Global Analysis and Optimal Control Model of COVID-19
COVID-19 remains the concern of the globe as governments struggle to defeat the pandemic. Understanding the dynamics of the epidemic is as important as detecting and treatment of infected individuals. Mathematical models play a crucial role in exploring the dynamics of the outbreak by deducing strategies paramount for curtailing the disease. The research extensively studies the SEQIAHR compartmental model of COVID-19 to provide insight into the dynamics of the disease by underlying tailored strategies designed to minimize the pandemic. We first studied the noncontrol model’s dynamic behaviour by calculating the reproduction number and examining the two nonnegative equilibria’ existence. The model utilizes the Castillo-Chavez method and Lyapunov function to investigate the global stability of the disease at the disease-free and endemic equilibrium. Sensitivity analysis was carried on to determine the impact of some parameters on . We further examined the COVID model to determine the type of bifurcation that it exhibits. To help contain the spread of the disease, we formulated a new SEQIAHR compartmental optimal control model with time-dependent controls: personal protection and vaccination of the susceptible individuals. We solved it by utilizing Pontryagin’s maximum principle after studying the dynamical behaviour of the noncontrol model. We solved the model numerically by considering different simulation controls’ pairing and examined their effectiveness.
The unusual, life-threatening pneumonia condition affecting humanity remains the globe’s concern as governments struggle to defeat the pandemic. COVID-19, one of the most media campaigned viral diseases, emerged from Wuhan’s city in China in the last quarter of 2019. The epidemic has transcended the nations’ regions, and all sectors of the global economy have felt its adverse effects . The epidemic has been the source of setbacks in businesses, disruption in academic calendars, and closure of production companies and public services . The campaign against the disease has intensified due to the number of case counts and deaths recorded each day globally. The primary mode of transmissibility of the disease is airborne. Thus, individuals become infected by contact with the contaminated sneezes and droplets of the virus . The documented number of COVID-19-related deaths and cases in the first quarter of the year 2020 caused governments to respond accordingly by promoting and observing self-protective protocols and restrictions directed by the World Health Organization. However, the protocols and restrictions were gradually eased when evidence of the disease waning was confirmed by governments . Notably, the vast majority of the European countries are currently battling with the second wave of COVID-19 outbreaks after successfully minimizing COVID-19 disease in the early months of the year 2020 . The second wave has caused the European countries’ governments to respond to the second spike by introducing new measures that may help manage the disease and prevent the further spread of the outbreak. These measures range from strict new restrictions, such as minilockdown and compulsory mask wearing, to relaxed regulations, allowing the opening of bars and pubs with a set time for closure. These recommended restrictions vary from country to country regarding the number of cases and infection rates .
Since the detection of the virus in Wuhan, variants of COVID-19 mathematical models have been considered that have helped shape the pandemic, with inked preventive measures which could be adopted to flatten the curve, estimating the final epidemic size and prevent further infection (see [7–23]).
Mathematical modelling of infectious diseases has been an experimenting station where the vast knowledge of epidemic outbreaks, transmission dynamics, intervention, and alternative courses of action for controlling diseases are discovered. Integrating mathematical computation into the modelling of infectious diseases has evolved in an unparalleled achievement regarding intervention strategies and has been the spine of all explorations on infectious diseases .
