Abstract
The world is on its path from the post-COVID period, but a fresh wave of the coronavirus infection engulfing most European countries makes the pandemic catastrophic. Mathematical models are of significant importance in unveiling strategies that could stem the spread of the disease. In this paper, a deterministic mathematical model of COVID-19 is studied to characterize a range of feasible control strategies to mitigate the disease. We carried out an analytical investigation of the model’s dynamic behaviour at its equilibria and observed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number, is less than unity. The endemic equilibrium is also shown to be globally asymptotically stable when . Further, we showed that the model exhibits forward bifurcation around . Sensitivity analysis was carried out to determine the impact of various factors on the basic reproduction number and consequently, the spread of the disease. An optimal control problem was formulated from the sensitivity analysis. Cost-effectiveness analysis is conducted to determine the most cost-effective strategy that can be adopted to control the spread of COVID-19. The investigation revealed that combining self-protection and environmental control is the most cost-effective control strategy among the enlisted strategies.
1. Introduction
The world is on its path from the post-COVID fiscal year. However, a fresh wave of the coronavirus infection engulfing the European countries makes the pandemic catastrophic, with most regions grappling with their worst outbreaks since the pandemic began [1]. Since December 2019, when the coronavirus epidemic was reported in Hubei in China, the epidemic has threatened the global well-being of humans by significantly affecting human resources through sickness and death [2]. Asian countries such as India, Sri Lanka, and Thailand are battling COVID-19 containment. The rapid resurgence of the virus has put enormous pressure on these countries’ health systems and medical supplies, with some calling for foreign assistance amid the escalating crisis. Generally, the availability of vaccines and the introduction of various vaccination programs have lowered the global daily infection and death counts, yet the deaths and infections have not entirely abated [3]. Mathematical models have made an indelible mark on the fight against infectious diseases by identifying protocols to respond to hikes in the transmission dynamics of epidemics [4–12]. Notable works in the case of COVID-19 have been considered by several authors. Yang et al. [13] made a generalized prediction of the coronavirus epidemic in mainland China by considering an SEIR model that utilized the available data. Zuo et al. [14] presented a data analysis and comparison method that concerned institutions could employ for proper decision making.calibrated the Brazil coronavirus pandemic that worked on determining the peak time of the epidemic. Wu et al. [2] used the available data from China to establish the possibility of symptomatic people dying from the coronavirus epidemic. Götz and Heidrich [15] considered a mathematical model of the COVID-19 epidemic that used data from Germany to estimate the model’s parameters and established the ascertainment of official infected cases. In [16], a nonlinear coronavirus model was considered and utilized to fit China’s data to estimate the model’s parameters. Abbasi et al. developed a mathematical model for the novel coronavirus disease and proved through stability analysis studies that, mitigating measures alone could not eradicate the disease. Their conclusion was based on the fact that the endemic equilibrium was globally asymptotically stable. Gu et al. [17] proposed a mathematical model that primarily focused on isolation, quarantine, and hospitalization to study the transmission dynamics of the novel coronavirus epidemic. They exploited nonpharmaceutical intervention measures: social distancing, isolation, contact tracing, and quarantine to mitigate the spreading of the disease. The authors further rebuilt the model into a stochastic model and compared the stochastic model analysis to the deterministic one. Currie et al. [18] developed a mathematical COVID-19 model that identified how mathematical models could influence management decision-making. In a related study, Huang et al. [19] constructed a mathematical model with several compartments that calibrated the model’s parameters with the available data and predicted the transmissibility of COVID-19. Senapati et al. [20] designed a nonlinear model that considered India’s data to estimate the model’s parameters and established the effectiveness of various protocols ushered in to suppress the disease. Zhou et al. [21] considered a SEIR COVID-19 model and applied the available data to estimate the reproduction number. In [22], a SEIR model was calibrated to the available data to derive some vital model parameters and determine the propagation trend of the epidemic. Hou et al. [23] proposed a nonlinear SEIR compartmental model that assessed the potency of Wuhan’s quarantine measure during the peaks of the epidemic and estimated the model’s parameters by fitting the model to Wuhan’s data. The model proposed by Du and Yuan [24] examined the propagation trend of the SARS-CoV2 and validated the model with existing Influenza data. Petropoulos and Makridakis [25] made a generalized prediction of the SARS-CoV2 pandemic pattern in Cyprus and underlaid strategies for containing the epidemic.
