Global stability analysis of the role of multi-therapies and non-pharmaceutical treatment protocols for COVID-19 pandemic

Abstract

In this paper, we sought and presented an 8-Dimensional deterministic mathematical COVID-19 dynamic model that accounted for the global stability analysis of the role of dual-bilinear treatment protocols of COVID-19 infection. The model, which is characterized by human-to-human transmission mode was investigated using dual non-pharmaceutical (face-masking and social distancing) and dual pharmaceutical (hydroxylchloroquine and azithromycin) as control functions following the interplay of susceptible population and varying infectious population. First, we investigated the model state-space and then established and computed the system reproduction number for both off-treatment ${\Re }_{0\left(1\right)}=10.94$ and for onset-treatment ${\Re }_{0\left(2\right)}=3.224$. We considered the model for off-treatment and thereafter by incorporating the theory of LaSalle's invariant principle into the classical method of Lyapunov functions, we presented an approach for global stability analysis of COVID-19 dynamics. Numerical verification of system theoretical predictions was computed using in-built Runge-Kutta of order of precision 4 in a Mathcad surface. The set approach produces highly significant results in the main text. For example, while rapid population extinction was observed by the susceptible under off-treatment scenario in the first $t_f\le 18$ days, the application of non-pharmaceuticals at early stage of infection proved very effective strategy in curtailing the spread of the virus. Moreso, the implementation of dual pharmacotherapies in conjunction with non-pharmaceuticals yields tremendous rejuvenation of susceptible population ($0.5\le S_p\left(t\right)\le 3.143cells/m{l}^3$) with maximal reduction in the rates of isolation, super spreaders and hospitalization of the infectives. Thus, experimental results of investigation affirm the suitability of proposed model for the control and treatment of the deadly disease provided individuals adheres to treatment protocols.

1. Introduction

Following the upsurge of what could be considered the most dreaded transmittable infectious disease in the history of virological infections in human race, the infectious disease known as coronavirus disease 2019 (COVID-19) has taken an unprecedented dehumanization of mankind with over 213 nations of the world falling prey to the deadly disease. COVID-19 in its nature, is a negative-gram of ribonucleic acid (RNA) virus that have both human and animals as its prey. Zoonotic scientists have shown that coronavirus infection have been identified as causing the disease of the type – severe acute respiratory syndrome (SARS-CoV-2), middle eastern respiratory syndrome (MERSS-CoV) and the most recent COIVID-19. Among these, ASRS-CoV-2 have been found to be the causative agent of the later – COVID-19 [1]. The human coronavirus is rated among the most rapidly evolving viruses due to their genetic makeup with its origin traced to the bats, palm civet and rodents [2]. Prior to the outbreak of COVID-19, it is important to note that the international alarm about COVID-19 pandemic was first sounded not by human, rather by a HealthMap computer (a website run by Boston children's Hospital) operated with the aid of artificial intelligence (AI) [3]. Even when the outbreak was first noticed in Wuhan, China, December, 2019, the first documented confirmed case of COVID-19 was reported in the USA on January 20, 2020 [1].

In reality, as at August, 2020, the exact casualties, which is still ongoing cannot be affirmed but having an estimated matching value of 24.6 million infected population and death toll of over 833,556 world-wide with epicentric burden in USA, Italy, Spain, Brazil, Iran, Russia, Egypt, South Africa and Nigeria [4,5]. Like other coronaviruses, COVID-19 is noted to be transmitted from animal-to-human and from human-to-human, which is often characterized by its droplets and finer aerosol transmission prowess. The incubation period is of varying range of 2 – 14 days, making it possible to accommodate large number of asymptotic patients who could be infectious but with less clinical manifestations [3,6]. Generally, COVID-19 exhibits similar clinical symptoms as SARS-CoV-2 and MERS-CoV, which appear as typical pneumonia marked by cough, fever, headache, dry throat and subsequent onset acute respiratory syndrome – coronavirus 2 (SARS-CoV 2) with life-threating respiratory failure [7]. Victims of COVID-19 appears to cut across all human race with most vulnerable and severe cases occurring among the elderly (adults of age $\ge 65$ years).

Like most other infectious disease with peculiarities of non-availability of outright medical cure and vaccines, varying non-pharmaceutical preventive and control intervention measures (in the range of hand-sanitizers, regular washing of hands, face-masks, social distancing, quarantization, isolation, contact tracing, hospitalization and lockdown) have been explored as useful control measures. For severe cases, the aspect of isolation and hospitalization have resulted to some level of clinical trials following the recommendation of pharmaceutical therapies (in the range of Dexamethasone, chloroquine, lopinavir/ritonavir, hydroxylchloroquine, azithromycin and erythromycin). Perturbingly, efforts to demystify the disease transmission dynamics and to propagate clinical methodological treatment protocols have attracted the attention of mathematical modeling. For instance, in the wake of COVID-19 pandemic, innovative findings have been formulated starting with the extensive evaluation of the impact of mathematics and mathematical models in understanding and controlling the 2019 novel coronavirus pandemic [1].