In their paper, Hellewell et al.  assessed the effectiveness of a stochastic transmission model to control the new SARS-CoV-2 disease by utilizing the preventive measures of isolation and contact tracing. Qianying et al. , in their work, constructed SEIRNDC compartmental model of coronavirus-2 in Wuhan that examines the government’s responses towards the disease and the reaction from the people. The authors in  considered an eight compartmental nonlinear differential equation model of COVID-19 that uses time-dependent diagnosis and contact rates to refit their existing SEIR compartmental model  to a newly available data for better estimation of the basic reproduction number. In a related paper, Kang et al.  examined the COVID-19 epidemic dynamics in China. The authors applied Moran’s I spatial statistics in a test to ascertain the possibility of the disease’s spatial association. In , in their paper, the author adapted the growth model to study the spread pattern of MERS, COVID-19, and SARS by using the inhibition and growth rates to establish the model’s propagation. Benvenuto et al.  predicted the spread and trend of COVID-19 disease by using the ARIMA model. In a paper related to the current studies, Li et al.  investigated the COVID-19 transmission process using the official modelling data. The study examines the error between the model and the official data. In , in their work, the authors calibrated a COVID-19 epidemic model to deduce various characteristics such as age-dependent attack rates, length of incubation, generation periods, reproductive number, and growth rate of cumulative incidence. In , in their research, the authors analyzed the spread pattern of the COVID-19 outbreak by using a time series model. The authors in  considered variants of the SIR model with a parameter that factors into account the effects of social distancing. Fu et al.  applied Boltzmann-function-based regression analyses to estimate the number of SARS-CoV-2 confirmed cases in China. Shi et al.  considered a SEIR COVID-19 propagation model that assessed the variation in the length of the incubation period, weakness of the transmission ability of the incubation period, and the intervention of the government to detect and isolate the infected. In , the authors studied and predicted the pattern of the COVID-19 epidemic and estimated the various parameters involved in the model’s analyses. Liu et al.  considered a new QSEIR COVID-19 epidemic model that studied the effect of quarantine measures imposed in Mainland China during the peak of the outbreak. In , in their paper, a stochastic COVID-19 transmission model is examined that analyzed the intervention measures employed in Mainland China. Kissler et al.  applied a mathematical model to prove that the United States’ critical care capacity will not be sufficient to be maintained with a one-time intervention strategy. In , the authors fitted a coronavirus-2 age-structured model to data from six countries affected by the pandemic to determine the age gradient in observed cases. In , the authors studied a SEIR compartmental model of COVID-19, with modifications to account for the spreading of the epidemic in the latent stage and the effects of varying containment’s proportions. Acuna et al.  considered mathematical models which examine the issues related to the spread of SARS-CoV-2 in Mexico and explore the effect of behaviour changes needed to wane the infection. Rong et al.  studied a new dynamical model of SSqEI1I2HRV for SARS-CoV-2 to assess the effect of delay in detecting an infected person. In a related article, Giordano et al. , considered a nonlinear ordinary differential equation model of SIDARTHE compartments that predicts the propagation of COV-19 epidemic in Italy and helps to diffuse the misperceptions of epidemic spread and case fatality rate. In , the authors in their work explored the impact of community mask wearing on COVID-19 transmission dynamics in the United States by formulating a new mathematical model for the assessment.
Asamoah et al.  applied an optimal control theory to nonlinear ordinary differential equations of SEAIRV compartmental model of coronavirus transmission that analyzed the cost-effective strategy of all the proposed methods. In the paper by , the authors constructed and researched a compartmental SIRU model that brings further insight into the propagation of the COVID-19 disease. The SIRU compartmental model is then converted to an optimal control problem, and the numerical solutions are presented. The authors in  formulated a mathematical model for coronavirus-2 disease to estimate the model parameters by fitting it to actual data. The authors further performed optimal control analysis on the modified model. In , the author constructed a mathematical model that is segregated into compartments of SEIRV for the coronavirus- epidemic and converted it to an optimal control problem by characterizing several control strategies by applying the maximum principle. Sasmita et al.  considered SEI1I2RS nonlinear compartmental model of coronavirus-2 infection to predict the disease’s peak in Indonesia. The authors considered five time-dependent controls in constructing the optimal control model to deduce strategies critical for policymakers to curb the pandemic. The authors in  applied an optimal control analysis to a mathematical model of SARS-CoV-19 to help deduce many possible strategies for the control of the disease. In , the authors developed an ordinary differential equation model of SEIRW compartments that describes the COVID-19 disease’s dynamics. The authors made a SEIRW compartmental model modification to convert the dynamical model into a new SEIRW control model to assess the chosen strategies thoroughly.
This research article presents a SEQIAHR compartmental model of COVID-19 to provide insight into the disease’s dynamics by utilizing tailored strategies to minimize the pandemic. The study is motivated by the available COVID-19 works and formulating a new SEQIAHR compartmental optimal control model that would add to the existing knowledge and help improve public health decision-making by providing scientific strategies to prevent the disease.
The remaining work is organized as follows: Section 2 formulates a deterministic SEQIAHR compartmental COVID-19 model. Section 3 studies the global stability of the model, sensitivity and bifurcation analysis. A new SEQIAHR compartmental optimal control model is constructed and analyzed in Section 4 with time-dependent control functions of personal protection and treatment of asymptomatic infected individuals. Finally, Section 5 discusses the simulated results of the models by using an iterative method of Runge-Kutta 4th-order method and Matlab.
2. The Model Formulation
This section formulates a compartmental SEQIAHR transmission model for COVID-19 disease to understand the dynamical behaviour of the disease and the strategy needed in curtailing it. Here, we modify the baseline model of  which is segregated into compartments of: susceptible , exposed , symptomatic infectious , asymptomatic infectious , hospitalized , recovered , and cumulative deaths , by considering recruitment into the population, and natural death rate , and ignoring the cumulative death compartment. Additionally, we assume that COVID-19 has a high level of transmission, and therefore, the main way of preventing the spread of the disease is to adopt a quarantine approach of the exposed individuals. This is considered in the formulation of the model by first modifying the original model system (1) of , with the inclusion of quarantine compartment, by assuming that the exposed individuals are quarantined at rate . Further, the model assumes that the quarantine individuals are hospitalized at a rate of . Also, we assume that the quarantined individuals may die naturally. We assumed a time-dependent population for the modified model. With these assumptions, the modified model equations are given as follows:
with , , , , , , and .