Optimal control models have received credit from the scientific community for their contribution in determining essential measures for pushing down cases of infections and the spike of new variants [26–33]. In December 2019, when WHO ascertained the existence of the pandemic, intriguing optimal control studies have been carried out that helped to wrestle the epidemic from spreading. Nana-Kyere et al. [34] presented a coronavirus model that mainly examined the epidemic’s stability and constructed an optimal control problem to deduce potent strategies to curb the disease. Asamoah et al. [35] calibrated their model to Ghana’s data to estimate some essential parameters that influence the dynamics of the disease. They extended the model to an optimal control problem to pinpoint strategies that will boost lowering the infected cases of the pandemic with minimal cost. Tsay et al. [36] studied the dynamical behaviour of the epidemic by designing a compartmental model for the disease. They redesigned the model into the optimal control problem to find strategies to help manage the virus. Kouidere et al. [37] characterized some feasible control strategies that come with desirable impact in containing the 2019 coronavirus epidemic. Perkins and Guido [38] considered an unorthodox method of handling epidemic without involving drugs to determine how the coronavirus infection could be managed. They reformulated the model into an optimal control problem to deduce strategies paramount for curtailing the disease. A nonlinear COVID-2019 model was developed to calibrate the model’s parameters by fitting the model to Ethiopia’s data [39]. The mode was extended into an optimal control problem in order to determine the effectiveness of treatment controls, personal protection, and public health education’s in suppressing the epidemic. Srivastav et al. [40] categorized a compartmental COVID-19 model into two main groups of the aged and young and used the data of Italy and Spain to estimate some of the model’s parameters. The authors then considered optimal control methods that developed strategies for managing the epidemic. Seidu [41] proposed a SARS-CoV2 mathematical model that captured the respective roles of mildly symptomatic, severely symptomatic, and exposed individuals in the transmission of the disease. He modified the model to optimal control and determined the most cost-effective strategy among the considered strategies through the cost-effectiveness method. In another related study, Nana-Kyere et al. [42] proposed a nonlinear mathematical model for the COVID-19 virus that examined the transmission dynamics of the disease. They modified the model into an optimal control problem and solved it to determine whether the proposed control strategies were potent enough to mitigate the epidemic from spreading. Agossou et al. [43] assessed the effects of pharmaceutical and nonpharmaceutical control strategies on the transmission dynamics of the COVID-19 epidemic.
The COVID-19 epidemic has attracted a voluminous amount of information from mathematical models since the time the disease was confirmed by WHO. Nevertheless, a significant amount of these pieces of information cannot be adequately relied on by public health to contain the virus. The unreliability of these models is because they are not epidemiologically well-composed and do not accurately describe the propagation trends of the disease for the correct management decision to be taken. This research mainly focuses on extending the model of Khan et al. [44] into an optimal control problem by identifying control strategies for the COVID epidemic and carrying out a socioeconomic analysis of the identified strategies to boost management decision-making on the containment of the virus. This has become necessary as the control model analysis would provide policy direction for public health. The paper will add to the existing knowledge on the epidemic dynamics by assessing the combined effect of treatment, vaccination, and disinfection of the environment on the transmission dynamics of the disease.
The rest of the research is presented as follows: Section 2 “The Model Development” presents the model formulation, model’s parameter definition, invariant region, boundedness, and stationary points. Section 3 “Qualitative Analysis” deals with the stability analysis, sensitivity and bifurcation analysis. In Section 4 “Development of Optimal Control Model”, we present an optimal control problem formulation and its qualitative analysis. Section 5 “Numerical Experimentation and Discussion” provides the numerical solution for the models. In Section 6 “Economic Cost Analysis of the Control Implementation”, a cost-effectiveness analysis is carried out to determine the most cost-effective strategy among the considered strategies. The section also discusses the simulated results of the work with a recommended conclusion.
