Resurgence in focus: Covid-19 dynamics and optimal control frameworks

Abstract

The resurgence of Covid-19, accompanied by various variants of the virus, highlights the fact that Covid-19 is still present within the population. The study proposed a Covid-19 dynamical model for analyzing the effect of vaccination and the continuous use of non-medical interventions for addressing Covid-19 transmission dynamics. The Lyaponov function and Jacobian matrix techniques were used to analyze the stability of the model's equilibria. The model was transformed into a problem of optimal control with time-dependent variables, aimed at managing efforts to prevent the spread of Covid-19. Numerical assessments were deployed to assess the effect of vaccination and the continuous use of non-medical intervention strategies to mitigate the spread of Covid-19. The global sensitivity analysis of the model was used to detect the key parameters influencing the behavior of the model. In addition, numerical results showed a significant decrease in the basic reproduction rate $\left({\mathrm{ℛ}}_0\right)$ when implementing $\sigma$ and $\xi$, either separately or together. The optimal control results suggested that the control measures should be consistently enforced without any relaxation.

2010 Mathematics Subject Classification: 92D30, 93C95, 49 N90, 34H05, 37 N25.

Introduction

The impact of coronavirus cannot be overemphasized. It has had a significant impact on people's lives, as well as social-economically [1]. Coronavirus is a group of viruses belonging to the same family. A small number of the viruses are behind the cause of people's common cold and others infect animals such as bats, camels and cattles. Severe acute respiratory syndrome coronavirus 2 is the virus that triggers Covid-19 and it is genetically associated to the virus that brought about the SARS outbreak in 2003 [2,3]. Covid-19 surfaced on a lighter note in November 2019 with the first large cluster surfacing in December 2019, in Wuhan city, China. As the virus started to intermittently spread both within and outside the globe on March 11, 2020, the World Health Organization (WHO) classified it to be an epidemic [3,4]. According to the report from WHO as at August 13, 2024, that they were more than 775 million confirmed cases, more than 7 million deaths and about 9.8 million persons who have been given a dose of the Covid-19 vaccine across the number of states mentioned [5].

Sometimes in August of last year, the United Nations (UN) announced the resurgence of Covid-19 infections reported in America, Europe, and the Western Pacific [6]. After the outbreak of Covid-19, different measures were employed to curtail the dynamics of the spread of the disease, the reproduction rates and deaths [7]. Some of these measures employed include making use of non-medical intervention strategies and vaccination. There were different types of vaccines certified by WHO to be administered ranging from Pfizer-BioTech, AstraZeneca, SinoVaC-CoronaVaC, CoVaxin, CoVavax, etc. [8]. There were breakthrough cases among fully vaccinated persons who potentially could also be infectious [9,10]. This has become worrisome such that vaccination alone is not sufficient to control Covid-19 in the absence of non-medical intervention strategies [11,12].

Various dynamical models were employed to evaluate the effects of infectious disease and proffering realistic interpretations to combat them. We will review some of these models to that effect. [13] developed a mathematical model to understand the spread pattern of dengue disease. [14] developed a model to examine the impact of Covid-19 transmission by integrating the idea of contact tracing and contaminated surfaces. The authors [[15], [16], [17], [18], [19], [20]] formulated models to investigate the spread patterns, the stability, and the effect of the Covid-19 spread through the use of vaccination and isolation. In an effort to gain understanding of the management and spread of Covid-19 by vaccination and non-pharmaceutical intervention strategies, [[21], [22], [23], [24]] developed mathematical models. [[25], [26], [27], [28]] proposed mathematical models of Covid-19 to examine the spread of the virus via contact tracing, treatment of infected persons, quarantined individuals and individuals who were resistant to Covid-19 infection. Fractional order mathematical models were developed by [[29], [30], [31], [32], [33]] to study the dynamics of Covid-19 outbreak with optimal control strategy. [[34], [35], [36]] designed models to study the spread of infectious diseases considering vaccination strategy and the treatment of infected individuals. [[37], [38], [39]] devised double-dose vaccination strategy models to examine the behavior of infectious diseases. The authors [[40], [41], [42], [43], [44]] developed optimal control mathematical models with vaccination strategy and preventive measures to effectively control the disease epidemic. Mathematical models with optimal control strategy were formulated by [[45], [46], [47]] incorporating the vaccination of vulnerable people and the treatment of quarantined infected people. The authors [48,49] formulated optimal control mathematical models considering non-pharmaceutical intervention strategies and the treatment of infected individuals to combat the spread of the disease. The dynamics of Covid-19 have been studied through the formulation and analysis of mathematical models incorporating non-pharmaceutical intervention measures and vaccination strategy as cited in this paper. The research work focuses on the observation that individuals who were vaccinated against Covid-19 still contracted the virus, and their potential to spread the disease in the context of studying Covid-19 dynamics. The study aims to evaluate the effects of the continuous use of preventive measures, vaccination, and proper medical care for infected individuals. In this research, we introduced into the force of infection individuals who were fully vaccinated and also accounted for their potential to infect others, as outlined in the study of [9,10]. The data for the simulation were carefully selected from existing literature on Covid-19. The manuscript is planned as follows: a Covid-19 dynamical model is proposed and analyzed in Sections 2 and 3. The model simulation with optimal control analysis, discussion of the results, and the concluding remarks can be found in Sections 4 and 5.

Model development

$T\left(t\right)$ represents the entire population at time t, divided into seven compartments that are mutually exclusive, namely: vulnerable individuals ($S\left(t\right)$), vaccinated individuals ($V\left(t\right)$), exposed Covid-19 infected individuals ($E\left(t\right)$), unvaccinated Covid-19 infected individuals ($I$), vaccinated Covid-19 infected individuals ($I_v\left(t\right)$), isolated individuals ($H\left(t\right)$) and recovered individuals ($R\left(t\right)$). The rate at which the vulnerable individuals and the vaccinated individuals developed Covid-19 is given as

$$\lambda =\frac{\beta \left(1-\xi \right)\left(I+\eta I_v\right)}{T}$$ (2.1)

where $\left(\xi \right)$ is the non-pharmaceutical intervention rate to decrease the Covid-19 transmission rate $\left(\beta \right)$, and $\left(\eta \right)$ is the modification parameter for reduced transmissibility of vaccinated Covid-19 infected individuals. We presume that $\eta <1$. The vaccination rate for the vulnerable individuals is presumed to be $\left(\sigma \right)$. The vulnerable individuals acquired Covid-19 at the rate of $\left(\lambda \right)$, and while vaccinated individuals also acquired Covid-19 due to vaccine inefficacy at the rate $\left(\mathrm{ϵ}\lambda \right)$, respectively. A portion $\left(\rho \right)$ of the exposed Covid-19 infected individuals at the rate $\left(\kappa \right)$ advance to the unvaccinated Covid-19 infected individuals compartment, while the remaining portion progresses towards the vaccinated Covid-19 infected individuals compartment. We also presume that $\left(\theta \right)$ is the immunity waning rate. ${\delta }_1$ and ${\delta }_2$ are the rates of isolation for unvaccinated Covid-19 infected individuals, and vaccinated Covid-19 infected individuals. The recovery rates for unvaccinated Covid-19 infected individuals, vaccinated Covid-19 infected individuals, and the isolated individuals are ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$, respectively. The natural mortality rate $\left(\mu \right)$ is presumed to be the same in all of the compartments, while the disease mortality rate $\left(d_c\right)$ is also presumed to be the same in compartments $\left(I,I_v,H\right)$. Hence, the entire population at time t is expressed as

$$T\left(t\right)=S\left(t\right)+V\left(t\right)+E\left(t\right)+I\left(t\right)+I_v\left(t\right)+H\left(t\right)+R\left(t\right)$$ (2.2)