Noting the lack of vaccine for the control of the virus, [6] proposed the use of optimal quarantine strategies for the control of COVID-19, accounting for the long-term cost effect of the strategy. The outcome was massive even without vaccination provided survival level of viral load is kept less than 1. The model [8] proposed and studied the use of X-ray images characterized by hybrid 2D curvelet transform chaotic salp swarm algorithm and deep learning technique (CASSA) as a major alternative testing kits to the earlier proclaimed real-time reverse transcription-polymerase chain reaction (RT-PCR) by the World Health Organization (WHO). The model was analyzed using EfficientNet-BO method in conjunction with 2D curvelet transformation. Results showed that the model proved to be faster and low computation cost when compared to TR-PCR. The study [9] had considered the dynamical behavior of COVID-19 with carrier effect to outbreak epidemic. The model was formulated as a 5-Dimensional mathematical differential equations and studied using suitable Lyapunov function for the system global stability conditions. Numerical results obtained showed that the awareness as a single factor was not sufficient in reducing Covid-19 epidemic. The study proposed in addition to awareness, the inclusion of incidence rate, prevention rate and carrier as indices for the reduction of the spread of the virus. Incorporating Least-squares method into Lyapunov function, [10] investigated the global stability and cost-effectiveness of COVID-19, taking into account environmental factors and adopting six non-pharmaceutical control measures. Results indicated that the strategy involving practicing proper coughing etiquette, maintain distancing, covering cough and sneezing with disposable tissues and washing of hands is the most cost-effective strategy. Other mathematical models on COVID-19 with varying mathematical concepts can be found in [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].

Amidst these novel literatures on COVID-19, it is noticed that a standard mathematical model that accounted for the clinical and explicit combination of both pharmaceutical and non-pharmaceutical control strategies have not been given the desired attention. Thus, the present study considering human-to-human transmission mode, seek to formulate and analyze a set of standard mathematical model using dual-bilinear control treatment functions arising from dual non-pharmaceutical controls – use of face masks and social distancing and dual pharmaceutical therapies – hydroxylchloroquine (HCQ) and Azithromycin (AZT).

Resourcefully, with the introductory aspect in Section 1, the mathematical and epidemiological presentation of the study is partition into seven sections. In Section 2, we present the material and methods constituted by the system problem statement and derivation of mathematical model equations. In Section 3, we discuss the mathematical analysis involving the investigation of the system state-space. Under system stability analysis in Section 4, we investigated the existence of equilibrium states, system reproduction number, local stability in terms of reproduction number, system endemic equilibrium and the system global stability conditions. Numerical investigation of system theoretical predictions are presented in Section 5. Section 6 is devoted to the discussion and analysis of obtained results. Finally, we domicile our succinct conclusion and incisive remarks in Section 7. Notably, the present study is anticipated to unveil novel findings towards the annihilation of the deadly disease.

2. Material and methods

The material and methods of this study is characterized by the problem statement and derivation of model equations as well as analysis of model basic mathematical properties.

2.1. Problem statement for untreated COVID-19 and model equations

Following the advent of the unparalleled coronavirus pandemic, a number of notable mathematical models have been formulated in a move to mathematically present insight to the virus historical background, infection transmission dynamics and possible intervention and control measures with most models skewed to geographical locations.

Taking advantage of the limitations of two seeming compactible models [12,16] in relation to the present investigation, we seek to formulate a holistic COVID-19 differential model, considered adaptable to any geographical location. For instance, the model by [16] had studied a mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Moreso, the model [12] had investigated the contribution of mathematical modeling of COVID-19 in the Niger Republic using 8-Dimensional differential equations. On critical review of these two models, we observed the following limitations:

• iFormulated models [12,16] are much peculiar to their identified localities.
• iiSince dead patients do not transmit the virus, it implies that they do not contribute to infection dynamics. Then, needless for the inclusion of dead compartment among the system model [16].
• iiiInfectious model that focuses only on varying exposed and infected variables may not give a true representation of disease transmission dynamics [12].
• ivLack of natural source rate in infectious model could lead to abrupt experimental result and untimely population extinction [16].
• vTreatment interventions/control measures were not parametrically identified or valued for models [12, 16].
• viThe model [12] is optimal in nature (being treated as functions of time variation) but the study was not optimally analyzed.

It is on this premise that we proposed what could be considered a standard mathematical COVID-19 dynamic model as anticipated in the next sub-section.