2. The Model Development
The aggressiveness of the COVID-19 disease on the population means that newly designed models need to identify measures that would help clamp down on the disease. In this research, we use the COVID-19 model studied by Khan et al. [44]. The work segregates the model into eight compartments of Susceptible , Exposed , Infected , asymptomatically infected , Quarantined , Hospitalized , Recovered , and virus concentration in the environment . The model assumes that Susceptible individuals are recruited into the population at a rate of persons per unit time. The contact between a naive susceptible individual on the one hand and the infected or the asymptomatically infected on the other hand is given by . The parameter represents the transmissibility factor associated with the Asymptomatically infected persons. The model assumes that, the mortality that occurs naturally in individuals is given by . The parameter denotes the infection following exposure. The parameters and are the incubation periods of the disease. The infected, asymptomatically infected, quarantined, and hospitalized individuals have recovery rates, respectively, given by . The infected and quarantined individuals are hospitalized at rates and . The COVID-19-induced death rates in the infected and hospitalized are and . Susceptible individuals contract the disease at a rate due to a visit to the seafood market and contacting a viral source there or eating from a Covid-19-infected animal. The model assumes that the infected and asymptomatically infected individuals contribute to viral deposits in the seafood market at rate and , respectively. The parameter denotes the rate of removal of coronaviruses from the seafood market. Finally, the exposed individuals are quarantined at a rate . Based on these assumptions, the proposed dynamical COVID-19 model is of the form:
2.1. Qualitative Analysis: Disease-Free Equilibrium and the
In an epidemiological setting, the basic reproduction number , crystallizes the ability of the infection to spread or die out in the population. The is a measure of the number of people, on average, each sick person will infect throughout its period of infectivity. When is above one, the epidemic is generally recognized to be growing, and when the number is less than one, the outbreak is said to be declining. According to Khan et al. [44], , the basic reproduction number is given by where
2.2. Endemic Equilibrium
Any system that is not characterized by an absence of infection but rather a high transmissibility rate will eventually result in an endemic state. The model Equation (1) has a unique endemic equilibrium given as with , and
Using the analysis from [44] with the notation the endemic equilibrium can be characterised in terms of as where
The Equation (5) is used to generate the bifurcation plot in Figure 1, which was not done in [44].
3. Qualitative Analysis: Global Stability-Disease Free Equilibrium
In performing the global qualitative analysis, we employed the method of Castillo-Chavez et al. [45] to investigate the global stability of the state Equation (1) at the disease-free equilibrium. Following Castillo-Chavez et al. [45], the COVID-19 model Equation (1) can be restructured as follows: with representing the uninfected population, and denoting the infected. The disease-free equilibrium point is given by.
If the following criteria are satisfied, the point is a globally asymptotically stable equilibrium for the model (1).
F1: Given , is globally asymptotically stable.
F2: , where for
Theorem 1. The point is globally asymptotically stable equilibrium whenever .
Proof. Considering the model Equation (1), we generate and as; With , , , and then becomes It can be shown from (9) that as , . Hence is globally asymptotically stable, which satisfies condition one. Now, ascertaining whether condition would be satisfied, we consider . Thus, we have where matrix given by It can be deduced from model Equation (1) that the total population is bounded by . Hence, , and which implies is positive definite. Additionally, matrix is undoubtedly an -matrix, with the off-diagonal entries positive. Hence, the requirement of the two conditions is met, proving the global asymptotic stability of .
3.1. Qualitative Analysis: Global Stability-Endemic Equilibrium
In assessing global asymptotic stability of the endemic equilibrium of the model system (1), we capitalize on the geometrical method [46] which uses Bendixson criterion as sufficient condition for determining model system (1) global stability.
Theorem 2. Given , the endemic equilibrium of model Equation (1) is globally asymptotically stable; otherwise, it is unstable.