Built upon the assumptions mentioned above, the Covid-19 model with the schematic diagram and the explanation of the state variables and parameters are shown below in Fig. 1 and Table 1, is given as

$$\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=\Lambda +\mathit{\theta V}-\mathit{\lambda S}-\left(\sigma +\mu \right)S,\\ \frac{\mathit{dV}}{\mathit{dt}}=\mathit{\sigma S}-\mathit{ϵ\lambda V}-\left(\theta +\mu \right)V,\\ \frac{\mathit{dE}}{\mathit{dt}}=\mathit{\lambda S}+\mathit{ϵ\lambda V}-\left(\kappa +\mu \right)E,\\ \frac{\mathit{dI}}{\mathit{dt}}=\mathit{\rho \kappa E}-\left({\delta }_1+{\gamma }_1+d_c+\mu \right)I,\\ \frac{d{I}_v}{\mathit{dt}}=\left(1-\rho \right)\mathit{\kappa E}-\left({\delta }_2+{\gamma }_2+d_c+\mu \right)I_v,\\ \frac{\mathit{dH}}{\mathit{dt}}={\delta }_{1}I+{\delta }_2{I}_v-\left({\gamma }_3+d_c+\mu \right)H,\\ \frac{\mathit{dR}}{\mathit{dt}}={\gamma }_{1}I+{\gamma }_2{I}_v+{\gamma }_{3}H-\mathit{\mu R}.\end{array}$$ (2.3)

(See Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11.) (See Table 2, Table 3, Table 4.)

Fig. 1.

Model (2.3) schematic diagram.

Table 1.

Explanation of variables and parameters of system (2.3).

Symbols	Description
$S$	Vulnerable individuals
$V$	Vaccinated individuals
$E$	Exposed Covd-19 infected individuals
$I$	Unvaccinated Covid-19 Infected individuals
$I_v$	Vaccinated Covid-19 infected individuals
$H$	Isolated individuals
$R$	Recovered individuals
Parameters	Interpretation
$\mathrm{\Lambda }$	Recruitment rate
$\beta$	Covid-19 transmission rate
$\sigma$	Vaccination rate
$\xi$	Non-Pharmaceutical interventions rate
$\kappa$	Incubation period
$\rho$	Fraction of unvaccinated Covid-19 infected individuals
${\delta }_1$	Isolation rate for unvaccinated Covid-19 infected individuals
${\delta }_2$	Isolation rate for vaccinated Covid-19 infected individuals
$\mathrm{ϵ}$	Vaccine inefficacy
$\eta$	Modification parameter for reduced infectiousness
${\gamma }_1$	Unvaccinated Covid-19 infected individuals recovery rate
${\gamma }_2$	Vaccinated Covid-19 infected individuals recovery rate
${\gamma }_3$	Isolated individuals recovery rate
$\theta$	Immunity waning rate
$d_c$	Disease mortality rate
$\mu$	Natural mortality rate

Fig. 2.

Sensitivity analysis values using compartment $E$, $I$, $I_v$, and H as the output functions.

Fig. 3.

The sensitivity analysis values with $\left({\mathrm{ℛ}}_0\right)$ and the total infected population as the output functions.

Fig. 4.

Simulation of model (2.3) showing Covid-19 cumulative incidence as a function of time.

Fig. 5.

Simulation of model (2.3) depicting cumulative incidence of Covid-19 over time.

Fig. 6.

The 3D plots of the effective reproduction number (${\mathrm{ℛ}}_0$).

Fig. 7.

The graphs of the optimal control strategy when $u_1\ne 0$ is implemented.

Fig. 8.

The graphs of the optimal control strategy when $u_1\ne 0$ is implemented.

Fig. 9.

The graphs of the optimal control strategy when $u_1=u_2=u_3\ne 0$ is implemented.

Fig. 10.

The graphs of the optimal control strategy when $u_1=u_2=u_3\ne 0$ is implemented.

Fig. 11.

The effect of the control profile on the dynamics of system (2.3).

Table 2.

The description of parameter values of the system (2.3).

Parameters	Baseline value	Range	Source
$\beta$	1.12	$0-1.2$	[40]
$\sigma$	0.64	$0-1.0$	[37]
$\xi$	0.311	$0-1.0$	Assumed
$\kappa$	1/5.2	$0-1.0$	[38]
$\rho$	0.7	$0-1.0$	[40]
${\delta }_1$	0.06	$0-0.1$	[20]
${\delta }_2$	0.02	$0-0.1$	[20]
$\mathrm{ϵ}$	0.2	$0-1.0$	[25]
$\eta$	0.8	$0-1.0$	[40]
${\gamma }_1$	0.16979	$0-1.0$	[30]
${\gamma }_2$	0.909	$0-1.0$	[34]
${\gamma }_3$	0.16979	$0-1.0$	[30]
$\theta$	0.610	$0-1.0$	[28]
$d_c$	0.0119	$0-0.1$	[45]
$\mu$	0.00005	$0-0.01$	Estimated from [40]

Table 3.

The sensitivity analysis values with the infected compartments and effective reproduction number (${\mathrm{ℛ}}_0$) as the response functions.

Parameters	E	$\mathbf{I}$	${\mathbf{I}}_{\mathbf{v}}$	$\mathbf{H}$	${\mathrm{ℛ}}_0$
$\beta$	0.8576	0.5743	0.5882	0.5434	0.7580
$\sigma$	−0.0248	−0.0333	−0.0804	−0.0623	−0.1729
$\xi$	−0.0937	−0.0988	−0.0324	−0.0871	−0.7572
$\kappa$	−0.8996	0.5562	0.5865	0.5241	0.0409
$\rho$	−0.0160	0.8396	−0.8439	0.0102	0.2917
${\delta }_1$	−0.0054	−0.0538	0.0133	0.5074	−0.0418
${\delta }_2$	0.0190	−0.0019	−0.1091	0.5473	−0.0611
$\mathrm{ϵ}$	0.5289	0.2189	0.2208	0.1945	0.3004
$\eta$	0.0173	−0.0872	0.0189	−0.0466	0.3163
${\gamma }_1$	0.0368	−0.6348	0.0828	−0.2924	−0.5140
${\gamma }_2$	0.0422	−0.0213	−0.6436	−0.3463	−0.3015
${\gamma }_3$	0.0187	0.0102	−0.0670	−0.6222	−0.0005
$\theta$	0.1891	0.0434	0.0691	0.0533	0.1941
$d_c$	0.0074	−0.0747	−0.0865	−0.1406	−0.0003
$\mu$	−0.0248	−0.0082	−0.0288	0.0060	−0.0388

Table 4.

The sensitivity analysis values using the entire infected population as the output function.

