Mathematical assessment of the role of denial on COVID-19 transmission with non-linear incidence and treatment functions

Abstract

A mathematical model describing the dynamics of Corona virus disease 2019 (COVID-19) is constructed and studied. The model assessed the role of denial on the spread of the pandemic in the world. Dynamic stability analyzes show that the equilibria, disease-free equilibrium (DFE) and endemic equilibrium point (EEP) of the model are globally asymptotically stable for $R_0<1$ and $R_0>1$, respectively. Again, the model is shown via numerical simulations to possess the backward bifurcation, where a stable DFE co-exists with one or more stable endemic equilibria when the control reproduction number, $R_0$ is less than unity and the rate of denial of COVID-19 is above its upper bound. We then apply the optimal control strategy for controlling the spread of the disease using the controllable variables such as COVID-19 prevention, hospitalization and maximum treatment efforts. Using the Pontryagin maximum principle, we derive analytically the optimal controls of the model. The aforementioned control strategies are performed numerically in the presence of denial and without denial rate. Among such experiments, results without denial have shown to be more productive in ending the pandemic than others where the denial of the disease invalidates the effectiveness of the controls causing the disease to continue ravaging the globe.

1. Introduction

Coronavirus disease 2019 (COVID-19) is a respiratory disease of the Coronavidae family. The virus has four types of strains, namely; $\alpha$-Coronavirus, $\beta$-Coronavirus, $\gamma$-Coronavirus and $\delta$-Coronavirus. The first two affects humans and the last two are known majorly for birds infection [1]. The origin and the reservoir of the virus still remain uncertain, even though Singh et al. [2] considers COVID-19 to be zoonotic disease and traced it to the bats family. COVID-19 is the 3rd novel coronavirus in the 21st century after SARS and MERS to have caused large epidemic that spread over 210 nations worldwide [3].

COVID-19 is a pandemic of all ages when compare to SARS and MERS. It can spread rapidly among the elderly and individuals with underlying medical conditions, who are particularly a severe risk population [4]. Human-to-human transmission can occur when the respiratory droplets of an asymptomatic, pre-symptomatic and symptomatic person touches the nose, eyes or mouth of a susceptible individual on a closed contact [5]. The clinical symptoms of the disease such as cough, breathing difficulty, fever among other malarial related signs begin to manifest within 5–6 days on average during the incubation period of the virus [6].

The cure for this disease is still a problem facing nations, and the hope for guaranteed vaccine formulation though not a lost one but yet to be made available. As a result, an index case that began in Wuhan, China has spread to many countries and is responsible for over 1 940 529 deaths globally, 72 834 deaths in Africa [7] and 1405 deaths in Nigeria [8] as of January 14, 2021.

In addition, the above number of human deaths in Nigeria is due to the lack of willingness of the populace to accept the existence of COVID-19 [9]. Some are of the opinion that COVID-19 cases in Nigeria have been mutilated for political gains. Based on such psychological thoughts, many vulnerable populations have lost confidence in getting protective measures as recommended by World Health Organization (WHO) [4] and Nigeria Center for Disease Control(NCDC) [8]. From the past epidemics, transmission indices have shown that the denial of existence of a disease indeed stands a challenge for epidemiologists in the fight against the disease prevention and control. See for example, the denial effect of HIV/AIDS [10], [11] on susceptible population. Apart from denial, people who use both licit and illicit drugs pause a greater problem in harm reduction interventions in the era of COVID-19 in Africa [12].

Non-linear incidence and treatment functions have been reported by several authors [13], [14], [15] to be invaluable tool used to reduce the spread of diseases, not limited to measles and tuberculosis. In epidemiological modelling of COVID-19, mass action function of the kind $\beta SI$, where $\beta$ is the infection rate was applied in model of [16] and a saturated incidence rate, $\frac{\alpha SI}{1+\beta S}$ was found in the models of [17], [18]. On the other hand, the application of Beddington-De Angelis incidence rate of type $\frac{\alpha SI}{1+\beta S+\gamma I}$ was adopted in the works of [19], [20], [21], [22], [23] for non-COVID-19 models.

Treatment has been the integral part of the solution of epidemic outbreak considered at different rates. So far, Adeniyi et al. [24] and Daniel [18] used a constant rate and Holling type II, $\frac{\beta I}{1+\gamma I}$ respectively as treatment functions, in corona virus transmission models. Other controls such as lock down strategy, social isolation on corona virus have been deliberated upon in the work of Ndam [16] without denial effect. A dual-purpose wheelchair was developed for containment of COVID-19 in paraplegic patients in Lagos, Nigeria [25], and knowledge and attitudinal behaviour among Sudanese population was also observed to be key in eradicating the pandemic [26].

In this paper, we intend to assess the role of denial on the optimal control model for COVID-19 transmission dynamics using Beddington-De Angelis incidence and treatment function of Holling type II. To the best knowledge of the authors, this has not been applied on any mathematical model as regards to COVID-19 pandemic. Our model made advances to existing works on COVID-19 in the following dimensions:

• (i)we incorporate denial as an erroneous human behaviour that triggers the spread of COVID-19 among Africans. In mathematical sense, the consideration of this invaluable factor is a novelty.
• (ii)The introduction of denial on Beddegton-De Angelis function is unique compared to other existing mathematical models on COVID-19.
• (iii)The inclusion of a non-linear maximal treatment function as a pharmaceutical strategy in the control of COVID-19 is also a new idea compared to other models in literature.

This paper is organized as follows: we introduce the model formulation in Section 2. Equilibria analysis of the proposed model is given in Sections 3 and 4. An optimal control analysis is done in Section 5. Meanwhile numerical results and discussion are given in Section 6. Section 7 describes the conclusion of the paper.

2. Materials and methods

2.1. Model formulation

In this study, we propose a mathematical framework of generic SEIR model for COVID-19 transmission dynamics using the Beddington-De Angelis functional response and Holling type II treatment as applied in [15], [16] on infectious model. The derivation of the differential equations is as a result of the under listed assumptions:

• a.The total population, N at any time, t, is divided into five classes: Susceptible people, S (who are under risk of contracting COVID-19), Exposed people, E (who are in close contact with infected people, but not yet infected), Asymptomatic people, A (who harbour the corona virus without clinical symptoms but capable of transmitting the disease), Symptomatic people, I (who are infected with the corona virus with clinical symptoms and capable of transmitting the disease), Hospitalized people, H (infectious people who are isolated for treatment) and Recovered people, R (who survived the COVID-19 infection). Thus, $N=S+E+A+I+H+R$.
• b.The susceptible people are recruited at the constant rate, $\widehat{B}$ and die naturally like others at a rate $\mu$.
• c.
  COVID-19 is transmitted from Asymptomatic people, Symptomatic people, Hospitalized/isolated people to susceptible people by Beddington-De Angelis incidence rate in Eq. 1,

  $$\widehat{f}=\frac{\beta S(c_{1}A+c_{2}I+c_{3}H)}{1+({\eta }_1-d)(S+E+A)+{\eta }_{2}I+{\eta }_{3}H},$$ (1)
  where $\beta$ is the average infection rate, $c_i,i=1,2,3$ ($c_1>c_2>c_3)$ is the contact rate of susceptible with A, I and H respectively, $d$ is the denial effect rate, ${\eta }_1$ refers to the measure of inhibition effect of preventive protocols such as social distancing, use of face masks and hand sanitizers taken by both susceptible and exposed people. Meanwhile, ${\eta }_2,{\eta }_3$ are measures of inhibition effect due to maximum treatment with respect to infectious people, I and H respectively. Note that $0<{\eta }_1-d<{\eta }_2<{\eta }_3$ when denial of the disease exist. It is of great interest to reveal that, four kinds of incidence rates can be obtainable from this Beddington-De Angelis incidence, (i. e; Eq. 1) in this proposed study:

  • i.
    if we set ${\eta }_1=d,{\eta }_2={\eta }_3=0$, then $\overline{f}(S,A,I,H)=\beta S(c_{1}A+c_{2}I+c_{3}H),$ which is bilinear incidence rate (force of mass action).
  • ii.
    if we set ${\eta }_2={\eta }_3=c_2=c_3=0$, then $\overline{f}(S,E,A)=\frac{\beta S{c}_{1}A}{1+({\eta }_1-d)(S+E+A)},$ which is saturation incidence rate with the sum of susceptible and exposed people (infectious of COVID-19 without symptoms). This inhibition effect is as a result of saturation parameter ${\eta }_1$ which is the preventive protocol to control the spread of the pandemic.
  • iii.
    if we set ${\eta }_1=d,{\eta }_3=c_1=c_3=0$, then $\overline{f}(S,I)=\frac{\beta S{c}_{2}I}{1+{\eta }_{2}I},$ which is saturation incidence rate with infectious people (yet to be isolated/hospitalized). Here, the contact between susceptible and infectious people, may saturate at high transmission level due to crowding of infectious people or protection from the susceptible people.
  • iv.
    if we set ${\eta }_1=d,{\eta }_2=c_1=c_2=0$, then $\overline{f}(S,H)=\frac{\beta S{c}_{3}H}{1+{\eta }_{3}H},$ which is saturation incidence rate with hospitalized/isolated people. As in case (iii), the contact between isolated and susceptible people, may saturate at high transmission level due to crowding of isolated people or protection taken by the susceptible people.
• d.
  the hospitalized people recovered by saturated Holling type II treatment rate is given by Eq. 2,

  $$\overline{g}\left(H\right)=\frac{\gamma H}{1+ϵH},$$ (2)

  with $\gamma$ being the maximum treatment and $ϵ$ measures the inhibition recovered rate due to long time response of hospitalized people to treatment. Note that $ϵ$ is a constant parameter that accounts for the limited resource available for the treatment of COVID-19. The Asymptomatic people may recover earlier at a rate, $\nu$.
• e.The recovered people gain permanent immunity since no case of COVID-19 re-infection has been reported.
• f.COVID-19 is a fatal disease with a death rate, $\delta$.
• g.The exposed progress to Asymptomatic class at a rate, ${\rho }_1$ before further developing symptoms of the disease at a rate, ${\rho }_2$. This assumption is supported by the works of Alshammari [27] and Deressa and Duressa [28].
• h.Asymptomatic individuals who are infected without symptoms also deny the existence of the disease.

The summary of model parameter values and description is given in Table 1 below.

Table 1.

Variables and parameters of the model .

Variables	Description	Value	Ref.
$S\left(t\right)$	Number of Susceptible individuals at time, $t$	$200\times {10}^6$	[18]
$E\left(t\right)$	Number of Exposed individuals at time, $t$	1565	[Assumed]
$A\left(t\right)$	Number of Asymptomatic individuals at time, $t$	80	[Assumed]
$I\left(t\right)$	Number of Symptomatic individuals at time, $t$	20	[Assumed]
$H\left(t\right)$	Number of Hospitalized individuals at time, $t$	10	[Assumed]
$R\left(t\right)$	Number of Recovered individuals at time, $t$	0	[Assumed]
Parameters	Description	Value	Ref.
$\widehat{B}$	Recruitment rate into Susceptible individuals	22655	[18]
$\beta$	Infection rate	0.05	[Assumed]
${\eta }_1,{\eta }_2,{\eta }_3$	Inhibition factors	0.55,0.3,0.5	[Assumed]
$c_1,c_2,c_3$	Contact rates of susceptibles with A, I and H respectively	0.5,0.3,0.1	[Assumed]
$d$	Rate of the denial of the disease	[0,0.4)	[Assumed]
$\mu$	Natural death rate	0.0182	[18]
$\gamma$	Maximum treatment rate of the hospitalized individuals	$1/15$	[18]
$\nu$	Rate of recovery of the asymptomatic individuals	$1/7$	[29]
$\delta$	Disease induced death rate	0.022	[18]
$ϵ$	Reduced transmission factor of recovered individuals	0.5	[Assumed]
$h$	Rate of hospitalization of Symptomatic individuals	0.025	[29]
${\rho }_1$	Rate of progression from E to A individuals	$1/5.1$	[29]
${\rho }_2$	Rate at which individuals in A develop symptoms	$1/7$	[18]

2.2. The model

Using the flow-diagram of the interacting population in Fig. 1 , we obtain the differential coefficient of the variables as indicated in system 3.

$$\begin{array}{cc}\frac{dS}{dt}\hfill & =\widehat{B}-\overline{f}S-\mu S,\hfill \\ \frac{dE}{dt}\hfill & =\overline{f}S-(\mu +{\rho }_1)E,\hfill \\ \frac{dA}{dt}\hfill & ={\rho }_{1}E-(\mu +\nu +{\rho }_2)A,\hfill \\ \frac{dI}{dt}\hfill & ={\rho }_{2}A-(\mu +\delta +h)I,\hfill \\ \frac{dH}{dt}\hfill & =hI-\overline{g}\left(H\right)-(\mu +\delta )H,\hfill \\ \frac{dR}{dt}\hfill & =\nu A+\overline{g}\left(H\right)-\mu R,\hfill \end{array}$$ (3)

where $\overline{f}$ and $\overline{g}\left(H\right)$ are as in Eqs. 1 and 2, with initial conditions

$$S\left(0\right)=S_0>0;E\left(0\right)=E_0\ge 0;A\left(0\right)=A_0\ge 0;I\left(0\right)=I_0\ge 0;H\left(0\right)=H_0\ge 0;R\left(0\right)=R_0\ge 0.$$

It is obvious from Eq. 1 that once the dynamics of $(S,E,A,I,H)$ are understood, then the dynamics of $R$ immediately follows from the Eq. 4

$$\frac{dR}{dt}=\nu A+\overline{g}\left(H\right)-\mu R.$$ (4)

Therefore, from onwards, our study shall be focused on the reduced system below

$$\begin{array}{cc}\frac{dS}{dt}\hfill & =\widehat{B}-\overline{f}S-\mu S\hfill \\ \frac{dE}{dt}\hfill & =\overline{f}S-(\mu +{\rho }_1)E\hfill \\ \frac{dA}{dt}\hfill & ={\rho }_{1}E-(\mu +\nu +{\rho }_2)A\hfill \\ \frac{dI}{dt}\hfill & ={\rho }_{2}A-(\mu +\delta +h)I\hfill \\ \frac{dH}{dt}\hfill & =hI-\overline{g}\left(H\right)-(\mu +\delta )H\hfill \end{array}$$ (5)

whose positivity and boundedness of solution can be verify in the next subsection.

Fig. 1.

Schematic diagram for COVID-19 transmission dynamics.

2.3. Existence and uniqueness of solution

The epidemiological validity of the mathematical model in Eq. 5 rely on the solution of the dynamic system being positive and bounded for all time $t>0$. This is to be established in the following theorems.

Theorem 2.1

(positivity) If$t>0$and the starting conditions$\Gamma \left(0\right)⩾0,$where

$$\Gamma \left(t\right)=\left(S\right(t)>0,E(t)>0,A(0)>0,I(t)>0,H(0)>0),$$

then the solution of Eq.5is non-negative provided they exist.

Proof

With the method in Alshammari [27], let $t_1=\{t>0:\Gamma \left(t\right)>0\in [0,t]\},$ then it follows from the first Eq. 5 that

$$\frac{dS}{dt}=\widehat{B}-\left(\frac{\beta S\left(t\right)(c_{1}A\left(t\right)+c_{2}I\left(t\right)+c_{3}H\left(t\right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(t\right)}\right)S\left(t\right)-\mu S\left(t\right)$$ (6)

Eq. 6 can be re-arranged as

$$\frac{dS}{dt}\left\{S\left(t\right)exp\right(\mu t+\beta {\int }_0^t\left(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\right)S\left(\tau \right)d\tau \left)\right\}$$

$$-\widehat{B}exp(\mu t+\beta {\int }_0^t(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\left)S\left(\tau \right)d\tau \right)=0$$

Thus,

$$S\left(t_1\right)exp(\mu t_1+\beta {\int }_0^{t_1}(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\left)S\left(\tau \right)d\tau \right)-S\left(0\right)$$

$$={\int }_0^{t_1}\widehat{B}exp(\mu \lambda +\beta {\int }_0^{\lambda }(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\left)S\left(\tau \right)d\tau \right)d\lambda$$

from which we get

$$S\left(t_1\right)=S\left(0\right)exp\{-(\mu t_1+\beta {\int }_0^{t_1}\left(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\right)S\left(\tau \right)d\tau \left)\right\}$$

$$+exp\{-(\mu t_1+\beta {\int }_0^{t_1}\left(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\right)S\left(\tau \right)d\tau \left)\right\}$$

$$\times {\int }_0^{t_1}\widehat{B}exp(\mu \lambda +\beta {\int }_0^{\lambda }(\frac{(c_{1}A\left(\tau \right)+c_{2}I\left(\tau \right)+c_{3}H\left(\tau \right))}{1+({\eta }_1-d)(S\left(t\right)+E\left(t\right)+A\left(t\right))+{\eta }_{2}I\left(t\right)+{\eta }_{3}H\left(\tau \right)}\left)S\left(\tau \right)d\tau \right)d\lambda >0.$$

