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We formulate and theoretically analyze a mathematical model of COVID-19 transmission mechanism incorporating vital dynamics of the disease and two key therapeutic measures—vaccination of susceptible individuals and recovery/treatment of infected individuals. Both the disease-free and endemic equilibrium are globally asymptotically stable when the effective reproduction number

The December 2019 outbreak of the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), causing COVID-19, was first reported in Wuhan, Hubei Province of China [

Starting with the work of Daniel Bernoulli in 1760 [

As COVID-19 vaccines are being deployed worldwide, we formulate and qualitatively analyze a COVID-19 mathematical model, taking into consideration available therapeutic measures, vaccination of susceptible and treatment of hospitalized/infected individuals. Our proposed model incorporates some key epidemiological and biological features of COVID-19, including demographic parameters (recruitment/birth and death). Optimal control is carried out using Pontryagin’s maximum principle as described in [

The rest of the paper is organized as follows. The proposed COVID-19 model is formulated in Section

Consider a homogeneous mixing within the population, i.e., individuals in the population have equal probability of contact with each other. Using a deterministic compartmental modeling approach to describe the disease transmission dynamics, at any time

Figure

Compartment diagram of the human component of the model.

Model parameter values and source.

Parameter | Description | Value | Reference |
---|---|---|---|

Recruitment rate of individuals into the population | [ | ||

Proportion of recruited individuals who are vaccinated | 0.0001 | Assumed | |

Vaccination rate | 0.4 | Assumed | |

Reduction in the transmission from asymptomatic | 0.3 | [ | |

Increase in the transmission from symptomatic | 1.8 | Assumed | |

Reduction in the transmission from hospitalized | 0.3 | Assumed | |

Natural death rate | [ | ||

Disease-induced death rate | 0.018 | [ | |

Infection reduction of vaccinated individuals | 0.8 | Assumed | |

Exit rate from the exposed class | 0.13 | [ | |

Exit rate from the infectious class | 0.0833 | [ | |

Proportion of infectious who recover naturally | 0.05 | [ | |

Fraction of exposed who become infected | 0.7 | [ | |

Effective contact rate | 1.12 | [ | |

Recovery rate of hospitalized individuals | 0.0701 | [ | |

Proportion of asymptomatic who recover naturally | 0.14 | [ | |

Exit rate from the asymptomatic class | 0.13978 | [ | |

Rate at which individuals lose immunity | 0.011 | [ |

Since COVID-19 vaccination is available, it is realistic to consider a specific vaccinated class

After close contacts with symptomatic, asymptomatic, and hospitalized individuals, susceptible become exposed with the disease. We assume that the rate of disease transmission from asymptomatic to susceptible individuals is less than that from symptomatic and hospitalized individuals. While outbreaks usually persist for a shorter period of time, the COVID-19 pandemic which started in December 2019 is still ongoing, and for this reason, we incorporate vital dynamics (recruitment and death). Let

From the aforementioned and the model flow diagram of the disease transmission mechanisms Figure

From the model flow diagram in Figure

For simplicity, let

All the model parameters and their description, values, and sources are presented in Table

Well-posedness, nonnegativity, and boundedness of solutions of the proposed model can be shown using basic theory of dynamical systems as described in [

Model system 1 admits a disease-free equilibrium (DFE) given by

The linear stability of

From [

Thus, the effective reproduction number is given by

The effective reproduction number

We now study the global stability of the DFE using the approach described in [

(H1) For

(H2)

The fixed point

The DFE

First, we rewrite model 1 in the form 6 by setting

Then, the DFE is given by

This equation has a unique equilibrium point

We now verify the second condition (H2). For model 1, we have

Clearly,

Global stability of the DFE precludes the model to exhibit bistability also known as backward bifurcation [

We investigate the critical vaccination coverage rate that could help eradicate the disease.

In fact,

Thus,

Taking the partial derivative of

Therefore,

This can be interpreted to mean that when the vaccine efficacy

Graphical representation of

The next result provides the critical vaccination threshold

Assume that the basic reproduction number

Figure

Graphical representation of

After some algebraic manipulations, the endemic equilibrium of model system 1 is obtained as

From the last equation of model system 1 and using the definition of

Thus,

If

It can also be shown using the theory or permanence as described in [

Consequently, the model system is uniformly persistent when

Figure

Endemic equilibrium regions when

We investigate the impact of implementing pharmaceutical interventions to mitigate the spread of COVID-19. To accomplish this, we introduce a set of time-dependent control variables

The proposed COVID-19 model with optimal control

We wish to find the controls that minimize the total infected individuals, that is, to find an optimal control for the two control strategies while reducing their relative costs. In other words, we want to find the optimal values of

To derive the necessary conditions that the two optimal controls and corresponding states must satisfy, we apply Theorem 5.1 (Pontryagin’s maximum principle [

Given an optimal control

The controls

The differential equations governing the adjoint variables are obtained by differentiation of the Hamiltonian function, evaluated at the optimal control. Then, the adjoint system can be written as

Thus,

To illustrate the theoretical results, numerical simulations are carried out. Model parameter values for the numerical simulations with their description and source are listed in Table

Following the approach described in [

Model fit with cumulative daily COVID-19 cases in Senegal, 29 March–29 April 2021.

We investigated the impact of vaccination and treatment on mitigating the spread of COVID-19. An iterative fourth-order Runge-Kutta method (both forward and backward algorithms) is employed to compute the optimal controls and state values used. For more details on this approach, see [

The baseline weight parameters

Figure

Profile of

Profile of

In the next set of figures generated from model system 14, optimal control strategies are implemented. Using the model parameter values in Table

Dynamics of the vaccinated class

Dynamics of the exposed class

Dynamics of the asymptomatic class

Dynamics of the infected class

Dynamics of the hospitalized class

Controls

Because mathematical models are symbolic/mechanistic representations of complex biological systems, some model parameter values are not often known with certainty due to natural and seasonal variations, potential measurement errors [

Contour plot of

Partial rank correlation describes the relationship between two variables while at the same time removing the effects of several other variables from the relationship [

PRCCs showing the effect of varying the input parameters on

Figures

Contour plot of

Contour plot of

We formulated a deterministic model of the transmission dynamics of COVID-19 with an imperfect vaccine. The model is theoretically analyzed; its effective and basic reproduction numbers are derived. The disease-free equilibrium is globally asymptotically stable, and the disease could be eradicated when the reproduction number is below unity. The critical vaccination threshold is derived, and it is noted that if the vaccine efficacy is low and the disease reproduction number is high, the disease may not be eradicated even if a large proportion of the population is vaccinated. That is, additional efforts will be needed to reduce

We then introduce into model system 1 time-dependent control variables

Finally, we performed a sensitivity analysis using the partial rank correlation coefficient in conjunction with the Latin hypercube sampling technique, to identify the model parameters that significantly influence the initial disease transmission

This study is not exhaustive, and future studies could investigate the impact of both therapeutic and (adherence to) nontherapeutic measures on the dynamics of COVID-19.

Expressions of the coefficients

There are no underlying data.

The authors declare that they have no conflicts of interest.