2.2. Derivation of mathematical model equations

Following the above aforementioned limitations, the present study in an attempt to formulate a broader COVID-19 dynamic model determined by specific state variables and parameters, is further guided by the following assumptions.

Assumption 1

• iOnly the infectives die due to infection, such that ${\alpha }_1\ll {\alpha }_2\ll {\alpha }_3\ll {\alpha }_4\ll {\alpha }_5$ for all ${\alpha }_{i(i=1,...,5)}>0$.
• iiDetermination of aware infective is by screening method or any other clinical technique i.e., $\theta \ge 0$.
• iiiContact rate of susceptible with super-spreaders is much less than isolated aware infectives, which in turn is much less than hospitalized infectives and subsequently much less than aware infected and much lesser than unaware asymptotic population (i.e., ${\beta }_5<{\beta }_4<{\beta }_3<{\beta }_2<{\beta }_1$).
• ivOnly the hospitalized and isolated aware infectives use pharmaceutical control functions i.e., $(a_1>a_2\gg 0)$.
• vLatent infection period and non-cytotoxic carrying process are ignored.
• viAge-structure in transmission is ignored.
• viiRecovery are recruited to susceptible population i.e., ${\sigma }_1,{\sigma }_2>0$.

Furthermore, suppose the population under study is denoted by $N\left(t\right)$ with population volume measure in $cells/m{l}^3$ and such that by subdividing the population, we let the $S_p\left(t\right)$ represent susceptible population who are not COVID-19 positive but may be infected if completely exposed, $X_p\left(t\right)$ depicts the exposed class, $A_u\left(t\right)$ defining the unaware asymptomatic infectious population, $I_a\left(t\right)$- number of COVID-19 aware infectives, $I_s\left(t\right)$- isolated infectious population, $S_s\left(t\right)$- population of super spreaders, $H_i\left(t\right)$- population hospitalized infectives and $R_p\left(t\right)$ representing recovered population, then by the existing limitations and assumption 1, the structured epidemiological differential equations of the present study is derived as:

$$\begin{array}{c}{\displaystyle \frac{d{S}_p}{\mathit{dt}}=b_p+{\sigma }_1{X}_p+{\sigma }_2{R}_p-{\beta }_i\left(\widehat{N}\right)S_p-\mu S_p,}\hfill \\ {\displaystyle \frac{d{X}_p}{\mathit{dt}}={\beta }_i\left(\widehat{N}\right)S_p-(1-u_1)\lambda X_p-(\mu +{\sigma }_1)X_p,}\hfill \\ {\displaystyle \frac{d{A}_u}{\mathit{dt}}=(1-u_1)\lambda X_p-(1-u_2)k\theta A_u-(1-a_1)(1-a_2){\phi }_1{A}_u-(\mu +{\alpha }_1+{\phi }_2)A_u,}\hfill \\ {\displaystyle \frac{d{I}_a}{\mathit{dt}}=(1-u_2)k\theta A_u-[(1-a_1)(1-a_2){\rho }_2+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)]I_a-(\mu +{\alpha }_2)I_a,}\hfill \\ {\displaystyle \frac{d{I}_s}{\mathit{dt}}=a_1{\tau }_1{\rho }_1{I}_a+a_1{\tau }_2{\gamma }_s{S}_s-\left[(1-a_1)(1-a_2){\delta }_h\right]I_s-(\mu +{\alpha }_4+{\eta }_2)I_s,}\hfill \\ {\displaystyle \frac{d{S}_s}{\mathit{dt}}=(1-{\rho }_1-{\rho }_2)I_a+{\phi }_2{A}_u-a_1{\tau }_2{\gamma }_s{S}_s-(\mu +{\alpha }_5+{\eta }_3)S_s,}\hfill \\ {\displaystyle \frac{d{H}_i}{\mathit{dt}}=(1-a_1)(1-a_2)[{\phi }_1{A}_u+{\rho }_2{I}_a+{\delta }_h{I}_s]-(\mu +{\alpha }_3+{\eta }_1)H_i,}\hfill \\ {\displaystyle \frac{d{R}_p}{\mathit{dt}}={\eta }_1{H}_i+{\eta }_2{I}_s+{\eta }_3{S}_s-(\mu +{\sigma }_2)R_p}\hfill \end{array}$$ (1)

with initial conditions $S_p\left(t_0\right)>0,$ $X_p\left(t_0\right)>0,$ $A_u\left(t_0\right)>0,$ $I_a\left(t_0\right)>0,$ $I_s\left(t_0\right)>0,$ $S_s\left(t_0\right)>0,$ $H_i\left(t_0\right)>0,$ $R_p\left(t_0\right)>0$ for all $t=t_0$. Here, the system force of infection denoted by ${\beta }_i\left(\widehat{N}\right)={\beta }_i(X_p,A_u,I_a,S_s,H_i)$ is given by

$${\beta }_i\left(\widehat{N}\right)=(1-u_1-u_2)\left[\frac{{\beta }_1{c}_1{X}_p+{\beta }_2{c}_2{A}_u+{\beta }_3{c}_3{I}_a+{\beta }_4{c}_4{S}_s+{\beta }_5{c}_5{H}_i}{N\left(t\right)}\right],i=1,....,5$$ (2)

where

$$N\left(t\right)=S_p+X_p+A_u+I_a+I_s+S_s+H_i+R_p=1.$$ (3)