Therefore, $S\left(t\right)$ is non-negative for $t>0$. In a similar fashion, it can be proven that $E>0,A>0,I>0$ and $H>0$ for $t>0.$  □

Theorem 2.2 boundedness —

$$\widehat{\psi }=\{(S\left(t\right),E\left(t\right),A\left(t\right),I\left(t\right),H\left(t\right))\in R_{+}^5\cup \left\{0\right\}:0<S\left(t\right)+E\left(t\right)+A\left(t\right)+I\left(t\right)+H\left(t\right)\le \frac{\widehat{B}}{\mu }\}$$

is the positive feasible region of Eq.5containing non-negative starting conditions in$R_{+}^5$

Proof

Assume $N_1=S\left(t\right)+E\left(t\right)+A\left(t\right)+I\left(t\right)+H\left(t\right)$ holds for $t>0$, then we obtain

$$\frac{dS}{dt}=\widehat{B}-\mu N_1\left(t\right)-\delta (I\left(t\right)+H\left(t\right))-\frac{\gamma H\left(t\right)}{1+ϵH\left(t\right)}⩽\widehat{B}-\mu N_1\left(t\right)$$ (7)

By integrating Eq. 7, we get

$$N_1\left(t\right)⩽\frac{\widehat{B}}{\mu }+(N_1\left(0\right)-\frac{\widehat{B}}{\mu })\exp (-\mu t)$$ (8)

Therefore ${\lim }_{t\to \infty }\sup N_1\left(t\right)⩽\frac{\widehat{B}}{\mu }$. In addition, $\frac{d{N}_t}{dt}<0$ if $N_1\left(0\right)>\frac{\widehat{B}}{\mu }$. This indicates that all solutions of the model 5 remain in $\psi$, which is uniformly bounded and well-behaved. □

Theorem 2.3 uniqueness —

For the Model system5if the starting values$S\left(0\right)>0,E\left(0\right)>0,A\left(0\right)>0,I\left(0\right)>0,H\left(0\right)>0$and$t_0>0$, then for all$t>0$the solutions$S\left(t\right),E\left(t\right),A\left(t\right),I\left(t\right)$and$H\left(t\right)$exist and is unique in$R_{+}^5$.

Proof

Based on the methodology in Deressa and Duressa [28], we rewrite the model system 5 in the form $x^{\prime }=g\left(x\right)$,where

$$x^{\prime }{=(S\left(t\right),E\left(t\right),A\left(t\right),I\left(t\right),H\left(t\right))}^T,$$

$$g\left(x\right)=\left(\begin{array}{c}\widehat{B}-\left(\frac{\beta (c_{1}A+c_{2}I+c_{3}H)}{1+({\eta }_1-d)(S+E+A)+{\eta }_{2}I+{\eta }_{3}H}\right)S-\mu S\\ \left(\frac{\beta (c_{1}A+c_{2}I+c_{3}H)}{1+({\eta }_1-d)(S+E+A)+{\eta }_{2}I+{\eta }_{3}H}\right)S-(\mu +{\rho }_1)E\\ {\rho }_{1}E-(\mu +\nu +{\rho }_2)A\\ {\rho }_{2}A-(\mu +\delta +h)I\\ hI-\frac{\gamma H}{1+ϵH}-(\mu +\delta )H\end{array}\right)$$

For a singular reason, that $g$ has a continuous first derivative in $R_{+}^5$, it is locally Lipschitz. With the fundamental theorem on existence and uniqueness of solution [30] and Theorems 2.1 and 2.2 established above, we can conclude that the Model system 5 has a solution that is unique, positive and bounded in $R_{+}^5.$  □

3. Equilibria analysis

3.1. The disease-free equilibrium and the basic reproduction number

At the steady state, the disease-free equilibrium $D_0(S_0,E_0,A_0,I_0,H_0)$ of the System 5 was found at the point COVID-19 does not persist, and is given by $D_0=(\frac{\widehat{B}}{\mu },0,0,0,0)$. Basic reproduction number which is computed at DFE is the strength of stability analysis of an epidemiological model. It is an essential threshold that predicts whether the spread of an infection continues or discontinues in a population. For our study, basic reproduction number is the average number of COVID-19 secondary cases generated by an index case when introduced in wholly COVID-19 virgin population. It is traditionally denoted by, $R_0$. For Model 5, we compute $R_0$ with the use of the next generation matrix method [31], focusing only on $A,I,H$ being the infected compartments. Based on the principle of this method, the infection matrix, $F$ and the transition matrix, V, evaluated at $DFE$ are given respectively in Eq. 9

$$F=\left(\begin{array}{cccc}0& \frac{\beta S_0{c}_1}{1+({\eta }_1-d)S_0}& \frac{\beta S_0{c}_2}{1+({\eta }_1-d)S_0}& \frac{\beta S_0{c}_3}{1+({\eta }_1-d)S_0}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\text{and}V=\left(\begin{array}{cccc}\mu +{\rho }_1& 0& 0& 0\\ -{\rho }_1& \mu +\nu +{\rho }_2& 0& 0\\ 0& -{\rho }_2& \mu +\delta +h& 0\\ 0& 0& -h& \mu +\delta +\gamma \end{array}\right)$$ (9)

With that, $R_0$ which is expressed mathematically as the spectral radius of the next generation matrix, $F{V}^{-1}$ is given by Eq. 10

$$\begin{array}{cc}{R}_0\hfill & =\rho \left(F{V}^{-1}\right)\hfill \\ & =\frac{\beta S_0{c}_1{\rho }_1}{(\mu +{\rho }_1)(\mu +\nu +{\rho }_2)(1+({\eta }_1-d)S_0)}+\frac{\beta S_0{c}_2{\rho }_1{\rho }_2}{(\mu +{\rho }_1)(\mu +\nu +{\rho }_2)(\mu +\delta +h)(1+({\eta }_1-d)S_0)}\hfill \\ & +\frac{\beta S_0{c}_3{\rho }_1{\rho }_{2}h}{(\mu +{\rho }_1)(\mu +\nu +{\rho }_2)(\mu +\delta +h)(\mu +\delta +\gamma )(1+({\eta }_1-d)S_0)}=R_{0A}+R_{0I}+R_{0H},\hfill \end{array}$$ (10)

where the three components of $R_0$ in Eq. 10 measure the contribution of COVID-19 infection from asymptomatic to susceptible ($R_{0A}$), symptomatic to susceptible ($R_{0I}$) and hospitalized to susceptible ($R_{0H}$), respectively.

3.2. Local stability of disease-free equilibrium

Theorem 3.1

The COVID-19 free equilibrium, $D_0=(\frac{\widehat{B}}{\mu },0,0,0)$ is locally asymptomatically stable if $R_0<1$ and unstable when $R_0>1$ .

Proof

The community matrix of the Model system (5) at disease-free state is given by Eq. 11.

$$J\left(D_0\right)=\left(\begin{array}{ccccc}-\mu & 0& -\frac{\beta S_0{c}_1}{1+({\eta }_1-d)S_0}& -\frac{\beta S_0{c}_2}{1+({\eta }_1-d)S_0}& -\frac{\beta S_0{c}_3}{1+({\eta }_1-d)S_0}\\ 0& -{\sigma }_1& \frac{\beta S_0{c}_1}{1+({\eta }_1-d)S_0}& \frac{\beta S_0{c}_2}{1+({\eta }_1-d)S_0}& \frac{\beta S_0{c}_3}{1+({\eta }_1-d)S_0}\\ 0& {\rho }_1& -{\sigma }_2& 0& 0\\ 0& 0& {\rho }_2& -{\sigma }_3& 0\\ 0& 0& 0& h& -{\sigma }_4\end{array}\right),$$ (11)

where

${\sigma }_1=\mu +{\rho }_1$, ${\sigma }_2=\mu +\nu +{\rho }_2$, ${\sigma }_3=\mu +\delta +h$, ${\sigma }_4=\mu +\delta +\gamma$.