Proof. In proving the global stability of the endemic equilibrium, the study of ([47], with referencing to Lemma 3, Lemma 4 and Theorem 5.) is considered. we consider the subsystem of model Equation (1) as
Considering subsystem (12) the Jacobian matrix becomes,
with
Then, , the second additive matrix of is given by
Now, we denote the function diag, hence we deduce diag Then, the time derivative of becomes diag and
diag. It follows that
Hence, matrix can be determined and written as;
where
with
Consider to be a vector in and the norm given by . Let represents the Lozinki measure with respect to this norm; we adopt the method of approximating the as explored in [48], and choose where and and are matrix norms with respect to vector norms , and represents the Lozinki measure with respect to the norm .
Then, Hence,
It could be verified that
which provides the following inequalities when substituted into and
Further,
Hence,
substitute (24) into the below equation
Now from Equation (25), we denote and , then if , it follows that where Now, for , it follows that
Therefore, from (26), we get . This verifies that the system of nonlinear differential equation of submodel (12) is GAS around the interior equilibrium of . Therefore, considering the remaining subsystem of model (1).
The limit of (27) is given by
When system (28) is solved with the initial conditions of , then as , , , , , and which is the GAS of the endemic equilibrium.
3.2. Bifurcation Analysis
It has been observed from stability analysis of various epidemic models that the is the sufficient condition for wiping out infections in a population. This condition usually holds when the system exhibits forward bifurcation. However, in considering a control intervention measure such as vaccination of the susceptibles, it has been observed that the condition may not be sufficient to control the disease. Thus, the model transmissibility asymptomatic pattern is now determined by a quantity defined as the effective reproduction number. It has been observed in some studies [49–53] that, there can be coexistence of the endemic and disease-free equilibrium, which are both stable for , which represents backwards bifurcation. In controlling diseases in the presence of backward bifurcation, the has to be lowered to a value less than one. This section employs the theorem propounded by Castillo-Chavez et al. [54] to explore the type of bifurcation exhibited by the model system (1). To determine whether the phenomenon of bifurcation exists in model system (1), the centre manifold theory [54] is employed. The following necessary substitutions are taken to conform to the theorem’s requirement for easier computation of the bifurcation coefficients.
We denote , , , , , , , and such that the total population here is given by . With vector notation of , the model system (1) can be redesigned in the form , with given by
The Jacobian of the new system (29) evaluated at the DFE, denoted by is given by, where , , , Given that is considered as the bifurcation parameter in the case when , then expressing in terms of the other parameters when becomes,
Hence has a right eigenvector denoted by , with where
Further, has a left eigenvector , where , , ,
The newly translated model Equation (29) with the bifurcation parameter has at least one eigenvalue zero, and it affirms the existence of bifurcation in the transformed model (29). As expounded by Castillo-Chavez and Song ([54], refer to Theorem 5.1.), the equations are the principal determinants of the direction of bifurcation (whether backwards or forward bifurcation) in the transformed system (29). When and , then the bifurcation is backward and when and , the bifurcation is forward.
The bifurcation coefficients are obtained as
Assuredly, , since . is always positive. Hence, it can be concluded that model (29) exhibits forward bifurcation as shown in Figure 1.
3.3. Local Sensitivity Analysis
Due to the fact that epidemic model’s parameter values change, qualitative sensitivity analysis is necessary in order to determine the influence of parameters on model’s behaviour. The analysis sensitivity reveals the criticality of each parameter of to the model system (1). Sensitivity analysis is usually employed to bring to bear, the model’s parameters with strong influence on the that needs to be worked on in ushering in control measures. The indices from sensitivity computation assist us in quantifying the relative change in a variable with a change in the parameter. Hence, we will consider the normalized forward sensitivity index for model system (1), defined as the ratio of the relative change in the variable to the change in the parameter.
Definition 3. For a given parameter , the normalized forward sensitivity index of is computed using the formula discussed in [55, 56] as;
We will use Equation (36) and parameter values from Table 1 to derive the sensitivity indices of the parameters of the model system (1) as presented in Table 2.