2.1. Analysis of Model: Positivity and Boundedness
Theorem 1. The set being the solution of the state Equation (1) with nonnegative parameters is positive with initial conditions given by the following:
Proof. With the method illustrated by , theorem one can be proofed by adopting the same approach as underlaid below.
We redefine , , and , with denoting the transposition. Hence, COVID-19 model Equation (1) in matrix form is represented as follows: where
In model Equation (1), rewriting the third equation into a first-order nonhomogenous differential equation gives
Now, from (5), adopting the method of integrating factor and applying it provides
Similarly, mimicking the same approach, the fourth equation of model (1) gives
Now, as can be determined in the work of , it follows that by intuition, , at , and , at , for . Hence, following the same steps, the same can be generalized for , , , , and , which ensures that the state variables stays positive in the entire time of the study. In addition, by inspection, it could be verified that the off-diagoanl entries of are nonnegative and , which confirms the property Metzler matrix . Hence, the state model Equation (1) is positively invariant in .
Theorem 2. The nonlinear equation model (1) has solutions bounded within the invariant region, given as
Proof. Here, we denote . Then, the simplified nonlinear differential equation of (1) is given as Then, from Equation (9), setting to be a solution of (9) gives a unique initial value problem Solving Equation (54) produces Hence, from the comparison theorem , it can be seen that Therefore, from (12), it follows that the possible solution set of the state variables is bounded and the state model Equation (1) is positively in variant in , implying that any trajectory with initial condition in will remain in . This guarantees that model (1) is mathematically and epidemiologically feasible and well posed.
2.2. Disease-Free Equilibrium and Reproduction Ratio
The basic reproduction number threshold parameter is key in determining whether an invading infection from an infected individual will have the potency to be endemic or die out in a naive population. It is defined as a new infection from an infected individual who enters into a naive susceptible population. When , , , , , and , then the COVID-19 model (1) has a disease-free equilibrium (DFE), determined by equating the right-hand side of the equations in model (1) to zero and solve for the state variables. Hence,
Hence, the basic reproduction number for the COVID-19 model (1) is obtained by the spectral radius of as follows:
2.3. Existence of Endemic Equilibrium
When , , , , , , and , then it follows that the endemic equilibrium exists. This unique equilibrium point for the COVID-19 model system (1) given by is determined as follows:
3. Disease-Free Equilibrium (DFE) and Its Stability
In this section, the global stability analysis of the COVID-19 model (1) at the disease-free equilibrium is studied. The Castillo-Chavez method  would be used to prove that model (1) is globally asymptotically stable at the disease-free equilibrium. Thus, it follows that in considering the method of Castillo-Chavez, the COVID-19 model (1) can be transformed as follows:
where denotes the uninfected population; thus, , and represents the infected, with, . The disease-free equilibrium point of (1) is given by .
The point is a globally stable asymptotically stable equilibrium for the model (1) provided , and the below criteria are satisfied.
D1. Given , is globally asymptotically stable.
D2. , where for
What happens next is if the model Equation (1) meets the above conditions; then, the following theorem holds.
Theorem 3. The point is globally asymptotically stable equilibrium given that and the conditions D1 and D2 are satisfied.
Proof. Concerning the model Equation (1), we derive and as
It follows that with , , , , and , then becomes
What follows next is that from (19), we notice that as , . Hence, is globally asymptotically stable, which verifies the first condition.
Now, in determining whether condition two would be satisfied, we utilize . And we get where , , , , , , and matrix given by with where . It can be ascertained from model Equation (1) that the total population is bounded by . Therefore, it follows that , and which implies is positive definite. Further, matrix is evidently an M-matrix, with the off-diagonal entries positive. Hence, condition two is satisfied which proves the global asymptotic stability of .
3.1. Endemic Equilibrium (EE) and Its Stability
This subsection presents the global stability analysis of the COVID-19 model (1) at the endemic equilibrium by applying a Lyapunov function theory  for the global stability analysis. The results are given as follows:
Theorem 4. The unique endemic equilibrium for the COVID-19 model (1) is globally asymptotically stable in whenever .
Proof. The Lyapunov function , where for are constants to be chosen in the course of the proof are defined.