Parameters	COVID-19 Incidence
$\beta$	0.5640
$\sigma$	−0.0693
$\xi$	−0.0164
$\kappa$	0.4965
$\rho$	−0.0404
${\delta }_1$	0.5375
${\delta }_2$	0.5186
$\mathrm{ϵ}$	0.2186
$\eta$	0.0520
${\gamma }_1$	−0.3907
${\gamma }_2$	−0.3175
${\gamma }_3$	−0.6123
$\theta$	0.0987
$d_c$	−0.1884
$\mu$	−0.0300

Basic properties of the model

Positivity of the model solutions

The system (2.3) from an epidemiological perspective is significant if every of the model solutions are shown to be positive for all values of time (t).

Theorem 2.1

Given the starting data $S\left(0\right)\ge 0$, $V\left(0\right)\ge 0$, $E\left(0\right)\ge 0$, $I\left(0\right)\ge 0$, $I_v\left(0\right)\ge 0$, $H\left(0\right)\ge 0$, $R\left(0\right)\ge 0$. Then, the solutions $\left(S,V,E,I,I_v,H,R\right)$ of the system (2.3) are non-negative for all values of time $t>0$.

Proof. Let

$$t_s=\mathit{sup}\left\{t>0:S\left(t\right)\ge 0,V\left(t\right)\ge 0,E\left(t\right)\ge 0,I\left(t\right)\ge 0,I_v\left(t\right)\ge 0,H\left(t\right)\ge 0,R\left(t\right)\ge 0\right\}.$$

From the first equation in (2.3),we have that

$$\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=\Lambda +\mathit{\theta V}-\left(\lambda +\sigma +\mu \right)S\\ \frac{\mathit{dS}}{\mathit{dt}}=\Lambda -\left(\lambda +\omega \right)S\end{array}$$

where $\omega$=$\sigma$+$\mu$.

Adopting the techniques of integrating factor, we obtain

$$\begin{array}{c}\frac{d}{\mathit{dt}}\left[S\left(t\right)\exp \left(\mathit{\omega t}+{\int }_0^t\lambda \left(\nu \right)\mathit{d\nu }\right)\right]=\Lambda \left[\exp \left(\mathit{\omega t}+{\int }_o^t\lambda \left(\nu \right)\mathit{d\nu }\right)\right]\\ S\left(t_s\right)\exp \left(\omega t_s+{\int }_0^{t_s}\lambda \left(\nu \right)\mathit{d\nu }\right)=S\left(0\right)+{\int }_0^{t_s}\Lambda \left[\exp \left(\mathit{\omega y}+{\int }_o^y\lambda \left(\nu \right)\mathit{d\nu }\right)\right]\mathit{dy}\\ S\left(t_s\right)=S\left(0\right)\exp \left(-\omega t_s-{\int }_0^{t_s}\lambda \left(\nu \right)\mathit{d\nu }\right)+\left[\exp \left(-\omega t_s-{\int }_0^{t_s}\lambda \left(\nu \right)\mathit{d\nu }\right)\right]\times {\int }_0^{t_s}\Lambda \left[\exp \left(\mathit{\omega y}+{\int }_o^y\lambda \left(\nu \right)\mathit{d\nu }\right)\right]\mathit{dy}>0\\ \times {\int }_0^{t_s}\Lambda \left[\exp \left(\mathit{\omega y}+{\int }_o^y\lambda \left(\nu \right)\mathit{d\nu }\right)\right]\mathit{dy}>0\end{array}$$

□

Therefore, $S\left(t\right)\ge 0$ $\forall$ values of $t>0$.

Likewise, it can also be demonstrated that: $V\ge 0,E\ge 0,I\ge 0,I_v\ge 0,H\ge 0,R\ge 0.$.

Boundedness

Let the closed set $\mathcal{Z}$ be given as

$$\mathcal{Z}=\left\{\left(S,V,E,I,I_v,H,R\right)\in {\mathrm{ℝ}}_{+}^7:S+V+E+I+I_v+H+R\le \frac{\mathrm{\Lambda }}{\mu }\right\}$$

Then, $\mathcal{Z}$ is said to be positive invariant with regards to model (2.3). Summing the equations of system (2.3), we derive

$$\begin{array}{c}\frac{\mathit{dT}}{\mathit{dt}}=\mathrm{\Lambda }-\mathit{\mu T}-\left(I+I_v+H\right)d_c\\ \le \mathrm{\Lambda }-\mathit{\mu T}\end{array}$$ (2.4)

Adopting the method of [50] to eq. (4) and further simplifying, it can be inferred that

$$T\left(t\right)\le \frac{\mathrm{\Lambda }}{\mu }+\left(T\left(0\right)-\frac{\mathrm{\Lambda }}{\mu }\right)\mathit{Exp}\left(-\mathit{\mu t}\right)$$ (2.5)

which suggests that

$$\underset{t\to \infty }{\lim \sup }T\left(t\right)\le \frac{\mathrm{\Lambda }}{\mu }$$

Consequently, $T\left(t\right)\le \frac{\mathrm{\Lambda }}{\mu }$ as $t\to \infty$. Therefore, the model (2.3) has solution in $\mathcal{Z}$ and it is positive invariant.

Model analysis

Effective reproduction number of the model

The model has a disease free equilibrium (DFE) established by equating the various disease compartments of the system (2.3) to zero and it is expressed as

$$\begin{array}{l}{\mathrm{ℰ}}_0=\left(S^{\ast },V^{\ast },E^{\ast },I^{\ast },I_v^{\ast },H^{\ast },R^{\ast }\right)=\left(\frac{\mathrm{\Lambda }{g}_2}{\mu \left(g_2+\sigma \right)},\frac{\sigma \mathrm{\Lambda }}{\mu \left(g_2+\sigma \right)},00000\right)\end{array}$$ (3.6)

Employing the techniques of [51], we derive the effective reproduction number of the system (2.3) as

$${\mathrm{ℛ}}_0=\frac{\beta \left(1-\xi \right)\kappa \left(g_2+\sigma \mathrm{ϵ}\right)\left(g_4\mathit{q\eta }+g_5\rho \right)}{g_3{g}_4{g}_5\left(g_2+\sigma \right)}$$ (3.7)

with

$$g_2=\theta +\mu ,g_3=\kappa +\mu ,g_4={\delta }_1+{\gamma }_1+d_c+\mu ,g_5={\delta }_2+{\gamma }_2+d_c+\mu ,q=1-\rho$$

Local stability of the DFE

Theorem 3.1

The DFE $\left({\mathrm{ℰ}}_0\right)$ of system (2.3) is locally stable if ${\mathrm{ℛ}}_0<1$ or otherwise.

Proof. The Jacobian matrix of the model derived at the equilibrium point with no disease $\left({\mathrm{ℰ}}_0\right)$ was imployed to examined the local stability of model (2.3), and it is stated as

$$J\left({\mathcal{E}}_0\right)=\left(\begin{array}{ccccccc}-g_1& \theta & 0& \frac{-\beta \left(1-\xi \right)g_2}{g_2+\sigma }& \frac{-\beta \left(1-\xi \right)\eta g_2}{g_2+\sigma }& 0& 0\\ \sigma & -g_2& 0& \frac{-\beta \left(1-\xi \right)\mathit{\sigma ϵ}}{g_2+\sigma }& \frac{-\beta \left(1-\xi \right)\mathit{\sigma ϵ\eta }}{g_2+\sigma }& 0& 0\\ 0& 0& -g_3& \frac{\beta \left(1-\xi \right)\left(g_2+\mathit{\sigma ϵ}\right)}{g_2+\sigma }& \frac{\beta \left(1-\xi \right)\eta \left(g_2+\mathit{\sigma ϵ}\right)}{g_2+\sigma }& 0& 0\\ 0& 0& \mathit{\rho \kappa }& -g_4& 0& 0& 0\\ 0& 0& \mathit{q\kappa }& 0& -g_5& 0& 0\\ 0& 0& 0& {\delta }_1& {\delta }_2& -g_6& 0\\ 0& 0& 0& {\gamma }_1& {\gamma }_2& {\gamma }_3& -\mu \end{array}\right)$$ (3.8)

with.