If ${\beta }_i>0$ such that our control functions $u_i=0$ $a_i=0$, where $i=1,2$, then system (1) completely represent a standard endemic COVID-19 infection dynamic model.

Furthermore, the epidemiological descriptions of model (1) are as follows: from the first equation, the first three terms $b_p,{\sigma }_1{X}_p,{\sigma }_2{R}_p$ represent the birth rate, recovery rate of exposed and the recovery rate from varying severe forms of the infection, which reunite with the susceptible population. The fourth term ${\beta }_i\left(\widehat{N}\right)S_p$ depicts the rate at which the susceptible interacts with the varying infectious compartments, which then serves as supply rate for the exposed class. The susceptible die at a rate $\mu S_p$. The second term $(1-u_1)\lambda X_p$ from the second equation define the population of exposed that successively use face mask yet becomes asymptomatically infectious, while the last term $(\mu +{\sigma }_1)X_p$ describe the sum of exposed that recovers due to coherent use of non-pharmaceutical control (face mask) measure couple with strong natural immune defense mechanism and natural clearance rate.

From the third equation, the first term $(1-u_1)\lambda X_p$ denotes the transmutation of the exposed under face mask for unaware asymptomatic class, while the second term $(1-u_2)k\theta A_u$ is the proportional rate of unaware asymptomatic that are screen to become aware infectives with promising social distancing. The third term $(1-a_1)(1-a_2){\phi }_1{A}_u$ represent the proportion of asymptomatic infectious that progress to be hospitalized under pharmaceutical control functions, while the last term $(\mu +{\alpha }_1+{\phi }_2)A_u$ depicts the clearance differential sum from the asymptomatic infectious patients arising from natural and infection death rate as well as the proportion that becomes super-spreaders. In equation four, the first term $(1-u_2)k\theta A_u$ supply the source rate from asymptotic that becomes aware following the application of screening method. The second term $[(1-a_1)(1-a_2){\rho }_2+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)]I_a$ defined the sum differential of aware infectives that move to $H_i$ and $I_s$ where they are subjected to treatment functions while the proportion $(1-{\rho }_1-{\rho }_2)I_a$ becomes super-spreaders. The last term here depicts the clearance rates due to nature and infection respectively.

The proportions of aware infectives and super-spreaders $a_1{\tau }_1{\rho }_1{I}_a,a_1{\tau }_2{\gamma }_s{S}_s$ under initial treatment function are seen in the first and second terms of the fifth equation of the isolated infective compartment. The third term $\left[(1-a_1)(1-a_2){\delta }_h\right]I_s$ represent severe isolated patients that progresses to hospitalization under multi-therapies. The last term $(\mu +{\alpha }_4+{\eta }_2)I_s$ defines the removal rates arising from both natural death, death due to infection and a proportion that recovered from the infection. From equation six, the proportion of aware infectives that transmutes to super-spreaders is given by the first term $(1-{\rho }_1-{\rho }_2)I_a$, which is busted by proportion of asymptomatic infective, ${\phi }_2{A}_u$ under no treatment function. The rate at which super-spreaders becomes isolated and then place under initial treatment is given by $a_1{\tau }_2{\gamma }_s{S}_s$. The clearance rates due to natural death, infection rate and recovery rate are given by $(\mu +{\alpha }_5+{\eta }_3)S_s$.

Taking on equation seven, the first term $(1-a_1)(1-a_2)[{\phi }_1{A}_u+{\rho }_2{I}_a+{\delta }_h{I}_s]$ describe the varying rates of severe infected population that are hospitalized under multi-therapies. The last term in this compartment $(\mu +{\alpha }_3+{\eta }_1)H_i$ depicts the sum clearance rates due to deaths (natural and infection) and recovery rate. The final and eight compartment define the sum recovery population coming from hospitalized compartment, recovery from isolated compartment and from super-spreaders denoted by ${\eta }_1{H}_i+{\eta }_2{I}_s+{\eta }_3{S}_s$. Here, clearance rate is due to natural death rate and the proliferation of recovered to the susceptible denoted by $(\mu +{\sigma }_2)R_p$. Thus, following the above model description, model (1) can be schematically represented as in Fig. 1 , below:

Fig. 1.