From the community matrix $J\left(D_0\right)$, we obtain the eigenvalue $\lambda =-\mu$ and Eq. 12

$${\lambda }^4+m_3{\lambda }^3+m_2{\lambda }^2+m_1\lambda +m_0=0,$$ (12)

where

$$\begin{array}{cc}{m}_3\hfill & ={\sigma }_1+{\sigma }_2+{\sigma }_3+{\sigma }_4,\hfill \\ m_2\hfill & ={\sigma }_1{\sigma }_2(1-R_{0A})+({\sigma }_1+{\sigma }_2)({\sigma }_3+{\sigma }_4)+{\sigma }_3{\sigma }_4,\hfill \\ m_1\hfill & ={\sigma }_1{\sigma }_2{\sigma }_3(1-R_{0A}-R_{0I})+{\sigma }_1{\sigma }_2{\sigma }_4(1-R_{0A})+{\sigma }_3{\sigma }_4({\sigma }_1+{\sigma }_2),\hfill \\ m_0\hfill & ={\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_4(1-R_0),\hfill \end{array}$$ (13)

By Routh-Hurwitz criteria [32], Eq. 12 has negative real roots if the coefficients $m_i,i=0,1,2,3$ and $m_1(m_2{m}_3-m_1)-m_3^2{m}_0$ are all positive. Clearly, it can be seen from Eq. 13 that $m_3>0,m_2>0,m_1>0$ and $m_0>0$ if $R_0<1$. This implies that $R_{0A}<1$ and $R_{0A}+R_{0I}<1$ since $R_0$ is the sum of $R_{0A},R_{0I}$ and $R_{0H}$. Intuitively, it follows that the disease-free equilibrium $D_0$ is locally asymptotically stable provided $m_1(m_2{m}_3-m_1)-m_3^2{m}_0$ is strictly positive. Otherwise unstable. □

3.3. Threshold analysis

To assess the impact of inhibition factor ${\eta }_1$ adopted by susceptible, exposed and asymptomatic individuals, hospitalization of individuals with COVID-9 symptoms $\left(h\right)$, treatment rate $\left(\gamma \right)$ and denial rate (d) of COVID-19, we compute the partial derivatives of $R_0$ with respect to ${\eta }_1,h,\gamma$ and $d$ as presented below:

$$\begin{array}{cc}\hfill \frac{\partial R_0}{\partial {\eta }_1}& =-\left(\frac{\sqrt{\beta }{S}_0}{(1+({\eta }_1-d)S_0)}\right){}^2(\frac{{\rho }_1{c}_1}{{\sigma }_1{\sigma }_2}+\frac{{\rho }_2{\rho }_1{c}_2}{{\sigma }_1{\sigma }_2{\sigma }_3}+\frac{h{\rho }_2{\rho }_1{c}_3}{{\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_4}),\hfill \\ \hfill \frac{\partial R_0}{\partial h}& =\frac{\beta S_0}{(1+({\eta }_1-d)S_0)}\left(\frac{(\mu +\delta )c_3-(\mu +\delta +\gamma )c_2}{(\mu +{\rho }_1)(\mu +\nu +{\rho }_2)(\mu +\delta +\gamma ){(\mu +\delta +h)}^2}\right),\hfill \\ \hfill \frac{\partial R_0}{\partial \gamma }& =-\frac{\beta S_0}{(1+({\eta }_1-d)S_0)}\left(\frac{h{\rho }_2{\rho }_1{c}_3}{(\mu +{\rho }_1)(\mu +\nu +{\rho }_2)(\mu +\delta +h){(\mu +\delta +\gamma )}^2}\right),\hfill \\ \hfill \frac{\partial R_0}{\partial d}& =\left(\frac{\sqrt{\beta }{S}_0}{(1+({\eta }_1-d)S_0)}\right){}^2(\frac{{\rho }_1{c}_1}{{\sigma }_1{\sigma }_2}+\frac{{\rho }_2{\rho }_1{c}_2}{{\sigma }_1{\sigma }_2{\sigma }_3}+\frac{h{\rho }_2{\rho }_1{c}_3}{{\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_4}).\hfill \end{array}$$

From the above analysis, we observe that $\frac{\partial R_0}{\partial {\eta }_1}<0,\frac{\partial R_0}{\partial \gamma }<0,\frac{\partial R_0}{\partial h}<0$ if $(\mu +\delta )c_3<(\mu +\delta +\gamma )c_2$ and $\frac{\partial R_0}{\partial d}>0$. The last inequality clearly explains that the denial of COVID-19 existence will heavily contribute to the rapid spread of the virus and challenged the process of the disease prevention. The first three inequalities show that inhibition factors, hospitalization and treatment by Holling type II curtail the outgrowing cases of COVID-19 pandemic.

It is also invaluable to determine the upper bounds for infection and denial rates as well as the lower bound for inhibition factors.

If

$$R_0=\frac{\beta S_0\Omega }{1+({\eta }_1-d)S_0}<1,$$

$$\text{with}\Omega =\frac{{\rho }_1{c}_1}{{\sigma }_1{\sigma }_2}+\frac{{\rho }_2{\rho }_1{c}_2}{{\sigma }_1{\sigma }_2{\sigma }_3}+\frac{h{\rho }_2{\rho }_1{c}_3}{{\sigma }_1{\sigma }_2{\sigma }_3{\sigma }_4},$$

then,

$$\beta <\frac{1+({\eta }_1-d)S_0}{\beta S_0\Omega },d<\frac{1}{S_0}+{\eta }_1(1-\frac{\beta \Omega }{{\eta }_1})$$

and the lower bound for ${\eta }_1$ is:

$${\eta }_1>-\frac{1}{S_0}+d+\beta \Omega .$$

Using the values in Table 1, we obtain the Figures SM1 and SM2.

3.4. Existence of endemic equilibria and backward bifurcation

Before assessing the global asymptotic dynamics of the DFE, it is important to find number of equilibrium solutions of system 5. To do so, we can set the right hand sides of System 5 to zero (at steady state) to arrive at the following expressions for the endemic equilibrium point (EEP), $D_{*}(S_{*},E_{*},A_{*},I_{*},H_{*})$ as:

$$\begin{array}{cc}\hfill S_{*}& =\frac{\widehat{B}h{\rho }_2{\rho }_1(1+ϵH_{*})-{\sigma }_1{\sigma }_2{\sigma }_3(\gamma +q+qϵH_{*})H_{*}}{\mu h{\rho }_2{\rho }_1(1+ϵH_{*})},\hfill \\ \hfill E_{*}& =\frac{{\sigma }_2{\sigma }_3(\gamma +q+qϵH_{*})H_{*}}{h{\rho }_2{\rho }_1(1+ϵH_{*})},\hfill \\ \hfill A_{*}& =\frac{{\sigma }_3(\gamma +q+qϵH_{*})H_{*}}{h{\rho }_2(1+ϵH_{*})},\hfill \\ \hfill I_{*}& =\frac{(\gamma +q+qϵH_{*})H_{*}}{h(1+ϵH_{*})},\hfill \end{array}$$

and $H_{*}$ is the root of the following cubic polynomial equation

$$P{H}_{*}^3+Q{H}_{*}^2+R{H}_{*}+U=0,$$ (14)

where

$$\begin{array}{cc}P\hfill & ={\sigma }_1{\sigma }_2{\sigma }_3{q}^2{ϵ}^2[{\sigma }_2{\sigma }_3({\eta }_1-d)(\nu +{\rho }_1)+\mu {\rho }_1({\sigma }_3({\eta }_1-d)+{\rho }_2{\eta }_2+h{\rho }_2{\eta }_3)]\hfill \\ & +\beta {\sigma }_1{\sigma }_2{\sigma }_{3}q{ϵ}^2{\rho }_1({\sigma }_{3}q{c}_1+{\rho }_{2}q{c}_2+h{\rho }_2{c}_3),\hfill \\ Q\hfill & =Q_1-Q_2,\hfill \\ R\hfill & ={\sigma }_1{\sigma }_2{\sigma }_{3}h{\rho }_2{\rho }_1ϵ(\gamma +q)(\mu +({\eta }_1-d)\widehat{B})(1-R_0)-h{\rho }_2{\rho }_{1}qϵ[Q_3-Q_4],\hfill \\ U\hfill & ={\sigma }_1{\sigma }_2{\sigma }_{3}h{\rho }_2{\rho }_1(\gamma +q)(\mu +({\eta }_1-d)\widehat{B})(1-R_0),\hfill \end{array}$$ (15)