$g_1=\sigma +\mu$, $g_2=\theta +\mu ,$ $g_3=\kappa +\mu$, $g_4={\delta }_1+{\gamma }_1+d_c+\mu$, $g_5={\delta }_2+{\gamma }_2+d_c+\mu$, $g_6={\gamma }_3+d_c+\mu$.

We apply the method as presented in [37] to evaluate eq. (3.8) linearized at the DFE. Our concern is to prove that some of the eigenvalues of eq. (3.8) are negative. The eq. (3.8) includes diagonal elements that can be read off the diagonal. The sixth column and the seventh column are negative, and are $-g_6$ and $-\mu$. Then, eq. (3.8) is now transformed into the form

$$J\left({\mathcal{E}}_1\right)=\left(\begin{array}{ccccc}-g_1& \theta & 0& \frac{-\beta \left(1-\xi \right)g_2}{g_2+\sigma }& \frac{-\beta \left(1-\xi \right)\eta g_2}{g_2+\sigma }\\ \sigma & -g_2& 0& \frac{-\beta \left(1-\xi \right)\mathit{\sigma ϵ}}{g_2+\sigma }& \frac{-\beta \left(1-\xi \right)\mathit{\sigma ϵ\eta }}{g_2+\sigma }\\ 0& 0& -g_3& \frac{\beta \left(1-\xi \right)\left(g_2+\mathit{\sigma ϵ}\right)}{g_2+\sigma }& \frac{\beta \left(1-\xi \right)\eta \left(g_2+\mathit{\sigma ϵ}\right)}{g_2+\sigma }\\ 0& 0& \mathit{\rho \kappa }& -g_4& 0\\ 0& 0& \mathit{q\kappa }& 0& -g_5\end{array}\right)$$

The transformed $J\left({\mathrm{ℰ}}_1\right)$ is reduced again to the following sub-matrices below.

$$J\left({\mathrm{ℰ}}_2\right)=\left(\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right)$$

where

$$\begin{array}{c}{A}_1=\left(\begin{array}{cc}-g_1& \theta \\ \sigma & -g_2\end{array}\right),A_2=\left(\begin{array}{ccc}0& \frac{-\beta \left(1-\xi \right)g_2}{g_2+\sigma }& \frac{-\beta \left(1-\xi \right)\eta g_2}{g_2+\sigma }\\ 0& \frac{-\beta \left(1-\xi \right)\mathit{\sigma ϵ}}{g_2+\sigma }& \frac{-\beta \left(1-\xi \right)\mathit{\sigma ϵ\eta }}{g_2+\sigma }\end{array}\right),\\ A_3=\left(\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right),\mathit{and}{A}_4=\left(\begin{array}{ccc}-g_3& \frac{\beta \left(1-\xi \right)\left(g_2+\mathit{\sigma ϵ}\right)}{g_2+\sigma }& \frac{\beta \left(1-\xi \right)\eta \left(g_2+\mathit{\sigma ϵ}\right)}{g_2+\sigma }\\ \mathit{\rho \kappa }& -g_4& 0\\ \mathit{q\kappa }& 0& -g_5\end{array}\right)\end{array}$$

Since the elements of matrix $A_2$ are null and the number of rows does not match the number of columns in matrix $A_3$, then the $\det \left(A_1-\mathit{\lambda I}\right)$ multiplied by the $\det \left(A_4-\mathit{\lambda I}\right)$ is equal to 0. We implement the Routh-Hurwitz stability criterion on matrices $A_1$ and $A_4$ separately, as presented by [37], to prove that the traces of matrices $A_1$ and $A_4$ are less than 0, the $\det \left(A_1\right)>0$ and the $\det \left(A_4\right)<0$.

$$A_1=\left(\begin{array}{cc}-g_1& \theta \\ \sigma & -g_2\end{array}\right)$$

$\mathit{Trace}\left(A_1\right)=-g_1-g_2<0$ and $\mathit{det}\left(A_1\right)=g_1{g}_2-\mathit{\sigma \theta }>0$

$$A_4=\left(\begin{array}{ccc}-g_3& \frac{\beta \left(1-\xi \right)\left(g_2+\sigma \mathrm{ϵ}\right)}{g_2+\sigma }& \frac{\beta \left(1-\xi \right)\eta \left(g_2+\sigma \mathrm{ϵ}\right)}{g_2+\sigma }\\ \mathit{\rho \kappa }& -g_4& 0\\ \mathit{q\kappa }& 0& -g_5\end{array}\right)$$

$$\mathit{Trace}\left(A_4\right)=-\left(g_3+g_4+g_5\right)<0$$

$$\begin{array}{c}\mathit{det}\left(A_4\right)=\frac{\beta \left(1-\xi \right)\left(g_2+\mathit{\sigma ϵ}\right)\mathit{\rho \kappa }{g}_5}{g_2+\sigma }+\frac{\beta \left(1-\xi \right)\eta \left(g_2+\mathit{\sigma ϵ}\right)\mathit{q\kappa }{g}_4}{g_2+\sigma }-g_3{g}_4{g}_5\\ ={\mathcal{R}}_0-1\end{array}$$

Therefore, for all values of ${\mathrm{ℛ}}_0$ less than 1, the determinant of matrix $A_4$ will always be less than 0. Hence, the system (2.3) exhibits local stability whenever ${\mathrm{ℛ}}_0$ is less than 1 or otherwise.

Existence of endemic equilibrium point(EEP) of the model

Let the EEP of system (2.3) be represented by

$${\mathrm{ℰ}}_p=\left(S^{\ast \ast },V^{\ast \ast },E^{\ast \ast },I^{\ast \ast },I_v^{\ast \ast },H^{\ast \ast },R^{\ast \ast }\right).$$

The solutions of the steady state variables of system (2.3) are:

$$\begin{array}{c}{S}^{\ast \ast }=\frac{\Lambda \left(g_2+\mathit{ϵ\lambda }\right)}{g_1\mathit{ϵ\lambda }+g_2\lambda +ϵ{\lambda }^2+g_1{g}_2-\mathit{\sigma \theta }},V^{\ast \ast }=\frac{\mathit{\Lambda \sigma }}{g_1\mathit{ϵ\lambda }+g_2\lambda +ϵ{\lambda }^2+g_1{g}_2-\mathit{\sigma \theta }},\\ E^{\ast \ast }=\frac{\Lambda \left(\mathit{\sigma ϵ\lambda }+g_2\lambda +\mathit{ϵ\lambda }\ast 2\right)}{g_1{g}_2{g}_3+g_1{g}_3\mathit{ϵ\lambda }+g_2{g}_3\lambda +g_3ϵ{\lambda }^2-\mathit{\sigma \theta }{g}_3},I^{\ast \ast }=\frac{\mathit{\Lambda \kappa \rho }\left(\mathit{\sigma ϵ\lambda }+g_2\lambda +\mathit{ϵ\lambda }\ast 2\right)}{g_1{g}_2{g}_3{g}_4+g_1{g}_3{g}_4\mathit{ϵ\lambda }+g_2{g}_3{g}_4\lambda +g_3{g}_4ϵ{\lambda }^2-\mathit{\sigma \theta }{g}_3{g}_4},\\ I_v^{\ast \ast }=\frac{\mathit{\Lambda \kappa q}\left(\mathit{\sigma ϵ\lambda }+g_2\lambda +\mathit{ϵ\lambda }\ast 2\right)}{g_1{g}_2{g}_3{g}_5+g_1{g}_3{g}_5\mathit{ϵ\lambda }+g_2{g}_3{g}_4\lambda +g_3{g}_5ϵ{\lambda }^2-\mathit{\sigma \theta }{g}_3{g}_5},H^{\ast \ast }=\frac{\mathit{\Lambda \kappa }\left(\rho g_5{\delta }_1+q{g}_4{\delta }_2\right)\left(\mathit{\sigma ϵ\lambda }+g_2\lambda +ϵ{\lambda }^2\right)}{g_3{g}_4{g}_5{g}_6\left(g_1\mathit{ϵ\lambda }+g_2\lambda +ϵ{\lambda }^2+g_1{g}_2-\mathit{\sigma \theta }\right)},\\ R^{\ast \ast }=\frac{\mathit{\Lambda \kappa }\left(\rho g_5{g}_6{\gamma }_1+q{g}_4{g}_6{\gamma }_2+\rho g_5{\gamma }_3{\delta }_1+q{g}_4{\gamma }_3{\delta }_2\right)\left(\mathit{\sigma ϵ\lambda }+g_2\lambda +ϵ{\lambda }^2\right)}{\mu g_3{g}_4{g}_5{g}_6\left(g_1\mathit{ϵ\lambda }+g_2\lambda +ϵ{\lambda }^2+g_1{g}_2-\mathit{\sigma \theta }\right)},\\ T^{\ast \ast }=\frac{\Lambda (\mu g_3{\gamma }_4{\gamma }_5{\gamma }_6\left(\sigma +g_2+\mathit{ϵ\lambda }\right)+B\left(g_2\lambda +\mathit{ϵ\lambda }\left(\sigma +\lambda \right)\right)}{\mu g_3{g}_4{g}_5{g}_6\left(\left(g_2+\mathit{ϵ\lambda }\right)\left(g_2+\lambda \right)-\mathit{\sigma \theta }\right)},\end{array}$$

where

$$B=\left(\mathit{\kappa \rho }{g}_5\left(g_6\left(\mu +{\gamma }_1\right)+\left(\mu +{\gamma }_3\right){\delta }_1\right)+g_4\left(g_6\left(\mu g_5+\mathit{q\kappa }\left(\mu +{\gamma }_2\right)\right)+\mathit{q\kappa }\left(\mu +{\gamma }_3\right){\delta }_2\right)\right).$$

Inputting the values of $I^{\ast \ast }$,$I_v^{\ast \ast }$ and $T^{\ast \ast }$ into eq. (2.1), we have that

$$\begin{array}{l}{A}_0{\left({\lambda }^{\ast \ast }\right)}^2+A_1\left({\lambda }^{\ast \ast }\right)+A_2=0\end{array}$$ (3.9)

where

$$\begin{array}{c}{A}_0=\mathit{\mu \rho \kappa ϵ}{g}_5{g}_6+\mathit{\kappa \rho }{\gamma }_1ϵg_5{g}_6+\mathit{\kappa \rho \mu }{\delta }_1ϵg_5+\mathit{\kappa \rho }{\gamma }_3{\delta }_1ϵg_5+\mathit{\mu ϵ}{g}_4{g}_5{g}_6+\mathit{\mu q\kappa ϵ}{g}_4{g}_6+\mathit{\kappa q}{\gamma }_2ϵg_4{g}_6+\\ \mathit{\kappa q\mu }{\delta }_2ϵg_4+\mathit{\kappa q}{\gamma }_3{\delta }_2ϵg_4,A_1=\mathit{\mu \rho \kappa }{g}_2{g}_5{g}_6+\mathit{\mu \rho \kappa \sigma ϵ}{g}_5{g}_6+\mathit{\kappa \rho }{\gamma }_1{g}_2{g}_5{g}_6+\mathit{\kappa \rho }{\gamma }_1\mathit{\sigma ϵ}{g}_5{g}_6+\mathit{\mu \kappa \rho }{\delta }_1{g}_2{g}_5+\\ \mathit{\mu \kappa \rho }{\delta }_1\mathit{\sigma ϵ}{g}_5+\mathit{\kappa \rho }{\gamma }_3{\delta }_1{g}_2{g}_5+\mathit{\kappa \rho }{\gamma }_3{\delta }_1\mathit{\sigma ϵ}{g}_5+\mu g_2{g}_4{g}_5{g}_6+\mathit{\mu \sigma ϵ}{g}_4{g}_5{g}_6+\mathit{\mu q\kappa }{g}_2{g}_4{g}_6+\mathit{\mu q\kappa \sigma ϵ}{g}_4{g}_6+\\ \mathit{\kappa q}{\gamma }_2{g}_2{g}_4{g}_6+\mathit{\kappa q}{\gamma }_2\mathit{\sigma ϵ}{g}_4{g}_6+\mathit{\mu \kappa q}{\delta }_2{g}_2{g}_4+\mathit{\mu \kappa q}{\delta }_2\mathit{\sigma ϵ}{g}_4+\mathit{\kappa q}{\gamma }_3{\delta }_2{g}_2{g}_4+\mathit{\kappa q}{\gamma }_3{\delta }_2\mathit{\sigma ϵ}{g}_4+\mathit{\mu ϵ}{g}_3{g}_4{g}_5{g}_6-\\ \mathit{\mu \kappa q\eta \beta }\left(1-\xi \right)ϵg_4-\mathit{\mu \kappa \rho \beta }\left(1-\xi \right)ϵg_5{g}_6,A_2=\mu g_6\left[1-{\mathcal{R}}_0\right]\end{array}$$

When the disease mortality rate is not equal to zero, if ${\mathrm{ℛ}}_0$<1 and $A_1>0$, then there is no positive root suggesting that, there is no EEP and only the DFE will exist.

Global stability of the DFE

Theorem 3.2

The model (2.3) remains globally stable whenever the basic reproduction number ${\mathrm{ℛ}}_0\le 1$.