Schematic flow-chart of COVID-19 transmission dynamics with multi-control functions.

A critical review of system (1) shows that the present model is characterized by system force of infection rate depicted by Eq. (2). This force of infection gives a biological meaning to the system reproduction number, (see Section 4). Moreso, the design of present model explicitly outline the methodological application of designated control functions following the introduction of dual pharmaceuticals at severe stages of COVID-19 infection. Furthermore, we perform the numerical simulations using generated initial values for the state-space and parameters of the system in relation to model (1) and Fig. 1. That is, with modified data from related compactible models, Tables 1 and 2 below depicts the simulating data for the present investigation.

Table 1.

Description of state variables with values – model (1).

Variables	Dependent variables Description	Initial values	Units
$S_p$	Susceptible population to COVID-19 infection	0.5	cells/ml3
$X_p$	Exposed population	0.3
$A_u$	Unaware asymptotic infectious population	0.1
$I_a$	Aware infective population	0.15
$I_s$	Isolated infectious population	0.0
$S_s$	Super-spreaders infectious population	0.05
$H_i$	Hospitalized infectious population	0.0
$R_p$	COVID-19 recovered population	0.0

Note: Table 1 is extracted and modified from models [12,16].

Table 2.

Description of constants and parameter values for model (1).

Parameter symbols	Parameters and constants	Initial values	Units
	Description
$b_p$	Source rate of susceptible population	$b_p\le 10.5$	$m{l}^3{d}^{-1}$
$\mu$	Natural death rate for all sub-population	0.1	day-1
$k$	Clearance rate of virus	0.25
${\alpha }_{i(i=1,..,5)}$	Death rates due infection at varying stages	0.2;0.3;0.1;0.4;0.5
${\tau }_{i=1,2}$	Rate at which $I_a$ progresses to $I_s$ and $S_s$	0.3, 0.5
$c_{i(i=1,..,5)}$	Rates of contact of susceptible with various infectious stages	0.5;0.4;0.3;0.2;0.1
${\eta }_{i=1,2,3}$	Rates of recovery from $H_i$, $I_s$ and $S_s$	0.5; 0.27;0.13
${\beta }_{i(1,..,5)}$	Probability of interactions of susceptible with varying infectious classes	0.32;0.27;0.175; 0.125;0.05	$m{l}^{3}vi{r}^{-1}{d}^{-1}$
${\phi }_{i=1,2}$	Proportions of $A_u$ that progresses to $H_i$ and $S_s$	0.3;0.18	ml3d-1
$\lambda$	Proportion of $X_p$ becoming $A_u$	0.58
$\theta$	Proportion of $A_u$ becoming $I_a$	0.32
${\sigma }_{i=1,2}$	Proliferations of recovered population to susceptible	0.14;0.6
${\gamma }_s$	Proportionof $S_s$ progressing to $I_s$	0.22
${\delta }_h$	Proportion of $I_s$ progressing to $H_i$	0.14
${\rho }_1$	Proportion of $I_a$ that progresses to $I_s$	0.34
${\rho }_2$	Proportion of $I_a$ that progresses to $H_i$	0.48
$(1-{\rho }_1-{\rho }_2)$	Proportion of $I_a$ that progresses to $S_s$	0.08
$u_{i=1,2}\left(t\right)$	Rates of use of face mask and social distancing	$u_i\in [0,1]$
$a_{i=1,2}\left(t\right)$	Treatment control functions (HCQ and AZT)	$a_i\in [0,1]$

Note: Tables 2 is clinically generated from certified data of [9,16].

Obviously, since model (1) represent a set of living organism, then we must verify the model well-posedness, which involves the mathematical properties of derived system.

3. Mathematical analysis of model properties

Here, we quantitatively verify the characteristic properties of the model constituted by the system invariant region and non-negativity of system solutions. Notably, if the sum population under study is defined by Eq. (3), then the differential sum of model (1) is given by

$$\begin{array}{c}{\displaystyle \frac{dN}{dt}=\frac{d{S}_p}{dt}+\frac{d{X}_p}{dt}+\frac{d{A}_u}{dt}+\frac{d{I}_a}{dt}+\frac{d{I}_s}{dt}+\frac{d{S}_s}{dt}+\frac{d{H}_i}{dt}+\frac{d{R}_p}{dt}}\hfill \\ \\ =b_p-\mu (S_p+X_p+A_u+I_a+I_s+S_s+H_i+R_p)-\widehat{\alpha }(X_p+A_u+I_a+S_s+H_i)\hfill \end{array},$$

where $\widehat{\alpha }={\alpha }_1,...,{\alpha }_5$. This imply that

$$\frac{dN}{dt}=b_p-\mu N-\widehat{\alpha }(X_p+A_u+I_a+S_s+H_i)$$

or

$$\frac{dN}{dt}=b_p-\mu N-\widehat{\alpha }\widehat{N},$$ (4)

where $\widehat{N}=(X_p+A_u+I_a+S_s+H_i)$ and $N\left(t\right)$ defined by Eq. (3). Clearly, Eq. (4) shows that the differential sum of system (1) is a function of the system birth rate, natural death rate and death due to infection. We also use Eq. (4) as the index for the verification of the system invariant region.