The sensitivity indices computation from Table 1 gives fascinating values for different parameters, and we can make some inferences. Thus, parameters with positive indices have the strength to make the number of the COVID-19 cases rise since they increase the . Additionally, the negative sensitivity indices’ parameters help lower the COVID-19 epidemic because they lower the . Further, parameters with sensitivity indices increase or decrease by the same percentage as the parameters’ percentages. Figures 2 and 3 further presents the influence of various combinations of parameters on the basic reproduction number, .
(a) Sensitivity plot of in terms of and
(b) Sensitivity plot of in terms of and
(c) Sensitivity plot of in terms of and
(d) Sensitivity plot of in terms of and
(a) Sensitivity plot of in terms of and
(b) Sensitivity plot of in terms of and
(c) Sensitivity plot of in terms of and
(d) Sensitivity plot of in terms of and
4. Development of Optimal Control Model
To generate control strategies to wrestle the disease from existence, we employ optimal control theory. For the assessment of the conditions necessary to determine the feasible optimal control in the COVID-19 model (1), we apply Pontryagin’s maximum principle [57]. We note that, personal protection, vaccination of the susceptible population, and spraying the environment with disinfectant have significantly been encouraged during the COVID era. Hence, to add control strategies to the study, we rebuild the model (1) by reducing the transmission rate by a factor . We consider further the vaccination control denoted by for the susceptibles and a control for disinfection of the environment represented by . Hence, the new formulated optimal control model of model system (1) becomes;
The controls are bounded and Lebegue measurable.
The objective here is to minimize given by
subject to model system (37). We assign factors in the integrand of (18) as coefficients that serve as weight that balance Equation (38). The coefficients estimate the cost of ushered in interventions that need to be taken over the time of . We determine an optimal control such that
with subject to the model system (37). We use the most explored method in optimal control modelling, Pontryagin maximum principle [57], and find the solution and the necessary conditions of the optimal control problem (37).
4.1. Existence of the Optimal Controls
Related to the result of Fleming and Rishel as provided in [58], the existence of optimal control triple that minimizes (38) subject to (37) is shown.
Theorem 4. There exists an optimal control that minimizes the objective functional (38).
The existence of the optimal control can be proved by showing that, the following conditions hold.
(D1) The set of controls is convex and closed.
(D2) The system (37) is bounded by a linear function in both the state and control variables.
(D3) The integrand in the objective functional (38) is convex with respect to the control.
(D4) There exist constants that make the integrand in the objective functional (38) bounded by .
Proof. We construct the proof in steps as follows:
(D1): The control set is closed. Again, we let , so that and . Then, for every , we have , confirming the convexity property of the control set.
(D2): The control system (37) and its related solution are represented by and such that becomes
where , , , , , , , and
Hence the Lipschitz continuous uniformity of is proved.
(D3): We let the Langrangian of the objective functional (38) be . Then,
with, and . The convexity of , which is a linear combination of the control function has to be shown. We let for . Then, . Hence, it follows that the below inequality holds.
This confirms the convexity of the Lagrangian with respect to the control.
(D4): By inspection, we noticed in (41) that . Hence, grounded on this, we conclude that
with min, and . This completes the proof.
4.2. Characterization of Optimal Controls
Using the Principle translates the COVID model (37) and its associated objective functional (38) into a problem of minimizing a Hamiltonian , with respect to the controls , for . The Hamiltonian for system (17) is given by;
where the in Equation (44) are the right-hand sides of system (37) and are the respective adjoint variables for the state .
Theorem 5. Consider the triple optimal control satisfying the Condition (39), then there exist adjoint variables satisfying adjoint system below;
with transversality conditions
The control functions satisfies the optimality condition and is given by where
Proof. Referring to the Hamiltonian (49), given by
we generate the adjoint system (45) by differentiating partially the Hamiltonain (49) with respect to the corresponding state variables as
The characterization of the controls as in (39) is determined by applying the equation below;
Utilizing bounds on the controls by standard argument, we determine the characterization
where
This completes the proof.