with

$$\begin{array}{cc}\hfill Q_1& =\frac{2{\sigma }_1^2{\sigma }_2^2{\sigma }_3^{2}q{ϵ}^2}{\widehat{B}}(\gamma +q)(\mu +({\eta }_1-d)\widehat{B})R_0+2{\sigma }_1{\sigma }_2{\sigma }_3\mu {\rho }_1ϵ(\gamma +q)({\sigma }_{3}q({\eta }_1-d)+q{\rho }_2{\eta }_2+h{\rho }_2{\eta }_3)\hfill \\ & +{\sigma }_1{\sigma }_2{\sigma }_{3}h{\rho }_2{\rho }_1ϵ(\beta \gamma c_3+qϵ(\mu +({\eta }_1-d)\widehat{B})),\hfill \\ \hfill Q_2& =\mu {\sigma }_1{\sigma }_2{\sigma }_{3}h{\rho }_2{\rho }_1ϵ\gamma {\eta }_3+2{\sigma }_1{\sigma }_2^2{\sigma }_3^{2}qϵ({\eta }_1-d)(\nu +{\rho }_1)(\gamma +q)\hfill \\ & +\beta \widehat{B}h{q}^2{\rho }_2{\rho }_1^2({\sigma }_{3}q{c}_1+{\rho }_{2}q{c}_2+h{\rho }_2{c}_3),\hfill \\ \hfill Q_3& =\beta \widehat{B}{\rho }_1({\sigma }_{3}q{c}_1+{\rho }_{2}q{c}_2+h{\rho }_2{c}_3),\hfill \\ \hfill Q_4& =h{\rho }_2{\rho }_1(\mu +({\eta }_1-d)\widehat{B})\text{and}\hfill \\ \hfill q& =\mu +\delta .\hfill \end{array}$$

It is obvious from Eq. 15 that $P>0$ (since all the parameters of the model are non-negative with $0\le d<{\eta }_1$) and $U>0$ provided that $R_0<1$. Therefore, the number of possible positive roots of Eq. 14 can depends on the signs of $Q$ and $R$. Following the work of Gumel and Sharomi [33], this can be analyzed using the Descarte’s rule of signs on the cubic $g\left(x\right)=P{x}^3+Q{x}^2+Rx+U$, given Eq. 14 (with $x=H_{*}$). The results (Theorem 3.2 and Conjuncture 1) below follow from the various cases listed in Table 2 .

Theorem 3.2

The COVID-19 Model5

• (a)has a unique endemic equilibrium point if$R_0>1$and whenever the cases 1, 2 and 3 are satisfied;
• (b)could have at least one endemic equilibrium point if$R_0>1$and case 4 is satisfied;
• (c)could have at least two endemic equilibria if$R_0<1$and cases 2–4 are satisfied.

Conjuncture 1

The COVID-19 Model 5 has more than one positive endemic equilibrium point when $R_0>1$ and could have zero or two positive equilibria when $R_0<1$.

Table 2.

Number of possible positive real roots of $g\left(x\right)$ for $R_0<1$ and $R_0>1$ .

Cases	P	Q	R	U	$R_0$	No. of sign changes	No. of endemic Points
1	+	+	+	+	$R_0<1$	0	0
	+	+	+	-	$R_0>1$	1	1
2	+	-	-	+	$R_0<1$	2	0, 2
	+	-	-	-	$R_0>1$	1	1
3	+	+	-	+	$R_0<1$	2	0, 2
	+	+	-	-	$R_0>1$	1	1
4	+	-	+	+	$R_0<1$	2	0, 2
	+	-	+	-	$R_0>1$	3	1, 3

The occurrence of multiple endemic equilibria at $R_0<1$ (as indicated in Table 2) suggests the likelihood of backward bifurcation (see [33]), where the stable DFE co-exists with a stable endemic equilibrium point(EEP), in a scenario when the basic reproduction number is below unity. This is explored via numerical simulations (rigorous result can be obtained using centre manifold theory (see [34]) as illustrated in the Fig. 2 and Figure SM3. Fig. 2 depicts the associated backward bifurcation diagram. Furthermore Figure SM3 shows convergence to both the DFE and the EEP for the total asymptomatic and symptomatic individuals when $R_0<1$ depending on the initial sizes of the sub-populations. The biological consequence of this result is that the effective control of COVID-19 in a population (when $R_0<1$) is independent on the initial sizes of the sub-populations of the Model 5 within the bounds of denial rate.

Fig. 2.

Bifurcation diagram for the Model 5. Parameter values used are: $\beta =0.009,\widehat{B}=180,\delta =0.8\text{and}d=0.51$ (so that $R_{0A}=0.3378752329,R_{0I}=0.03434622182,R_{0H}=0.0003234594838$ and $R_0=0.3725449142$). All other parameter values are in Table 1. It is essential to note that the values of the parameters were used for illustrative purpose only, and may not be realistic epidemiologically.

4. Global analysis

Theorem 4.1

The COVID-19-free equilibrium $D_0$ is globally asymptotically stable if $R_0<1$ .

Proof

The work of Naji and Abdulateef [14] is the basis by which this theorem can be verified. Adopting a similiar real-valued function

$$L_0=\frac{{(S-S_0)}^2}{2}+E+A+I+H$$

with the derivative

$$L_0^{{}^{\prime }}\left(t\right)=(S-S_0)\frac{dS}{dt}+\frac{dE}{dt}+\frac{dA}{dt}+\frac{dI}{dt}+\frac{dH}{dt}.$$ (16)

Upon substitution of System 5 in Eq. 16, we have

$$\begin{array}{cc}\hfill L_0^{{}^{\prime }}\left(t\right)& =(S-S_0)(\widehat{B}-\overline{f}S-\mu S)+(\overline{f}S-(\mu +{\rho }_1)E)+({\rho }_{1}E-(\mu +\nu +{\rho }_2)A)\hfill \\ & +({\rho }_{2}A-(\mu +\delta +h)I)+(hI-\overline{g}\left(H\right)-(\mu +\delta )H).\hfill \end{array}$$

Taking $\widehat{B}=\mu S_0$, we obtain

$$\begin{array}{cc}\hfill L_0^{{}^{\prime }}\left(t\right)& =-\mu {(S-S_0)}^2-\overline{f}{S}^2+(1+S_0)\overline{f}S-\mu E-(\mu +\nu )A-(\mu +\delta )I\hfill \\ & -(\mu +\delta +\frac{\gamma }{1+ϵH})H\hfill \\ & \le -\mu {(S-S_0)}^2+(1+S_0)\left(\frac{\beta (c_{1}A+c_{2}I+c_{3}H)}{1+({\eta }_1-d)(S+E+A)+{\eta }_{2}I+{\eta }_{3}H}\right)S\hfill \\ & -(\mu +\nu )A-(\mu +\delta )I-(\mu +\delta +\frac{\gamma }{1+ϵH})H\hfill \end{array}$$

If $\overline{a}=min({\eta }_1-d,{\eta }_2,{\eta }_3,{\eta }_4)={\eta }_1-d$ (by assumption), then

$$\begin{array}{cc}\hfill L_0^{{}^{\prime }}\left(t\right)& =-\mu {(S-S_0)}^2-\overline{f}{S}^2+(1+S_0)\overline{f}S-\mu E-(\mu +\nu )A-(\mu +\delta )I\hfill \\ & -(\mu +\delta +\frac{\gamma }{1+ϵH})H\hfill \\ & \le -\mu {(S-S_0)}^2+(1+S_0)\left(\frac{\beta (c_{1}A+c_{2}I+c_{3}H)}{1+\overline{a}N}\right)S\hfill \\ & -(\mu +\nu )A-(\mu +\delta )I-(\mu +\delta +\frac{\gamma }{1+ϵH})H.\hfill \end{array}$$

But, $N\left(t\right)\le \frac{\widehat{B}}{\widehat{\mu }},S\left(t\right)\le \frac{\widehat{B}}{\mu }$ and $S_0=\frac{\widehat{B}}{\mu }$

$$\begin{array}{cc}{L}_0^{{}^{\prime }}\left(t\right)\hfill & \le -\mu {(S-S_0)}^2-(\mu +\nu )(1-\frac{\beta c_1{S}_0(\mu +\widehat{B})}{(\mu +\overline{a}\widehat{B})(\mu +\nu )})A-(\mu +\delta )(1-\frac{\beta c_2{S}_0(\mu +\widehat{B})}{(\mu +\overline{a}\widehat{B})(\mu +\delta )})I\hfill \\ & -(\mu +\delta +\frac{\gamma }{1+ϵH})(1-\frac{\beta c_3{S}_0(\mu +\widehat{B})}{(\mu +\overline{a}\widehat{B})(\mu +\delta +\frac{\gamma }{1+ϵH})})H.\hfill \end{array}$$ (17)

Now replacing $\beta c_1{S}_0,\beta c_2{S}_0$ and $\beta c_3{S}_0$ in terms of $R_0$ into Eq. 17, we have

$$\begin{array}{cc}\hfill L_0^{{}^{\prime }}\left(t\right)& \le -\mu {(S-S_0)}^2-(\mu +\nu )(1-R_{0A}\frac{(\mu +\widehat{B})}{(\mu +\overline{a}\widehat{B})(\mu +\nu )})A-(\mu +\delta )(1-R_{0I}\frac{(\mu +\widehat{B})}{(\mu +\overline{a}\widehat{B})(\mu +\delta )})I\hfill \\ & -(\mu +\delta +\frac{\gamma }{1+ϵH})(1-R_{0H}\frac{(\mu +\widehat{B})}{(\mu +\overline{a}\widehat{B})(\mu +\delta +\frac{\gamma }{1+ϵH})})H.\hfill \end{array}$$

Therefore, $L_0^{{}^{\prime }}\left(t\right)<0$ if $R_{0A}<1,R_{0I}<1,R_{0H}<1$ implying that $R_0<1$ and $L_0^{{}^{\prime }}\left(t\right)=0$ if $A=I=H=0$ and $S=S_0=\frac{\widehat{B}}{\mu }$.