3.1. System invariant region

For the fact that the system represents a set of living organism, then it is sufficient to assume that for $t>0$, the state variables and parameters are non-negative at any given region. This region we shall investigate using the following theorem.

Theorem 1

Let the system state variables be bounded in a closed set ${\Re }_D$ such that

${\Re }_D=\left\{(S_p,X_p,A_u,I_a,I_s,S_s,H_i,R_p)\in {\Re }_{+}^8:N\le \frac{b_p}{\mu }\right\}$. Then, the closed set ${\Re }_D$ is positively invariant and attracting with respect to system (1).

Proof

We invoke results from [21,22], then by Eq. (6), we have

$$\frac{dN}{dt}=b_p-\mu N-\widehat{\alpha }\widehat{N}.$$

In the absence of mortality due to COVID-19 infection, the population is said to be free from the virus i.e., $\widehat{\alpha }=0$. That is,

$$\frac{dN}{dt}=b_p-\mu N,$$

or

$$\frac{dN}{dt}+\mu N\le b_p,$$

which gives a first order homogeneous differential inequality. The integration in the presence of its initial conditions yields

$$N\left(t\right)\le {\displaystyle \frac{b_p}{\mu }}+\left(N\left(0\right)-\frac{b_p}{\mu }\right)e^{-\mu t},$$

where $N\left(0\right)$ is the initial population at $t=t_0=0$. This gives $N\left(t\right)\le N\left(0\right)$ as $t\to 0$ and $N\left(t\right)\le \frac{b_p}{\mu }$ as $t\to \infty$.

Then, in line with Birkhof and Rota's theorem on differential inequality as applied by [23], when $t\to \infty$, we arrive at $0\le N\left(t\right)\le \frac{b_p}{\mu }$ for all $t\ge 0$. But from model (1),

$$\frac{d{S}_p}{dt}+\frac{d{X}_p}{dt}+\frac{d{A}_u}{dt}+\frac{d{I}_a}{dt}+\frac{d{I}_s}{dt}+\frac{d{S}_s}{dt}+\frac{d{H}_i}{dt}+\frac{d{R}_p}{dt}=0,$$

which gives

$$\frac{dN}{dt}=0.$$

The resulting integral gives

$$N=C,$$

where $C$, is the constant of integration. We know from Eq. (3) that

$$N=S_p+X_p+A_u+I_a+I_s+S_s+H_i+R_p=1.$$

It follows that $C=1$, implying that the population is constant, positive and equal 1. Hence, all the feasible solutions of the system (1) enter the invariant region

$${\Re }_D=\left\{\left(S_p,X_p,A_u,I_a,I_s,S_s,H_i,R_p\right)\in {\Re }_{+}^8:N\le \frac{b_p}{\mu }:S_p+X_p+A_u+I_a+I_s+S_s+H_i+R_p=1\right\}.$$

Therefore, the region is positive and attracting implying that the model is both mathematically well-posed in the region ${\Re }_D$ and epidemiologically meaningful. □

3.2. Positivity of system solutions

The following theorem is use to show that the solutions of the system state variables remain positive for all $t>0$

Theorem 2

Let the initial conditions $\left\{S_p\left(0\right),X_p\left(0\right),A_u\left(0\right),I_a\left(0\right),I_s\left(0\right),S_s\left(0\right),H_i\left(0\right),R_p\left(0\right)\ge 0\right\}\in {\Re }_{+}^8$. Then, the solution set $\left\{S_p\left(t\right),X_p\left(t\right),A_u\left(t\right),I_a\left(t\right),I_s\left(t\right),S_s\left(t\right),H_i\left(t\right),R_p\left(t\right)\right\}$ of system (1) is non-negative for all $t>0$.