5. Numerical Experimentation and Discussion
With the aid of the forward-backwards sweep method as derived by Lenhart and Workman [59], a numerical solution to the optimality system of the control model was obtained. We offer the scenarios of different feasible control strategies to identify their effect on the dynamics of the COVID model. The backwards scheme solves the adjoint system (45) backwards in time, and the forward scheme solves (37) forward in time. In addition, the structured scheme considered the set of initial conditions related to the state variables and the boundary condition to solve the control optimality system in a simulated interval of . With this set in play, we consider the balancing factors of the objective functional (38) as: , and initial population for the simulation purpose. We concluded that , as we assume that the cost of vaccines and implementing a vaccination program would be higher than the cost of personal protection and disinfecting the environment. Even though the model in [44] is a Fractional-order model, the authors estimated the model’s parameters at . However, at , the model becomes an integer order model. Furthermore, since the study modifies model 1 to optimal control which simulated results are always compared to the noncontrol model, it becomes necessary that we consider the same parameter values as used in model 1. Hence, we would consider model [44] parameter values for the simulation.
5.1. Strategy (Implementing Controls ,, and )
The strategy considered the controls . The simulation results are presented in Figures 4(a)–4(d) of Figure 4. We observe that, the exposed population (28) without control swiftly rose in the first five days and reached almost within the next five days. The dynamics of the exposed graph changed after day . The graph began to slowly decrease until day when the population seemed to have been wiped out. The exposed without control graph maintained this new level for the next days and began to surge to a height of for the next days when it began to decrease for the rest of the simulated time smoothly. Also, the infected population without control (See Figure 4(b)) gradually rose in the first days until day when it began to decrease with an estimated infected population of . It decreased slowly to in days and picked up after the . The asymptomatically infected graph of (See Figure 4(c)) sharply increased in days to reach a population of about and then decreased slowly after day and reached the lower bound at days. The level was maintained until days when it smoothly increased for the remaining simulated time. The exposed population under controls (Figure 4(a)) decreased slowly and was expected to wipe out in days completely. The infected population with control (See Figure 4(b)) increased in a similar pattern slowly in the first days and decreased slightly, with the population recognized to be a little below at the final time. The asymptomatically infected population with control (See Figure 4(c)) was the same as that without controls for the first 5 days but is reduced after days. From the control profile graph (Figure 4(d)), , which represents the personal protection control, starts from the upper bound but drops suddenly to the lower bound at the initial zero days until when it swiftly moves to the upper bound. The next days remained at the upper bound and then returned to the lower bound at the final time of days. The vaccination control, , remained at the lower bound until days when it moved to the upper bound. It stayed at the upper bound until days and then dropped to the lower bound and remained there for the rest of the time. , denoting disinfectant control, started at the lower bound and picked to the upper bound after days. It remained at the upper bound for the remaining time until days, when it dropped to the lower bound.
(a) The solution path of the exposed
(b) The solution path of the infected
(c) The solution path of the asymptomatically infected
(d) Control profile
5.2. Strategy (Implementing Controls and )
The strategy sets , the disinfectant control to zero and experiments with the controls and . The simulation results are presented in Figure 5. The exposed population without control (See Figure 5(a)) shows a sudden rise in the first days. The graph rises to a peak of about and after days, decreases steeply to a minimal population of about within the next days and maintains this new level until days when it begins to increase smoothly. The infected population without control (See Figure 5(b)) gently increases for the first days to reach a maximum height of . The new height of the infected became unstable after the next days, and that resulted in the population slowly decreasing to in the next days, after which it began to rise for the remaining time. With the asymptotically infected without control (See Figure 5(c)), we noted a quick rise of the curve in the first days and reached the pinnacle at in days at the infection hotspots.
(a) The solution path of the exposed
(b) The solution path of the infected
(c) The solution path of the asymptomatically infected
(d) Control profile
The exposed population yielded to the design experiment strategy . The exposed population starts with population and gently reduces until days when it seems to have been wiped from the system. The infected population under control strategy 2 slightly increased in the early days and began to lower after days. The graph decreased in that pattern and reached a lower height of at the final time. Also, the asymptotically infected population increased to about in the first days, decreased slowly to zero in days, and maintained that level for the remaining period of the simulation time. Figure 5(d) presents the control profile for Strategy 2. By inspection, we observed that the personal protection control falls to the lower bound at the initial zero time. It remained on the lower bound for days when it suddenly moved to the upper bound. It then remained at the upper bound until days when it dropped to the lower bound again. Control , representing the vaccination control, remained at the lower bound for the first days when it swiftly moved to the upper bound. It stayed at the upper bound for the next days when it dropped to the lower bound and remained there for the remaining time.