We conclude that the largest compact invariant set in $\{(S,E,A,I,H)\in \widehat{\psi }:L_0^{{}^{\prime }}\left(t\right)=0\}$ is the only set $\left\{D_0\right\}$. By La Salle’s invariance principle [35], the COVID-19-free equilibrium is globally asymptotically stable if $R_0<1$  □

The time series plots showing the number of exposed, asymptomatic, symptomatic and hospitalized people when the condition $R_0<1$ in the above theorem is given by Figure SM4.

Theorem 4.2

The endemic equilibrium point$D_{*}$is globally asymptotically stable if$R_0>1$and$\gamma =0$.

Proof

Using the work in [36], let $\lambda \left(I\right)=(C_{1}A+C_{2}I+C_{3}H)\Psi \left(I\right)$, and $\Psi \left(H\right)=\frac{1}{1+ϵH}$, where $\Psi \left(I\right)=\frac{1}{1+({\eta }_1-d)(S+E+A)+{\eta }_{2}I+{\eta }_{3}H}$

Then, we adopt the following Lyapunov function

$$\begin{array}{cc}L\left(x\right)\hfill & =(S-S_{*}-S_{*}\ln \frac{S}{S_{*}})+(E-E_{*}-E_{*}\ln \frac{E}{E_{*}})+\frac{\lambda \left(I_{*}\right)\beta S_{*}}{{\rho }_1{E}_{*}}(A-A_{*}-A_{*}\ln \frac{A}{A_{*}})\hfill \\ & +\frac{\lambda \left(I_{*}\right)\beta S_{*}}{{\rho }_2{A}_{*}}(I-I_{*}-I_{*}\ln \frac{I}{I_{*}})+\frac{\lambda \left(I_{*}\right)\beta S_{*}}{h{I}_{*}}(H-H_{*}-H_{*}\ln \frac{H}{H_{*}}).\hfill \end{array}$$ (18)

Putting System 5 into the derivative of the Lyapunov function in Eq. 18 gives

$$\begin{array}{cc}{L}^{\prime }\hfill & =(1-\frac{S_{*}}{S})(\widehat{B}-\lambda \left(I\right)\beta S-\mu S)+(1-\frac{E_{*}}{E})(\lambda \left(I\right)\beta S-(\mu +{\rho }_1)E)\hfill \\ & +\frac{\lambda \left(I_{*}\right)\beta S_{*}}{{\rho }_1{E}_{*}}(1-\frac{A_{*}}{A})({\rho }_{1}E-(\mu +\nu +{\rho }_2)A)+\frac{\lambda \left(I_{*}\right)\beta S_{*}}{{\rho }_2{A}_{*}}(1-\frac{I_{*}}{I})({\rho }_{2}A-(\mu +\delta +h)I)\hfill \\ & +\frac{\lambda \left(I_{*}\right)\beta S_{*}}{h{I}_{*}}(1-\frac{H_{*}}{H})(hI-\gamma H\Psi \left(H\right)-(\mu +\delta )H).\hfill \end{array}$$ (19)

At equilibrium state

$$\widehat{B}=\lambda \left(I_{*}\right)\beta S_{*}+\mu S_{*},\mu +{\rho }_1=\frac{\lambda \left(I_{*}\right)\beta S_{*}}{E_{*}},\mu +\nu +{\rho }_2=\frac{{\rho }_1{E}_{*}}{A_{*}},$$

$$\mu +\delta +h=\frac{{\rho }_2{A}_{*}}{I_{*}},\mu +\delta =\frac{h{I}_{*}}{H_{*}}-\gamma \Psi \left(H_{*}\right).$$

Expanding Eq. 19 at endemic equilibrium point and after simplifying, we get

$$\begin{array}{cc}\hfill L^{\prime }& =\mu S_{*}(2-\frac{S_{*}}{S}-\frac{S}{S_{*}})+\beta S_{*}\lambda \left(I_{*}\right)[5+\frac{\lambda \left(I\right)}{\lambda \left(I_{*}\right)}-\frac{S_{*}}{S}-\frac{S}{S_{*}}\frac{E_{*}}{E}\frac{\lambda \left(I\right)}{\lambda \left(I_{*}\right)}-\frac{E}{E_{*}}\frac{A_{*}}{A}-\frac{A}{A_{*}}\frac{I_{*}}{I}\hfill \\ & -\frac{H}{H_{*}}-\frac{I}{I_{*}}\frac{H_{*}}{H}]+\frac{\lambda \left(I_{*}\right)\beta S_{*}}{h{I}_{*}}\gamma H_{*}\Psi \left(H_{*}\right)[-1+\frac{H}{H_{*}}-\frac{H}{H_{*}}\frac{\Psi \left(H\right)}{\Psi \left(H_{*}\right)}+\frac{\Psi \left(H\right)}{\Psi \left(H_{*}\right)}]\hfill \end{array}$$

But, setting $\gamma =0$ and $\frac{\lambda \left(I\right)}{\lambda \left(I_{*}\right)}\le 1$, since $\lambda \left(I\right)$ is an increasing function, we have

$$L^{\prime }=\mu S_{*}(2-\frac{S_{*}}{S}-\frac{S}{S_{*}})+\beta S_{*}\lambda \left(I_{*}\right)[6-\frac{S_{*}}{S}-\frac{S}{S_{*}}\frac{E_{*}}{E}-\frac{E}{E_{*}}\frac{A_{*}}{A}-\frac{A}{A_{*}}\frac{I_{*}}{I}-\frac{H}{H_{*}}-\frac{I}{I_{*}}\frac{H_{*}}{H}]$$

By arithmetic-geometric theorem,

$$2\le \frac{S_{*}}{S}+\frac{S}{S_{*}},6\le \frac{S_{*}}{S}+\frac{S}{S_{*}}\frac{E_{*}}{E}+\frac{E}{E_{*}}\frac{A_{*}}{A}+\frac{A}{A_{*}}\frac{I_{*}}{I}+\frac{H}{H_{*}}+\frac{I}{I_{*}}\frac{H_{*}}{H}$$

.

Therefore, $L^{\prime }\le 0$. Hence, the System 5 is globally asymptotically stable if $R_0>1$ by La Salle invariance principle. □

The biological significance of the above result is that, in the absence of maximum treatment for the hospitalized people $(\gamma =0)$, COVID-19 pandemic will not be eliminated from the community if the basic reproduction number, $R_0$ rise above unity.

