

Table 2. Number of reported daily COVID-19 cases in Pakistan from 1 January 2022 to 10 April 2022. Source: [37].

Date (January)	Cases	Date (February)	Cases	Date (March)	Cases	Date (April)	Cases
01/01/2022	1,296,527	01/02/2022	1,436,413	01/03/2022	1,510,986	01/04/2022	1,525,181
02/01/2022	1,297,235	01/02/2022	1,442,263	02/03/2022	1,511,754	02/04/2022	1,525,466
03/01/2022	1,297,865	03/02/2022	1,448,663	03/03/2022	1,512,707	03/04/2022	1,525,620
04/01/2022	1,298,763	04/02/2022	1,454,800	04/03/2022	1,513,503	04/04/2022	1,525,775
05/01/2025	1,299,848	05/02/2022	1,459,773	05/03/2022	1,514,258	05/04/2022	1,525,923
06/01/2022	1,301,141	06/02/2022	1,463,111	06/03/2022	1,515,014	06/04/2022	1,526,093
07/01/2022	1,302,486	07/02/2022	1,465,910	07/03/2022	1,515,392	07/04/2022	1,526,234
08/01/2022	1,304,058	08/02/2022	1,470,161	08/03/2022	1,516,150	08/04/2022	1,526,472
09/01/2022	1,305,707	09/02/2022	1,474,075	09/03/2022	1,516,789	09/04/2022	1,526,568
10/01/2022	1,307,174	10/02/2022	1,477,573	10/03/2022	1,517,512	10/04/2022	1,526,666
11/01/2022	1,309,248	11/02/2022	1,480,592	11/03/2022	1,518,083
12/01/2022	1,312,267	12/02/2022	1,483,798	12/03/2022	1,518,692
13/01/2022	1,315,834	13/02/2022	1,486,361	13/03/2022	1,519,154
14/01/2022	1,320,120	14/02/2022	1,488,958	14/03/2022	1,519,627
15/01/2022	1,324,147	15/02/2022	1,491,423	15/03/2022	1,520,120
16/01/2022	1,328,487	16/02/2022	1,494,293	16/03/2022	1,520,634
17/01/2022	1,333,521	17/02/2022	1,496,693	17/03/2022	1,520,817
18/01/2022	1,338,993	18/02/2022	1,498,676	18/03/2022	1,521,513
19/01/2022	1,345,801	19/02/2022	1,500,320	19/03/2022	1,521,888
20/01/2025	1,353,479	20/02/2022	1,501,680	20/03/2022	1,522,191
21/01/2022	1,360,019	21/02/2022	1,502,641	21/03/2022	1,522,419
22/01/2022	1,367,605	22/02/2022	1,503,873	22/03/2022	1,522,862
23/01/2022	1,374,800	23/02/2022	1,505,328	23/03/2022	1,523,072
24/01/2022	1,381,152	24/02/2022	1,506,450	24/03/2022	1,523,401
25/01/2022	1,386,348	25/02/2022	1,507,657	25/03/2022	1,523,590
26/01/2022	1,393,887	26/02/2022	1,508,504	26/03/2022	1,523,900
27/01/2022	1,402,070	27/02/2022	1,509,360	27/03/2022	1,524,086
28/01/2022	1,410,033	28/02/2022	1,510,221	28/03/2022	1,524,355
29/01/2022	1,417,991			29/03/2022	1,524,549
30/01/2022	1,425,039			30/03/2022	1,524,793
31/01/2022	1,430,366			31/03/2022	1,524,973

). The model fits well to the real data when the order of derivative $\Im =0.95$, as can be observed in Figure 1.

Analyses investigating the most sensitive parameters of the model are presented in Figure 2. 3D plots of the COVID-19-associated reproduction number, ${\mathcal{R}}_{0f}$, as a function of the contact rate and vaccination parameters are presented in Figure 2a,b, respectively. It is observed from both figures that an increase in the contact rate, ${\beta }_f$, results in a subsequent rise in the value of the reproduction number, as expected. However, from Figure 2a, it is noticed that increasing the COVID-19 vaccination rate, $\psi$, causes a decrease in the reproduction number and also the disease burden within the population. However, increasing the vaccine inefficacy, $\sigma$, causes an increase in the reproduction number, as observed in Figure 2b.

3D plots of the viral hepatitis B effective reproduction number, ${\mathcal{R}}_{0g}$, as a function of vaccine-related parameters and contact rate are depicted in Figure 2c,d, respectively. It is seen from both figures that increasing the effective transmission rate causes resulting increments in the reproduction number and disease burden within the population. Nonetheless, as shown in Figure 2c, an increase in the vaccination rate, $\rho$, leads to a subsequent decrease in the reproduction number, while an increase in the viral hepatitis B vaccine inefficacy $\gamma$ causes a large spike in the reproduction number (as can be seen in Figure 2d).