5.3. Strategy (Implementing Controls and )
With set to zero, strategy uses the controls and for the experimentation by simulating the optimality system with the structured interval. The simulation results for this strategy are presented in Figure 6. We observe that, the exposed population without control (See Figure 6(a)) moved sharply to in day of the simulated time and then decreased gradually to almost and maintained the level until days when it began to rise for the remaining time. The infected population without control (See Figure 6(b)) increased slowly and moved to a height of for days. The population then decreased slowly for the next days and began to pick up after the days. The asymptomatically infected population without control (See Figure 6(c)) also shoots quickly in the first days to its maximum of . The dynamics changed, and the population began to reverse slowly to a level of about , and then picked up after days and smoothly increased to the final time. The optimal control plot of the exposed produced a graph that seemed to wane in the early days but began to rise after days to meet the noncontrol graph at days. The level of the controlled exposed population was maintained until the final time. The infected population with control Strategy 3 (See Figure 6(b)) got lowered in the early days and continued till the final time of the simulation. The controlled population of asymptotically infected persons (See Figure 6(c)) got depleted in the first days. However, for the uncontrolled problem, the population resurged and maintained the level for the rest of the simulation time. From Figure 6(d), denoting personal protection control drops suddenly from the upper bound to the lower bound at time . It then remained at the lower bound for days and shot to the upper bound, where it remained for days and dropped to the lower bound. It remained in the lower bound for another days and moved to the upper bound. It then stayed at the upper bound for days and dropped to the lower bound. It picked after days to the upper bound and stayed there until it dropped to the lower bound. denoting disinfectant control started from the lower bound and moved to the upper bound after days. It dropped to the lower bound after days and again moved to the upper bound after . It stayed at the upper bound for another days and then dropped to the lower bound for days. It picked to the upper bound of days and remained until the final time before dropping.
(a) The solution path of the exposed
(b) The solution path of the infected
(c) The solution path of the asymptomatically infected
(d) Control profile
5.4. Strategy (Implementing Controls and )
The strategy set and utilized the control and in the numerical simulation for assigned interval. The simulation results are presented in Figures 7(a)–7(d). The Strategy 4 is observed to also do well in reducing the exposed, infected and asymptomatically infected populations. From the control profile (Figure 7(d)), the vaccination control, , shot to the upper bound at time , stayed at the upper bound for days, and then dropped to the lower bound. It remained at the lower bound for the rest of the simulated time. The disinfectant spraying control stayed at the lower bound from day to , rocketed to the upper bound and remained there until days when it returned to the lower bound.
(a) The solution path of the exposed
(b) The solution path of the infected
(c) The solution path of the asymptomatically infected
(d) Control profile
6. Economic Cost Analysis of the Control Implementation
With the simulation results of the optimality control problem, the present section seeks to identify the strategy among the considered strategies that minimize the cost involved in executing the strategy yet give a satisfying performance output. To identify the most cost-effective strategy, we compare strategies 1, 2, 3, and 4. The approach outlined in related previous studies [60–63] is adopted to examine the average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER) of strategies 1, 2, 3, and 4. The ICER is the ratio of the differences in cost of two control strategies to the differences in infection averted by executing that strategy. On the other hand, the ACER is the ratio of the total cost involved in executing the strategy to the total infected prevented. The mathematical formulation
is used to exhibit the number of infections averted, and the related cost of executing the strategies under consideration in increasing order of magnitude of the infection averted is presented in Table 3.