Proof. Considering the Lyapunov function by adopting the methods of [52,53], where the coefficients of the infected classes are $A=\frac{\left(g_4\mathit{\kappa q\eta }+g_5\mathit{\kappa \rho }\right)}{g_3{g}_4\eta }$, $B=\frac{g_5}{g_4\eta }$, C = 1, and D = 0. We have that

$$\begin{array}{c}\mathcal{V}=\mathit{AE}+\mathit{BI}+C{I}_v+\mathit{DH}\\ \dot{\mathcal{V}}=\frac{\left(g_4\mathit{\kappa q\eta }+g_5\mathit{\kappa \rho }\right)}{g_3{g}_4\eta }E\left[\mathit{\lambda S}+\mathit{ϵ\lambda V}-g_{3}E\right]+\frac{g_5}{g_4\eta }\left[\mathit{\rho \kappa E}-g_{4}I\right]+\mathit{q\kappa E}-g_5{I}_v\\ \dot{\mathcal{V}}=\frac{\left(g_4\mathit{\kappa q\eta }+g_5\mathit{\kappa \rho }\right)}{g_3{g}_4\eta }E-\frac{g_5}{\eta }\left[I+I_v\right]\\ \dot{\mathcal{V}}=\frac{\left(I+\eta I_v\right)}{\eta g_3{g}_4}\left[\frac{\beta \left(1-\xi \right)\kappa (g_4\mathit{q\eta }+g_5\rho }{g_2+\sigma }-g_3{g}_4{g}_5\right]\\ \dot{\mathcal{V}}\le \frac{\left(I+I_v\right)}{\eta g_3{g}_4}\left({\mathcal{R}}_0-1\right)\end{array}$$

The DFE is globally stable as long as ${\mathrm{ℛ}}_0\le 1$ and not stable otherwise. When ${\mathrm{ℛ}}_0<1$, $\dot{\mathcal{V}}<0$ and if ${\mathrm{ℛ}}_0=1$, $\dot{\mathcal{V}}=0$.

Numerical simulation and discussion

Uncertainty and global sensitivity analysis

Since uncertainties are anticipated to occur from parameter estimate in the model's numerical simulations, so we adopted and applied Latin Hypercube Sampling (LHS) on the model's parameter as presented by [54]. For the purpose of sensitivity analysis, the Partial Ranked Correlation Coefficient was computed between the parameter values in the output function, and the results of the output function are deduced from the global sensitivity analysis. 1000 simulations were run for the model.

The parameters values of system (2.3) in Table (2) were used for the numerical assessments of the system. It is germane we establish here that the most dominant and common PRCC parameter value that is positively correlated and also influences the model's dynamics in Table (3) and Table (4) is the Covid-19 transmission rate $\left(\beta \right)$. This actually suggests that measures should be taken to reduce or minimize the Covid-19 transmission rate. By making use of compartment $\left(E\right)$ as the output function in Fig. (2(a)) and Table (3), we observed that the vaccine inefficacy $\left(\mathrm{ϵ}\right)$ drives the dynamics of the model. This suggests that vaccines with minimal inefficacy, designed to provide better protection against Covid-19, should be administered to individuals. Also, using the compartment $\left(I\right)$ as the output function in Fig. (2(b)) and Table (3), the parameter values influencing the model's dynamics are the incubation period $\left(\kappa \right)$ and the fraction of unvaccinated Covid-19 infected individuals $\left(\rho \right)$. Based on Fig. (2(c)) and Table (3), the parameter that is positively correlated and also influences the dynamics of the system is the incubation period $\left(\kappa \right)$, when compartment $\left(I_v\right)$ is used as the response function. Also, when compartment $H$ is used as the response function, looking at Fig. (2(d)) and Table (3), the PRCC-ranked parameter values that impact the model's dynamics are the incubation period $\left(\kappa \right)$, the isolation rate $\left({\delta }_1\right)$ for unvaccinated Covid-19 infected individuals, and the isolation rate $\left({\delta }_2\right)$ for vaccinated Covid-19 infected individuals, respectively. This implies that after isolation, proper medical care and attention should be given to these infected individuals for swift recovery. The main PRCC parameter value positively associated with the effective reproduction number $\left({\mathrm{ℛ}}_0\right)$, as the output function from Fig. (3(a)) and Table (3) as earlier stated, is the Covid-19 transmission rate $\left(\beta \right)$. When the total infected population was taken as the response function, we deduced that the parameter values that are positively impacting the model's dynamics, according to Fig. (3(b)) and Table (4) are the isolation rate $\left({\delta }_1\right)$ for unvaccinated Covid-19 infected individuals and the isolation rate $\left({\delta }_2\right)$ for vaccinated Covid-19 infected individuals, respectively.

We observed that by varying the recovery rates $\left({\gamma }_1,{\gamma }_2\right)$ for compartments $I$ and $I_v$, according to Fig. (4(a)) and (4(b)), there is a significant reduction in the total incidence of the disease. Also, varying the recovery rate $\left({\gamma }_3\right)$ for compartment $H$ in Fig. (5(a)) results in a notable decrease in the cumulative incidence of the disease. According to Fig. (5(b)), there is an aggregate increase in the incidence of the disease due to vaccine inefficacy.

The contour plots in Fig. (6(a)) and Fig. (6(b)) show that increasing the vaccination rate $\left(\sigma \right)$ and the non-pharmaceutical intervention rate $\left(\xi \right)$ against the Covid-19 transmission rate $\left(\beta \right)$ results in an associated decrease in the value of the effective reproduction number (${\mathrm{ℛ}}_0$). We also discovered that by concurrently increasing the vaccination rate $\left(\sigma \right)$ with the non-pharmaceutical intervention rate $\left(\xi \right)$ against the effective reproduction number (${\mathrm{ℛ}}_0$) in Fig. (6(c)), there is a notable decrease in the effective reproduction number (${\mathrm{ℛ}}_0$). This suggests that the continuous vaccination program and implementation of non-pharmaceutical interventions over time may eliminate Covid-19 in the population.

Optimal control analysis

In this subsection, we will apply Pontryagin's Maximum Principle (PMP) to determine the necessary and sufficient conditions for the optimal control of the system as presented by [[55], [56], [57]]. The model (2.3) optimal control analysis considers time-dependent controls $u_1\left(t\right)$, $u_2\left(t\right)$, and $u_3\left(t\right)$, which will help us in obtaining the most effective intervention strategies to combat the circulation of Covid-19. The controls $u_1\left(t\right)$, $u_2\left(t\right)$, and $u_3\left(t\right)$ are assumed to be defined between 0 and 1 and also lebesgue integrable. Control $u_1\left(t\right)$ is geared towards the continuous implementation of preventive measures, which entails the use of face-masks, keeping of physical distance, maintaining personal hygiene, washing hands with sanitizers, etc., to effectively reduce the contact rate. Control $u_2\left(t\right)$ is geared towards the use of the vaccine, in which the vaccine inefficacy is minimal and aims to enhance protection for individuals against Covid-19. Control $u_3\left(t\right)$ is also geared towards isolation and proper medical care for infected individuals for quick recovery. We went further to replace the vaccination rate($\sigma$) with $u_2\left(t\right)$. The optimal control system is given below as:

$$\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=\Lambda +\mathit{\theta V}-\left(1-u_1\right)\mathit{\lambda S}-\left(u_2+\mu \right)S,\\ \frac{\mathit{dV}}{\mathit{dt}}=u_{2}S-\left(1-u_1\right)\mathit{ϵ\lambda V}-\left(\theta +\mu \right)V,\\ \frac{\mathit{dE}}{\mathit{dt}}=\left(1-u_1\right)\mathit{\lambda S}+\left(1-u_1\right)\mathit{ϵ\lambda V}-\left(\kappa +\mu \right)E,\\ \frac{\mathit{dI}}{\mathit{dt}}=\mathit{\rho \kappa E}-\left({\delta }_1+{\gamma }_1+d_c+\mu \right)I,\\ \frac{d{I}_v}{\mathit{dt}}=\left(1-\rho \right)\mathit{\kappa E}-\left({\delta }_2+{\gamma }_2+d_c+\mu \right)I_v,\\ \frac{\mathit{dH}}{\mathit{dt}}={\delta }_{1}I+{\delta }_2{I}_v-\left({\gamma }_3+d_c+\mu +u_3\right)H,\\ \frac{\mathit{dR}}{\mathit{dt}}={\gamma }_{1}I+{\gamma }_2{I}_v+\left({\gamma }_3+u_3\right)H-\mathit{\mu R},\end{array}$$ (4.10)

which is subjected to the starting conditions $S=S\left(0\right)$, $V=V\left(0\right)$, $E=E\left(0\right)$, $I=I\left(0\right)$, $I_v=I_v\left(0\right)$, $H=H\left(0\right)$, $R=R\left(0\right)$.