The derivative of along the solution of (1) is given by Choosing , , , , , , and such that and gives It follows that is positive definite, and is negative definite. Therefore, the function is a Lyapunov function for model system (1), and by Lyapunov asymptotic stability theorem , the endemic equilibrium is globally asymptotically stable.
3.2. Sensitivity Analysis
Sensitivity analysis is one of the essential subjects that has been explored by many researchers and is of great importance to epidemiological modelling. Sensitivity analysis study assists us in ascertaining parameters that impact the and allows epidemiologists to improve the design of the control strategies. The results from sensitivity index computation indicate the effect of involving parameters that contribute to the spreading of the epidemic and inform us of the relative change of and other parameters.
As noted from Table 2, , , and are the parameters with positive indices contributing to the spreading of the epidemic. The positive parameters contribute to the spreading of the outbreak since they increase the . On the other hand, the parameters with a negative index contribute to controlling the disease since they have reduced the . Further, as noted, the parameter has a sensitivity index of , which implies increasing or decreasing by a specific percentage increases or reduces by the same percentage.
3.3. Bifurcation Analysis
In an attempt to ascertain whether model system (1) exhibits backward bifurcation or not, we analyze model system (1) with the theory of centre manifold as credited to Castillo-Chavez and Song in ( see Theorem 6).
where is continously differentiable at least twice in and is the bifurcation parameters.
Equations and are denoted by as follows: and
are the determinants of the existence of bifurcation in model system (1). Thus, when and , then backward bifurcation exists and occurs at . When and , forward bifurcation exists and occurs at . Now, considering as a bifurcation parameter, then be equivalent to
We alter the initial design of the state model (1) for easy computations as follows; , , , , , , and , such that . Also, and are vector notations of the model system (1) and can be rewritten in the form where
Now, the Jacobian matrix of model system (1) was evaluated at , when is provided as
The Jacobian matrix has a right eigenvector corressponding to a simple zero eigenvalue given by and a left eigenvector corressponding to a simple eigenvalue given by . Deducing the eigenvectors of and gives
The derivation of the nonzero partial derivatives is given by
Hence, we obtain
4. COVID-19 Optimal Control Model
In formulating the optimal control model, we restructure the compartmental model (1) into an optimal control model with admissible controls that are considered to be continuous in time. The controls that are identified for the new structured control model are defined as follows:
F1. The personal protection control rate varies with time and is given by .
F2. The vaccination control rate varies with time and is denoted as .
F3. The considered time is given by , where is the final time and relatively short.
Hence, the new restructured system for Equation (1) is provided below.
with , , , , , , and .
We usher in a measurable control set:
The target of the considered control strategy is to (a)Lower the COVID exposed, asymptomatic, and symptomatic infectious persons(b)Make intervention cost small as possible
The constants , , and are weight related to exposed, symptomatic infectious, and asymptomatic infectious individuals, respectively. Additionally, the weights and are positive and in association with time-dependent control functions , respectively.
The main objective of the control mode is to identify an optimal control pair that makes
The fundamental concept of the optimal control problem requires that we verify the existence and uniqueness of the optimal controls to characterize them.
4.1. Existence of Optimal Controls
Theorem 6. An optimal control duple exists that minimizes the objective functional (40) subject to the system (38), given that the below properties are met. (a)The set of control is convex and closed(b)The system (38) is bounded by a linear function in both the state and control variable(c)The objective functional (40) integrand is convex with respect to the control(d)There exist constants , , and that make the objective functional (40) integrand bounded by the below quantity
Proof. (a)It is sufficient to write by definition of (39). Hence, is bounded and convex . In addition, we choose , so that and . Then, , and we have , satisfying the convexity property of the control set(b)We denote the right side of system (38) and the associated solution by and ; then,where where , , , , , , , and .
Hence, is uniformly Lipschitz continuous. (c)The Langrangian defined as is the integrand of the objective functional (40). Thus, we rewrite in the formwith
and . The convexity of , which is a linear combination of the control function , needs to be proved. We prove the convexity by letting be . Then, and . Hence, it follows that the below inequality holds. This confirms the convexity of the Langrangian with respect to the control. (d)When observed from (45), we easily see that . Hence, we conclude thatwith , , and . This completes the proof.
4.2. Characterization of Optimal Controls
Pontryagin’s maximum principle has been the wheel on which the necessary condition for the COVID (38) duple control needs to meet. With the principle, we convert the COVID (38) and the objective functional (40) into a problem of minimizing the Hamiltonian with respect to the controls . Hence, the Hamiltonian is given by
Theorem 7. With the optimal control duple satisfying the condition (41), there exist adjoint variables satisfying the adjoint system below.