Proof

Invoking existing results from [22,24], then system (1) can be confined in compact subset as

$$\Omega =\left\{\begin{array}{c}{\displaystyle (S_p,X_p,A_u,I_a,I_s,S_s,H_i,R_p)\in {\Re }_{+}^8:}\hfill \\ {\displaystyle N=S_p\left(t\right)+X_p\left(t\right)+A_u\left(t\right)+I_a\left(t\right)+I_s\left(t\right)+S_s\left(t\right)+H_i\left(t\right)+R_p\left(t\right)\le \frac{b_p}{\mu }}\hfill \end{array}\right\}.$$

Let $(S_p\left(t\right)+X_p\left(t\right)+A_u\left(t\right)+I_a\left(t\right)+I_s\left(t\right)+S_s\left(t\right)+H_i\left(t\right)+R_p\left(t\right))$ be any solution with positive initial conditions such that

$$N\left(t\right)=S_p\left(t\right)+X_p\left(t\right)+A_u\left(t\right)+I_a\left(t\right)+I_s\left(t\right)+S_s\left(t\right)+H_i\left(t\right)+R_p\left(t\right).$$

Then, the time derivative of $N\left(t\right)$ along solution of system (1) with zero mortality rate is

$$\frac{dN}{dt}=b_p-\mu (S_p+X_p+A_u+I_a+I_s+S_s+H_i+R_p)\le \frac{b_p}{\mu }-\mu N\left(t\right),\text{for}\text{all}\widehat{\alpha }=0.$$

Applying the theorem of differential equation, we obtain

$$N\left(t\right)\le N\left(0\right)e^{-\mu t}+\frac{b_p}{\mu }\left(1-e^{-\mu t}\right)$$

and for all $t\to \infty$, we have

$$\underset{t\to \infty }{lim}N\left(t\right)\le \frac{b_p}{\mu }.$$

Clearly, it has been proved that all the solutions of system (1), which is initiated in ${\Re }_{+}^8$ is confined in the region $\Omega$, implying the solutions are bounded and positive in the interval $[0,\infty )$. □

4. Stability analysis of derived model

In this section, we quantitatively investigate the model equilibria and explicitly study the model local and global stability conditions.

4.1. Existence of equilibrium states

For the existence of an equilibrium state, it is assumed that model (1) is at its steady state i.e.,

$$\frac{d{S}_p}{dt}+\frac{d{X}_p}{dt}+\frac{d{A}_u}{dt}+\frac{d{I}_a}{dt}+\frac{d{I}_s}{dt}+\frac{d{S}_s}{dt}+\frac{d{H}_i}{dt}+\frac{d{R}_p}{dt}=0.$$

This gives the system

$$\begin{array}{c}{\displaystyle 0=b_p+{\sigma }_1{X}_p+{\sigma }_2{R}_p-{\beta }_i\left(\widehat{N}\right)S_p-\mu S_p,}\hfill \\ {\displaystyle 0={\beta }_i\left(\widehat{N}\right)S_p-(1-u_1)\lambda X_p-(\mu +{\sigma }_1)X_p,}\hfill \\ {\displaystyle 0=(1-u_1)\lambda X_p-(1-u_2)k\theta A_u-(1-a_1)(1-a_2){\phi }_1{A}_u-(\mu +{\alpha }_1+{\phi }_2)A_u,}\hfill \\ {\displaystyle 0=(1-u_2)k\theta A_u-[(1-a_1)(1-a_2){\rho }_2+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)]I_a-(\mu +{\alpha }_2)I_a,}\hfill \\ {\displaystyle 0=a_1{\tau }_1{\rho }_1{I}_a+a_1{\tau }_2{\gamma }_s{S}_s-(1-a_1)(1-a_2){\delta }_h{I}_s-(\mu +{\alpha }_4+{\eta }_2)I_s,}\hfill \\ {\displaystyle 0=(1-{\rho }_1-{\rho }_2)I_a+{\phi }_2{A}_u-a_1{\tau }_2{\gamma }_s{S}_s-(\mu +{\alpha }_5+{\eta }_3)S_s,}\hfill \\ {\displaystyle 0=(1-a_1)(1-a_2)[{\phi }_1{A}_u+{\rho }_2{I}_a+{\delta }_h{I}_s]-(\mu +{\alpha }_3+{\eta }_1)H_i,}\hfill \\ {\displaystyle 0={\eta }_1{H}_i+{\eta }_2{I}_s+{\eta }_3{S}_s-(\mu +{\sigma }_2)R_p.}\hfill \end{array}$$ (5)

Then, from Eq. (5), the COVID-19 free equilibrium (C-19FE) for system (1) exists if $u_i=0,a_i=0$ for all $i=1,2$ with other controls held constant. Therefore, in computing the C-19FE, we let $E^0$ denotes C-19FE such that at C-19FE, there is no infection and thus, no recovery i.e.,

$$X_p^{*}=A_u^{*}=I_a^{*}=I_s^{*}=S_s^{*}=H_i^{*}=R_p^{*}=0.$$ (6)

Then, we define

$$E^0=(S_p^{*},X_p^{*},A_u^{*},I_a^{*},I_s^{*},S_s^{*},H_i^{*},R_p^{*})=0.$$ (7)