The ICER representations are derived as follows;
From Table 3, we notice that the ACER value of strategy 2 is the least among the strategies considered. The ACER values insinuate that strategy 2 is the most cost-effective. Further, we observe that strategy 1 has the highest ACER value, which is the least cost-effective measure. However, an in-depth analysis of the least cost-effective measure was required. The ICER of the strategies was then computed. The ICER computations depicted that strategy 1 is more costly but less effective than strategy 2. The analysis confirmed that ICER of strategy 1 is higher than ICER of strategy 2. Hence, the control strategy 1 is deleted from the considered strategies, and the remaining strategies ICERs are recalculated for further analysis on the three. The analysis of the remaining strategies is presented in Table 4.
From the computed ICERs of Table 5, we see that strategy 4 ICER is higher than strategy 2. The cross-examination of the two ICER shows the dominance of strategy 2 over strategy 4. Thus, strategy 2 costs less and performs best in managing the disease. Hence, strategy 4 is removed from the remaining strategies, and the ICER of strategy 2 and 3 are recalculated and analyzed, as depicted in Table 5.
The ICER computation from Table 5 indicated that strategy 3 is higher than strategy 2. The analysis further proves that strategy 2 is the most cost-effective control strategy among the listed strategies.
6.1. Conclusion
The research considered a dynamical study on a nonlinear autonomous COVID-19 model to unveil strategies that could assist in stemming the spreading of the disease. With the introduction of the control strategies: personal protection, vaccination, and disinfecting of the environment into the state model (1), the basic model was redesigned into an optimal control problem. We utilized Pontryagin’s maximum principle’s [57] analytic method to characterize possible controls that may help stem the epidemic from spreading. We supported the qualitative analyses with a numerical solution that employed the fourth-order Runge-Kutta method to generate numerical solutions of the control problem. The results from the executed strategies unveiled a satisfying impact on the dynamics of the exposed, asymptomatically infected, and infected individual. The strategy 1 considered the three controls of personal protection, vaccination, and disinfectant of the environment. Interestingly, the generated graphs showed the retrospective impact of the strategy on the disease’s dynamics. Thus, we see substantially minimized exposed, infected, and asymptomatically infected individuals. Strategy 2 employed personal protection and vaccination control as its intervention strategy. A similar result to using strategy 1 was obtained. Regarding strategy 3, personal protection and disinfectant controls were used. The simulation produced a pleasing effect on the dynamics of the infected. Thus, the graph depicts a convincing minimized exposed, infected, and asymptomatically infected with the strategy. The strategy of 4 chose the vaccination and disinfectant of the environment. The result minimized the exposed, infected, and asymptomatically infected. A careful examination of the strategies that employed vaccination as part of the controls substantially minimized the exposed, infected, and asymptomatically infected graphs. For instance, strategy 3, which did not consider vaccination control experienced an early rise in the asymptomatically infected graph but was minimized in the long run. The results suggest that vaccines are essential pharmaceutical control strategies needed to mitigate the disease but cannot be wholly relied on due to the resistance imposed by the mutation of the virus and the vaccines efficacies waning after a short while. This is a result of the leakiness of vaccines and limited vaccination capacity. The finding is consistent with the findings of the work [43, 64, 65], which recommended that policymakers should be pragmatic in relaxing other COVID-19 mitigating protocols since the vaccine efficacies are not , as evidence of reinfection after vaccination has been proved in several studies [66–68]. Finally, we assessed the various strategies’ economic costs with the cost-effectiveness analysis method. The method examined the effectiveness and cost of the listed strategies considered. The analysis unveiled that the strategy 2: personal protection and vaccination control strategy is the most cost-effective method among the considered strategies. From our investigations, we recommend that a deliberate attempt should be made by public health and all stakeholders to strengthen the personal protection protocols as recommended by WHO and encourage and educate society about the benefits of being vaccinated, as vaccines reduce hospitalization [66].
Data Availability
The authors confirm that the data supporting the findings of this study are available within the article.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
(i) The results in this paper form part of the PhD thesis of the first author in the Department of Mathematics of the C. K. Tedam University of Technology and Applied Sciences, Ghana. The second author is the lead supervisor. (ii) The author acknowledges the paper as part of a PhD thesis at C. K. Tedam University of Technology and Applied Sciences, titled “Mathematical modelling of COVID-19 and Vector-borne diseases with optimal control”.