Considering the objective functional:

$$\mathcal{J}\left[u_1,u_2,u_3\right]={\int }_0^N\left[E\left(t\right)+I\left(t\right)+I_v\left(t\right)+H\left(t\right)+\frac{w_1}{2}{u}_1^2+\frac{w_2}{2}{u}_2^2+\frac{w_3}{2}{u}_3^2\right]\mathit{dt},$$ (4.11)

where $w_1$, $w_2$, and $w_3$ represent the weight constants for implementing the controls, and it is assumed that they are all greater than zero, with N denoting the final time.

The optimal control is evaluated in a way in which

$$\mathcal{J}\left(u_1^{\ast },u_2^{\ast },u_3^{\ast }\right)=\mathit{min}\left\{\mathcal{J}\left(u_1,u_2,u_3\right)|u_1,u_2,u_3\in \mathbb{U}\right\},$$ (4.12)

with $\mathbb{U}=\left\{\left(u_1,u_2,u_3\right)\right\}$ is the control set and $u_1^{\ast },u_2^{\ast },u_3^{\ast }$ are measurable such that $0\le u_1^{\ast },u_2^{\ast },u_3^{\ast }\le 1$ for $t\in \left[0,N\right]$. The PMP transforms eqs. (10), (11), and (12) into a problem focused on minimizing the Hamiltonian point-wisely with respect to the control variables $u_1$,$u_2$, and $u_3$. Therefore, the Hamiltonian is given as:

$$\begin{array}{c}\mathcal{H}=E\left(t\right)+I\left(t\right)+I_v\left(t\right)+H\left(t\right)+\frac{w_1}{2}{u}_1^2+\frac{w_2}{2}{u}_2^2+\frac{w_3}{2}{u}_3^2\\ +{\lambda }_1\left(\Lambda +\mathit{\theta V}-\left(1-u_1\right)\mathit{\lambda S}-\left(u_2+\mu \right)S\right)\\ +{\lambda }_2\left(u_{2}S-\left(1-u_1\right)\mathit{ϵ\lambda V}-\left(\theta +\mu \right)V\right)\\ +{\lambda }_3\left(\left(1-u_1\right)\mathit{\lambda S}+\left(1-u_1\right)\mathit{ϵ\lambda V}-\left(\kappa +\mu \right)E\right)\\ +{\lambda }_4\left(\mathit{\rho \kappa E}-\left({\delta }_1+{\gamma }_1+d_c+\mu \right)I\right)\\ +{\lambda }_5\left(\left(1-\rho \right)\mathit{\kappa E}-\left({\delta }_2+{\gamma }_2+d_c+\mu \right)I_v\right)\\ +{\lambda }_6\left({\delta }_{1}I+{\delta }_2{I}_v-\left({\gamma }_3+d_c+\mu +u_3\right)H\right)\\ +{\lambda }_7\left({\gamma }_{1}I+{\gamma }_2{I}_v+\left({\gamma }_3+u_3\right)H-\mathit{\mu R}\right).\end{array}$$ (4.13)

Theorem 4.1

For a control set $u_1,u_2,u_3$ that optimally minimizes the objective functional $\mathcal{J}$ over the domain $\mathbb{U}$, there are adjoint variables ${\lambda }_1,\dots ,{\lambda }_7$ satisfying -$\frac{\partial {\lambda }_J}{\partial t}=\frac{\mathrm{\mathscr{H}}}{{\partial }_j}$ and with transversality condition ${\lambda }_j\left(t_f\right)=0$, where $j=S,V,E,I,I_v,H,R$.

In addition,

$$\begin{array}{c}{u}_1^{\ast }=\max \left\{0,\min \left(1,\frac{\left(I+\eta I_v\right)\left(\xi -1\right)\beta }{w_{1}T}\left(\left({\lambda }_1-{\lambda }_3\right)S+ϵ\left({\lambda }_2-{\lambda }_3\right)V\right)\right)\right\}\\ u_2^{\ast }=\max \left\{0,\min \left(1,\frac{\left({\lambda }_1-{\lambda }_2\right)S}{w_2}\right)\right\}\\ u_3^{\ast }=\max \left\{0,\min \left(1,\frac{\left({\lambda }_6-{\lambda }_7\right)H}{w_3}\right)\right\}\end{array}$$ (4.14)

Proof. Assume that $\mathbb{U}=\left(u_1^{\ast },u_2^{\ast },u_3^{\ast }\right)$ is an optimal control with associated state solutions $\left(S^{\ast \ast },V^{\ast \ast },E^{\ast \ast },I^{\ast \ast },I_v^{\ast \ast },H^{\ast \ast },R^{\ast \ast }\right)$. Then, adopting the method of PMP, there are adjoint variables satisfying:

$$\begin{array}{c}-\frac{d{\lambda }_S}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial S},{\lambda }_S\left(t_f\right)=0,\\ -\frac{d{\lambda }_V}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial V},{\lambda }_V\left(t_f\right)=0,\\ -\frac{d{\lambda }_E}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial E},{\lambda }_E\left(t_f\right)=0,\\ -\frac{d{\lambda }_I}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial I},{\lambda }_I\left(t_f\right)=0\\ -\frac{d{\lambda }_{I_v}}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial I_v},{\lambda }_{I_v}\left(t_f\right)=0,\\ -\frac{d{\lambda }_H}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial H},{\lambda }_H\left(t_f\right)=0\\ -\frac{d{\lambda }_R}{\mathit{dt}}=\frac{\partial \mathcal{H}}{\partial R},{\lambda }_R\left(t_f\right)=0,\end{array}$$ (4.15)

and the transversality conditions; ${\lambda }_S\left(t_f\right)={\lambda }_V\left(t_f\right)={\lambda }_E\left(t_f\right)={\lambda }_I\left(t_f\right)={\lambda }_{I_v}\left(t_f\right)={\lambda }_H\left(t_f\right)={\lambda }_R\left(t_f\right)=0$. We can understand the dynamics of the control by differentiating the Hamiltonian $\left(\mathrm{\mathscr{H}}\right)$ in respect with the controls $\left(u_1,u_2,u_3\right)$ at t. On the interior of the control set, with $0<i<1$ for all $\left(i=\mathrm{1,2,3}\right)$, we have that

$$\begin{array}{c}0=\frac{\partial \mathcal{H}}{\partial u_1}=u_1^{\ast }{w}_1+\frac{\left(I+\eta I_v\right)\left(\xi -1\right)\beta }{T}(\left({\lambda }_3-{\lambda }_1\right)S+\frac{ϵ\left(I+\eta I_v\right)\left(\xi -1\right)\beta }{T}(\left({\lambda }_3-{\lambda }_1\right)V\\ 0=\frac{\partial \mathcal{H}}{\partial u_2}=u_2^{\ast }{w}_2+{\lambda }_{2}S-{\lambda }_{1}S\\ 0=\frac{\partial \mathcal{H}}{\partial u_3}=u_3^{\ast }{w}_3+{\lambda }_{7}H-{\lambda }_{6}H\end{array}$$ (4.16)