5. Optimal control analysis

In this section, we denote mathematically the three time series controls as ${\xi }_1\left(t\right),{\xi }_2\left(t\right)=h\left(t\right)$ and ${\xi }_3\left(t\right)=\gamma \left(t\right)$ for $t\in [0,t_f]$, where $t_f$ is the time period of the intervention to represent the additional preventive protocol(creation of awareness about COVID-19) for the susceptible and exposed, testing and isolation (hospitalization) and treatment interventions. The application of preventive protocol reduces the force of COVID-19 infection as modelled in Eq. 20

$$\overline{f_0}\left(t\right)=\frac{\beta (1-{\xi }_1\left(t\right))(c_{1}A+c_{2}I+c_{3}H)}{1+({\eta }_1-d)(S+E+A)+{\eta }_{2}I+{\eta }_{3}H}.$$ (20)

The testing boost the chances of hospitalization and medical care of the infected people while treatment program may lead to their recovery. Our main motive which is to minimize the number of COVID-19 cases and the cost of control measures involved, can be mathematically translated into finding piecewise continuous controls ${\xi }_1^{*},{\xi }_2^{*},{\xi }_3^{*}$ and associated state variables $S^{*},E^{*},A^{*},I^{*},H^{*}$ and $R^{*}$ that minimize the objective functional $J$ given by Eq. 21

$$J({\xi }_1^{*},{\xi }_2^{*},{\xi }_3^{*})={\int }_0^{t_f}(kE+pA+qI+rH+\frac{m_1}{2}{\xi }_1^2+\frac{m_2}{2}{\xi }_2^2+\frac{m_3}{2}{\xi }_3^2)dt,$$ (21)

where $m_i>0$ for $i=1,2,3$ weighs constants that balance the optimal controls. The terms $m_1{\xi }_1^2,m_2{\xi }_2^2$ and $m_3{\xi }_3^2$ are the cost associated with preventive protocols for susceptible and exposed, testing and hospitalization for symptomatic individuals and attaining maximum treatment for hospitalized, respectively. Therefore, greater values of $m_1,m_2$ and $m_3$ will indicate higher implementation costs for the preventive protocols, testing and hospitalization and maximum treatment for COVID-19, respectively. The above terms are assumed non-linear and quadratic due to denial in observing preventive protocols, cost of testing and hospitalizing symptomatic individuals and achieving maximum treatment for the hospitalized individuals.

Optimal control problem is to minimize the objective functional subject to the non-linear system in Eq. 22

$$\begin{array}{cc}\frac{dS}{dt}\hfill & =\widehat{B}-\overline{f_0}\left(t\right)S-\mu S,\hfill \\ \frac{dE}{dt}\hfill & =\overline{f_0}\left(t\right)S-(\mu +{\rho }_1)E,\hfill \\ \frac{dA}{dt}\hfill & ={\rho }_{1}E-(\mu +\nu +{\rho }_2)A,\hfill \\ \frac{dI}{dt}\hfill & ={\rho }_{2}A-(\mu +\delta +{\xi }_2\left(t\right))I,\hfill \\ \frac{dH}{dt}\hfill & ={\xi }_2\left(t\right)I-\frac{{\xi }_3\left(t\right)H}{1+ϵH}-(\mu +\delta )H,\hfill \\ \frac{dR}{dt}\hfill & =\nu A+\frac{{\xi }_3\left(t\right)H}{1+ϵH}-\mu R.\hfill \end{array}$$ (22)

We set the optimal controls ${\xi }_i^{*},{\xi }_2^{*}$ and ${\xi }_3^{*}$ such that

$$J({\xi }_i^{*},{\xi }_2^{*},{\xi }_3^{*})=\underset{▽}{\underbrace{min}}J({\xi }_i,{\xi }_2,{\xi }_3),$$ (23)

where $▿=\left\{({\xi }_i,{\xi }_2,{\xi }_3)\right|0⩽{\xi }_i\left(t\right)⩽1,i=1,2,3\}$ is the set for the controls.

Remark 1

In this set $▿$, when the rate of a control is zero, then no investment in control has been made. On the other hand, when the value of a control is one, then a control effort has been achieved maximally.

At this point, we employ the Pontryagin’s maximum principle to obtain the necessary conditions for getting optimal controls ${\xi }_i^{*},{\xi }_2^{*}$ and ${\xi }_3^{*}$ that satisfy Eq. 23 with constraint model in Eq. 22. The principle converts Eqs. 21 - 23 into a minimization problem of Hamiltonian function $\overline{H_{\alpha }}$ in terms of $({\xi }_i,{\xi }_2,{\xi }_3)$

$$\overline{H_{\alpha }}=kE\left(t\right)+pA\left(t\right)+qI\left(t\right)+rH\left(t\right)+\sum _{i=1}^3\frac{m_i}{2}{\xi }_i^2\left(t\right)+\sum _{i=1}^6{\zeta }_i{g}_i,$$

where $g_i$ denote the $ith$ right side in Eq. 22 and ${\zeta }_i$ is the $ith$ adjoint that satisfy the co-state functions given in Eq. 24

$$\begin{array}{cc}{\zeta }_1^{\prime }\hfill & =-\frac{\partial {\overline{H}}_{\alpha }}{\partial S}=({\zeta }_1-{\zeta }_2)\beta (1-{\xi }_1\left(t\right))\left(\frac{L_1{L}_2}{T^2}\right)+\mu {\zeta }_1,\hfill \\ {\zeta }_2^{\prime }\hfill & =-\frac{\partial {\overline{H}}_{\alpha }}{\partial E}=-k+({\zeta }_2-{\zeta }_1)\beta (1-{\xi }_1\left(t\right))({\eta }_1-d)S\left(\frac{L_1}{T^2}\right)+({\zeta }_2-{\zeta }_3){\rho }_1+\mu {\zeta }_2,\hfill \\ {\zeta }_3^{\prime }\hfill & =-\frac{\partial {\overline{H}}_{\alpha }}{\partial A}=-p+({\zeta }_1-{\zeta }_2)\beta (1-{\xi }_1\left(t\right))S\left(\frac{L_3}{T^2}\right)+({\zeta }_3-{\zeta }_4){\rho }_2+({\zeta }_3-{\zeta }_6)\nu +\mu {\zeta }_3,\hfill \\ {\zeta }_4^{\prime }\hfill & =-\frac{\partial {\overline{H}}_{\alpha }}{\partial I}=-q+({\zeta }_1-{\zeta }_2)\beta (1-{\xi }_1\left(t\right))S\left(\frac{L_4}{T^2}\right)+({\zeta }_4-{\zeta }_5){\xi }_2\left(t\right)+(\mu +\delta ){\zeta }_4,\hfill \\ {\zeta }_5^{\prime }\hfill & =-\frac{\partial {\overline{H}}_{\alpha }}{\partial H}=-r+({\zeta }_1-{\zeta }_2)\beta (1-{\xi }_1\left(t\right))S\left(\frac{L_5}{T^2}\right)+({\zeta }_5-{\zeta }_6)\frac{{\xi }_3\left(t\right)}{{(1+ϵH)}^2}+(\mu +\delta ){\zeta }_5,\hfill \\ {\zeta }_6^{\prime }\hfill & =-\frac{\partial {\overline{H}}_{\alpha }}{\partial R}=\mu {\zeta }_6,\hfill \end{array}$$ (24)

where

$$L_1=c_1{A}^{*}+c_2{I}^{*}+c_3{H}^{*},L_2=1+({\eta }_1-d)(E^{*}+A^{*})+{\eta }_2{I}^{*}+{\eta }_3{H}^{*}$$

$$L_3=c_1(1+({\eta }_1-d)(S^{*}+E^{*}))+({\eta }_2{c}_1-({\eta }_1-d)c_2)I^{*}+({\eta }_3{c}_1-({\eta }_1-d)c_3)H^{*}$$

$$L_4=c_2(1+({\eta }_1-d)(S^{*}+E^{*}))+(({\eta }_1-d)c_2-{\eta }_2{c}_1)A^{*}+({\eta }_3{c}_2-{\eta }_2{c}_3)H^{*}$$

$$L_5=c_3(1+({\eta }_1-d)(S^{*}+E^{*}))+(({\eta }_1-d)c_3-{\eta }_3{c}_1)A^{*}+({\eta }_2{c}_3-{\eta }_3{c}_2)I^{*}$$

$$T=1+({\eta }_1-d)(S^{*}+E^{*}+A^{*})+{\eta }_2{I}^{*}+{\eta }_3{H}^{*}$$

and final time conditions ${\zeta }_i\left(t_f\right)=0,i=1,2,\dots ,6.$

The procedure for obtaining the optimal controls $\xi =({\xi }_i^{*},{\xi }_2^{*},{\xi }_3^{*})$ are mentioned below [37].

• Step 1.
  Minimize the Hamiltonian function ${\overline{H}}_{\alpha }$ with respect to $\xi$, we get

  $${\xi }_1^{*}=\{\begin{array}{cc}0,\hfill & \text{if}{\xi }_1⩽0\hfill \\ \frac{({\zeta }_2-{\zeta }_1)\beta (c_1{A}^{*}+c_2{I}^{*}+c_3{H}^{*})}{m_1(1+({\eta }_1-d)(S^{*}+E^{*}+A^{*})+{\eta }_2{I}^{*}+{\eta }_3{H}^{*})},\hfill & \text{if}0<{\xi }_1<1\hfill \\ 1,\hfill & \text{if}{\xi }_1⩾1\hfill \end{array}$$

  $${\xi }_2^{*}=\{\begin{array}{cc}0,\hfill & \text{if}{\xi }_2⩽0\hfill \\ \frac{({\zeta }_4-{\zeta }_5)I^{*}}{m_2},\hfill & \text{if}0<{\xi }_2<1\hfill \\ 1,\hfill & \text{if}{\xi }_2⩾1\hfill \end{array}$$

  $${\xi }_3^{*}=\{\begin{array}{cc}0,\hfill & \text{if}{\xi }_3⩽0\hfill \\ \frac{({\zeta }_5-{\zeta }_6)H^{*}}{m_3(1+ϵH^{*})},\hfill & \text{if}0<{\xi }_3<1\hfill \\ 1,\hfill & \text{if}{\xi }_3⩾1\hfill \end{array}$$
• Step 2.