Numerical experiments of the fractional-order system (1) at different orders of the derivative, ℑ, and when ${\mathcal{R}}_0<1$ are presented in Figure 3a–i. It is observed from these figures that variation in the order of the derivative greatly influences the dynamics of the disease in each of the epidemiological states. Moreover, the different solution paths for each compartment as the order of the derivative is varied tend towards the infection-free steady states. Phase portraits confirming the behaviour of the three infected components of the model, $I_f,I_g$ and $I_{fg}$, are depicted in Figure 4a–e. One important observation from these figures is that, for different initial conditions assumed for the disease classes, their trajectories always point towards the infection-free steady states over the passage of time. These results also confirm the local asymptotic stability and the global Ulam–Hyers stability results established in Section 2.2 and Section 4, respectively.

Numerical experiments of the fractional-order system (1) at different orders of the derivative, ℑ, and when ${\mathcal{R}}_0>1$ are presented in Figure 5a–i. It is observed from these figures that variation in the order of the derivative greatly influences the dynamics of the diseases in each of the epidemiological states. This is expected due to the effect of memory, which is inherent in the definition of the fractional derivative considered for this system. In addition, the different solution paths of each state at different orders of the derivative tend towards the infection-present equilibrium. Trajectory diagrams confirming the behaviour of the three infected components of the system, $I_f,I_g$ and $I_{fg}$, for different orders of the derivative, ℑ, are depicted in Figure 6a–e. One striking point that can be deduced from these figures is that, for different initial conditions assumed for the disease classes, their trajectories always move towards the infection-present steady state over the passage of time. These results also confirm the local asymptotic stability and the global Ulam–Hyers stability results established in Section 2.2 and Section 4, respectively.

6. Conclusions

To investigate the epidemiological impact of vaccination measures on viral hepatitis B and COVID-19 co-dynamics, a novel fractional-order model is presented in this paper. Real observations from Pakistan are used to fit and verify the model. A fixed point theory approach is used to investigate the existence and stability of the new model. From the fitting, important thresholds for COVID-19 and viral hepatitis B are estimated. To illustrate the impact of fractional derivatives on solution paths for each epidemiological state, as well as trajectory diagrams for different initial conditions, numerical analyses are performed. We also present contour plots to illustrate the relationship between the transmission coefficients and key vaccine-related parameters. The following are some of the most important analytical findings from the paper:

(i.)

The basic analysis/properties of the fractional-order system (1) are investigated.

(ii.)

The derivation of appropriate conditions for the existence of a unique solution of the formulated fractional-order system is carried out in Section 3, with the help of Theorem 2.

(iii.)

Stability analyses of the formulated model in the framework of Ulam–Hyers are discussed in Section 4, with the help of Theorem 3.

(iv.)

A numerical scheme using the two-step Lagrange polynomial approach to approximate the solution of the designed fractional system (1) is derived and presented in Section 5.

Quantitative analyses to validate the qualitative analysis presented in the paper are performed in Section 5.1. The highlights are listed below:

(i.)

The proposed fractional-order co-dynamical model fits well to Pakistan’s data for COVID-19 [37] when the order of the derivative is $\Im =0.95$. Other important parameters estimated from the fitting are also given in Table 1.

(ii.)

The 3D plots of the COVID-19-associated reproduction number ${\mathcal{R}}_{0f}$ as a function of the transmission rate and vaccination parameters are presented in Figure 2a,b, respectively. It was observed from both figures that an increase in the transmission rate ${\beta }_f$ results in a subsequent rise in the value of the reproduction number, as expected. However, from Figure 2a, it is noticed that increasing the COVID-19 vaccination rate $\psi$ causes a decrease in the reproduction number, and also in the disease burden within the population. However, increasing the vaccine inefficacy $\sigma$ causes an increase in the reproduction number, as observed in Figure 2b.

(iii.)

The different trajectories for the epidemiological states at various orders of the derivative, when the reproduction number ${\mathcal{R}}_0<1$, are presented in Figure 3a–i. It was observed that the trajectory diagrams tend towards the infection-free steady state. Phase portraits confirming the behavior of the infected components of the model are also depicted in Figure 4a–e. One important observation from these figures is that for different initial conditions assumed for the disease classes, their trajectories still point towards the infection-free steady states over the passage of time, regardless of the order of the fractional derivative. These results also confirm the local asymptotic stability and the Ulam–Hyers stability results established in Section 2.2 and Section 4, respectively.

There are also limitations and shortcomings to this study. The current formulation does not assume exposed (latent), asymptomatic, or quarantine classes for COVID-19 or viral hepatitis B. In a further study, we hope to incorporate these assumptions to capture reality. Additionally, there is little information available regarding vaccine or infection-acquired cross-immunity between COVID-19 and viral hepatitis B. Further, there are no detailed clinical reports on whether the current COVID-19 or viral hepatitis B vaccines can cross-protect. With such detailed information, further research on cross-immunity is much desired. Research on multiple strains of viral hepatitis B and COVID-19 is also anticipated because mutations are possible in diseases such as COVID-19 and viral hepatitis B. For viral hepatitis B, it was challenging to obtain properly documented daily records. In a subsequent study, we plan to fit an improved version of the model to both the COVID-19 and viral hepatitis B data sets in order to obtain better and more precise parameter estimates.