Therefore,

$$\begin{array}{c}{u}_1^{\ast }=\frac{\left(I+\eta I_v\right)\left(\xi -1\right)\beta }{w_{1}T}\left(\left({\lambda }_1-{\lambda }_3\right)S+ϵ\left({\lambda }_2-{\lambda }_3\right)V\right)\\ u_2^{\ast }=\frac{\left({\lambda }_1-{\lambda }_2\right)S}{w_2}\\ u_3^{\ast }=\frac{\left({\lambda }_6-{\lambda }_7\right)H}{w_3}\end{array}$$ (4.17)

$$\begin{array}{c}{u}_1^{\ast }=\max \left\{0,\min \left(1,\frac{\left(I+\eta I_v\right)\left(\xi -1\right)\beta }{w_{1}T}\left(\left({\lambda }_1-{\lambda }_3\right)S+ϵ\left({\lambda }_2-{\lambda }_3\right)V\right)\right)\right\}\\ u_2^{\ast }=\max \left\{0,\min \left(1,\frac{\left({\lambda }_1-{\lambda }_2\right)S}{w_2}\right)\right\}\\ u_3^{\ast }=\max \left\{0,\min \left(1,\frac{\left({\lambda }_6-{\lambda }_7\right)H}{w_3}\right)\right\}\end{array}$$ (4.18)

□

The simulations executed on the system (4.10) optimal control, the control characterizations (4.14) and the adjoint eqs. (4.15) are executed in MATLAB by employing the forward-backward sweep method as presented by [56]. The scheme makes use by combining the forward application of the same approach to the adjoint variables.

When the control $u_1\ne 0$ which is geared towards the continuous use of preventive measures was implemented in compartments $E$ and $I$, according Fig. (7(a)) and Fig. (7(b)), there is a substantial decrease in the number of Covid-19 infected individuals. However, without control measures $\left(u_1=u_2=u_3=0\right)$ in place, we noticed a substantial rise in the number of people infected with Covid-19. Also, When the control $u_1\ne 0$ which is geared towards the continuous use of preventive measures was implemented in compartment $I_v$ and $H$, according Fig. (8(a)) and Fig. (8(b)), there is a corresponding decrease in the number of Covid-19 infected individuals, while in the absence of control measures $\left(u_1=u_2=u_3=0\right)$ we observed a marked rise in the number of infected individuals. The implication for public health is that the continuous implementation of preventive measures will lead to a substantial decline in the number of individuals infected within the population.

When the three control strategies $\left(u_1=u_2=u_3\ne 0\right)$ are implemented in compartments $E$ and $I$ in Fig. (9(a)) and (9(b)), there is really no significant difference in the graphs when implementing control $u_1\ne 0$ in the both compartments. This is pointing to the fact that the continuous use of preventive measures should not be jettisoned. The impact of the three control strategies simultaneously executed in minimizing the number of infected individuals in compartments $I_v$ and $H$, according to Fig. (10(a)) and Fig. (10(b)) is highly significant. There is a significant decline in the cumulative number of vaccinated Covid-19 infected individuals and isolated individuals who are infected with Covid-19.

The control profile for the intervention strategy $\left(u_1\right)$ in Fig. (11(a)) is stable for 140 days before the haphazard movement during the simulation. This could be attributed to the relaxation in the proper use of preventive strategies to curb the circulation of Covid-19. As shown in Fig. (11(b)), the intervention strategy of the control profile $\left(u_2\right)$, which is focused on vaccine use, where vaccine inefficacy is minimal and aims to provide maximum protection against Covid-19, remains at its peak throughout the simulation. it implies that vaccines offering the least amount of ineffectiveness while delivering optimal protection against Covid-19 should be administered. Also, looking at Fig. (11(c)), the control profile $\left(u_3\right)$ focused on isolation and providing appropriate medical care for infected individuals to ensure a swift recovery, rises and falls within the first 10 days to attain its peak value and maintained that peak value for about 190 days during the course of the simulation.

Conclusion

Conclusively, a Covid-19 dynamical model was designed to assess the implications of vaccination and the continuous use of non-pharmaceutical interventions strategy on the population dynamics of Covid-19 with optimal control. The control reproduction number was calculated and employed to establish the stability of the equilibria. We carried out numerical assessments utilizing Covid-19 data published in literature. Numerical results depicted that by varying the different recovery rates $\left({\gamma }_1,{\gamma }_2,{\gamma }_3\right)$, there were corresponding decreases in Covid-19 cumulative incidence. The $3D$ contour plots also revealed that by varying the use of the non-pharmaceutical intervention rate and the vaccination rate $\left(\xi ,\sigma \right)$ against the basic reproduction number, there was significant decrease in the value of the basic reproduction number. Simultaneously varying the use of non-pharmaceutical intervention rate and vaccination rate $\left(\xi ,\sigma \right)$ against the effective reproduction number, there was also a significant decrease. Based on the global sensitivity analysis, there were key parameters steering the dynamics of the model which include the Covid-19 transmission rate $\left(\beta \right)$, the vaccine inefficacy $\left(\mathrm{ϵ}\right)$, incubation period $\left(\kappa \right)$, the fraction of unvaccinated infected individuals with Covid-19 $\left(\rho \right)$, the isolation rate $\left({\delta }_1\right)$ for unvaccinated infected individuals with Covid-19, and the isolation rate $\left({\delta }_2\right)$ for vaccinated and Covid-19 infected individuals. The control profile $u_1$ which is geared towards the continuous use of preventive measures such as the use of face-masks, keeping of physical distance, maintaining personal hygiene, washing hands with sanitizer, etc., to effectively reduce the contact rate with infected individuals is stable for 140 days before the haphazard movement. The control $u_2$ geared towards the use of the vaccine, in which the vaccine inefficacy is minimal and aims to enhance protection for individuals against Covid-19 remains at its peak value through out the simulation period. The control $u_3$ geared towards isolation and proper medical care for quick recovery of infected individuals was falling and rising for like 10 days and now attained a peak value which was maintained for about 190 days. We observed that implementing the three control variables in compartments ($I_v$ and $H$), there was a decrease in the number of infected individuals. The research findings showed that vaccines with the least level of ineffectiveness while providing the best protection against Covid-19 should be administered. It was also observed that the continuous implementation of preventive measures would significantly reduce the overall incidence of Covid-19. Additionally, infected individuals in isolation shoulb receive appropriate medical treatment for faster recovery. We want to state clearly that, the data employed for the model's simulations are not based on real-world data but were carefully sourced from literature.

CRediT authorship contribution statement

Evans O. Omorogie: Writing – original draft, Methodology, Formal analysis, Conceptualization. Kolade M. Owolabi: Writing – original draft, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Bola T. Olabode: Writing – review & editing, Supervision, Investigation, Conceptualization. Tunde T. Yusuf: Writing – review & editing, Supervision, Investigation, Conceptualization. Edson Pindza: Writing – review & editing, Supervision, Investigation, Formal analysis.

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.