  Obtain the solutions of the system $\frac{dx}{dt}=\frac{\partial {\overline{H}}_{\alpha }}{\partial \zeta }$, where $x=(S,E,A,I,H,R),\zeta =({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_6)$ using the initial condition $x_0=(S\left(0\right),E\left(0\right),A\left(0\right),I\left(0\right),H\left(0\right),R\left(0\right))$.

• Step 3.

  Determine the solution of the co-state system $\frac{d\zeta }{dt}=-\frac{\partial {\overline{H}}_{\alpha }}{\partial x}$ with the final time conditions ${\zeta }_i\left(t_f\right)=0$, $i=1,2,\dots ,6$.

In line with the above procedure, the optimal control $({\xi }_i^{*},{\xi }_2^{*},{\xi }_3^{*})$ is given in the next theorem.

Theorem 5.1

The optimal controls$({\xi }_i^{*},{\xi }_2^{*},{\xi }_3^{*})$that minimizes the cost function$J({\xi }_i,{\xi }_2,{\xi }_3)$on$▿$subject to the system in Eq.22is:

$${\xi }_1^{*}=max\{0,min\{1,\frac{({\zeta }_2-{\zeta }_1)\beta (c_1{A}^{*}+c_2{I}^{*}+c_3{H}^{*})}{m_1(1+({\eta }_1-d)(S^{*}+E^{*}+A^{*})+{\eta }_2{I}^{*}+{\eta }_3{H}^{*})}\left\}\right\},$$

$${\xi }_2^{*}=max\{0,min\{1,\frac{({\zeta }_4-{\zeta }_5)I^{*}}{m_2}\left\}\right\},$$

$${\xi }_3^{*}=max\{0,min\{1,\frac{({\zeta }_5-{\zeta }_6)H^{*}}{m_3(1+ϵH^{*})}\left\}\right\},$$

where${\zeta }_i,i=1,2,\dots ,6$remain the solutions of the co-state system in Eq.24.

6. Numerical results and discussions

In this section, numerical simulations are carried out using the parameters values in Table 1. We make the comparison of the optimal control system in Eq. 22 in terms of denial and without denial. The fourth-order Runge-Kutta (RK4) scheme is employed to solve the optimal control strategy; and adopted the forward RK4 scheme and backward RK4 scheme to solve the state system and co-state system, respectively. Apart from the initial conditions and other parameters values given in Table 1, we take $m_3=50,m_2=30$ and $m_1=15$ since the cost of treating hospitalized people is higher than the cost of isolating symptomatic people which in turn is higher than the cost of protecting susceptible. We also consider $k=p=0.1,q=0.15$ and $r=0.015$ for the numerical simulations.

It is evident from Figure SM1 that cases of COVID-19 grow exponentially in the population when denial rate exceeds 0.45 since $R_0>1$. However, when $d$ satisfy the inequality $0\le d<0.45$, $R_0<1$ and the disease can be curtailed. Figure SM2 displays the level set of $R_0$ in $\beta$ and d planes. They show that the number of cases of COVID-19 increases as d increases. This implies that with denial: preventable strategies, hospitalization and treatment measures may not be enough for crumbing COVID-19.

The profile of the optimal controls ${\xi }_1^{*},{\xi }_2^{*},{\xi }_3^{*}$ is given by Fig. 3 . Implementation of this strategy without denial, COVID-19 prevention, hospitalization and treatment of symptomatic individuals should be done intensively for almost 50, 200 and 600 days, respectively, and then decline to the lower bound. However, with denial we observed from the graph on the right in Fig. 3 that each control needed to be applied seriously for 600 days. Likewise, Fig. 4 shows that each population; Exposed, Asymptomatic and Symptomatic has been reduced successfully without the denial of the disease. On the other hand, denial of the pandemic has rendered these controls ineffective since the number of infectives Asymptomatic, Symptomatic and the exposed persist uniformly. This means that control in the presence of denial cannot eliminate the ongoing COVID-19 pandemic. This result is consistent with the outcome of the work by Sayedahmed et al. [26] that says good level of knowledge (awareness creation) is associated with good practice towards COVID-19 elimination. The outcome is also in line with the report on the effect of denial on other diseases like HIV/AIDS [10], [11] that says denial triggers the spread of HIV/AIDS. We observed from Figure SM4 that in the presence of maximum treatment and preventive protocols (i.e. use of face masks, hand sanitizers, social distancing) with out denial actually reduced the number of COVID-19 cases recorded against exposed, asymptomatic, symptomatic and hospitalized people. This result coincides with the work of Deressa and Duressa [28] that emphasises that public health education, protective measures and treatment of hospitalized are effective to significantly decrease the number of COVID-19 cases among the infected population. Furthermore, We display in Figure SM5 the necessity of time in eradicating COVID-19 in the population. These figures show that the number of Exposed, Asymptomatic and Symptomatic individuals (E(t), A(t), I(t)) will decrease drastically when the controls are time-dependent compare to when the controls are constant (do not dependent on time). This implies that even when tested control strategies are in place, it is crucial to implement them on time in order to gain effective outcome. For instance, hospitalization of Symptomatic individuals on time will prevent them from infecting susceptible individuals while treating them urgently will avert deaths that could have been recorded. This conforms to the result of Madubueze et al. [38] that says time dependent interventions reduced the number of exposed and infected individuals compared to constant interventions. From our findings, we conclude that people should accept the fact that COVID-19 exists, and take necessary preventive protocols, and maximum treatment for the hospitalized people. Since human altitude towards disease outbreak can not completely changed, the denial rate of COVID-19 existence should be less than 0.45 in order to control the spread of the pandemic.

Fig. 3.

Profile of optimal controls, ${\xi }_1^{*},{\xi }_2^{*}$ and ${\xi }_3^{*}$, when $d=0$ (Figure on the left) and $d=0.55$ (Figure on the right). Parameter values are as given in Table 1.

Fig. 4.

The population dynamics of (a) Exposed, (b) Asymptomatic and (c) Symptomatic individuals to COVID-19 using optimal controls, ${\xi }_1^{*},{\xi }_2^{*}$ and ${\xi }_3^{*}$, with denial effect. Parameter values are as given in Table 1.

7. Conclusion

In this paper, we assess the role of denial on the spread of COVID-19 pandemic through the construction of an optimal control model with Beddington-De Angelis incidence function. From our analyzes, we derived the control reproduction number that determines the existence and stability of the model equilibria. Also, we found that to put the pandemic under control, the infection and denial rates should not exceed the upper bounds 0.15 and 0.45 on average. By implication, the disease will continue to live with us if 45 people in every 100 deny the existence of the disease. The presence of denial rate in the system induces backward bifurcation that causes complexity in the disease control even if the control reproduction number is less than unity. We simulated the optimal control system with respect to denial and without denial. The numerical examples show that strategies with inhibition factors (without denial effect) are capable of reducing corona virus in the population. However, introducing denial into the system destabilizes it and renders the controls ineffective. Again, the results obtain from the comparison of the optimal control with constant control systems shows that timely implementation of these strategies is key in controlling the rapid spread of Corona virus disease. In Africa, COVID-19 is erroneously assumed to be a disease of White men and rich men who travel abroad. Thus most people in urban slums and rural areas refuse to adhere to control measures such as social distancing, wearing face masks, hand washing with water and soups, hand sanitizers e.t.c. This poses a challenge to the control of COVID-19 in Africa. It is therefore advocate that governments at all levels should carry out public health education to dispel this erroneous behaviour of their citizens that drive the spread of COVID-19. Eradicating the denial of the existence of COVID-19 is the first line of control before any form of interventions that will help to limit the spread of the disease.

Funding

This research did not receive any specific grant from funding agencies, commercial, or not-for-profit sectors.

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.

Editor: DR B. Gyampoh

Appendix A. Supplementary materials

Supplementary Data S1

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/