We solve Eq. (7) using Eqs. (5) and (6) to yield

$$0=b_p-({\beta }_i^{*}+\mu )S_p^{*},$$

which gives

$$S_p^{*}=\frac{b_p}{({\beta }_i^{*}+\mu )}.$$

From the second equation, substituting $S_p^{*}$ we have

$$X_p^{*}=\left(\frac{b_p}{\lambda +m_1}\right)\frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}.$$

In a similar approach, we solve for $A_u^{*},I_a^{*},I_s^{*},S_s^{*},H_i^{*}$ and $R_p^{*}$, which gives the comprehensive result for $E^0$ as:

$$\{\begin{array}{c}{\displaystyle S_p^{*}=\frac{b_p}{({\beta }_i^{*}+\mu )}}\hfill \\ {\displaystyle X_p^{*}=\left(\frac{b_p}{\lambda +m_1}\right)\frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}}\hfill \\ {\displaystyle A_u^{*}=\left(\frac{b_p\lambda }{Q_1}\right)\frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}}\hfill \\ {\displaystyle I_a^{*}=\left(\frac{b_p\lambda k\theta }{Q_2}\right)\frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}}\hfill \\ {\displaystyle I_s^{*}=0}\hfill \\ {\displaystyle S_s^{*}=\left(\frac{k\theta (1-{\rho }_1-{\rho }_2)+{\phi }_2}{Q_3{m}_5}\right)b_p\lambda \frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}}\hfill \\ {\displaystyle H_i^{*}=\left(\frac{k\theta {\rho }_2+{\phi }_1}{Q_4{m}_6}\right)b_p\lambda \frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}}\hfill \\ {\displaystyle R_p^{*}=\frac{1}{m_7}\left(\frac{{\eta }_1(k\theta {\rho }_2+{\phi }_1)}{Q_4}+\frac{{\eta }_{3}k\theta (1-{\rho }_1-{\rho }_2)+{\phi }_2}{Q_3}\right)b_p\lambda \frac{{\beta }_i^{*}}{({\beta }_i^{*}+\mu )}}\hfill \end{array},$$ (8)

where, $Q_1=(\lambda +m_1)(k\theta +{\phi }_1+m_2)$, $Q_2=(\lambda +m_1)(k\theta +{\phi }_1+m_2)(1-{\rho }_2+m_3)$, $Q_3=(1-{\rho }_2+m_3)Q_1+(\lambda +m_1)$ and $Q_4=Q_1+Q_2$ with

$$\{\begin{array}{ccc}{\displaystyle m_1=\mu +{\sigma }_1}& ,& m_2=\mu +{\alpha }_1+{\phi }_2\\ {\displaystyle m_3=\mu +{\alpha }_2}& ,& m_4=\mu +{\alpha }_4+{\eta }_2\\ {\displaystyle m_5=\mu +{\alpha }_5+{\eta }_3}& ,& m_6=\mu +{\alpha }_3+{\eta }_1\\ {\displaystyle m_7=\mu +{\sigma }_2}& & \end{array}.$$ (9)

Now, since ${\beta }_i^{*}=0$ define the point at which C-19FE exists, then from Eqs. (5) and (8), the equilibrium point given by Eq. (7) is obtained as:

$$E^0=(\frac{b_p}{\mu },0,0,0,0,0,0,0).$$ (10)

Clearly, Eq. (10) depicts C-19FE, where no infection exists. But with the introduction of COVID-19 into the system, we are the required to investigate the pattern of transmission dynamics. This process invariably demands that we establish the system reproduction number denoted by ${\Re }_0$.

4.2. COVID-19 control reproduction number, ${\Re }_0$

Focusing on infectious diseases, the basic reproduction number is a critical mathematical quantity considered paramount to the public health sector. For a typical infectious disease like the COVID-19, we defined the reproduction number as the average number of new cases of the infection generated by an infected individual (i.e., super-spreaders) following the interaction with the susceptible population. Here, we adopt the approach by [25], which explore the Next generation matrix defined by

$$\left[\frac{\partial F_i}{\partial x_j}\left(E^0\right)\right]·{\left[\frac{\partial V_i}{\partial x_j}\left(E^0\right)\right]}^{-1},$$

where the notations $F_i$ and $V_i$, denotes the matrices of new infections in compartment $i$ and the transfer terms at COVID-19 free equilibrium into compartment $i$ while $E^0$ is the C-19FE for finding ${\Re }_0$. Now, since reproduction number is about the rate of infection, then we required to rewrite system (1) starting with the exposed class and substituting Eq. (9) to have,

$$F_i=\left(\begin{array}{c}{\displaystyle F_1}\\ {\displaystyle F_2}\\ {\displaystyle F_3}\\ {\displaystyle F_4}\\ {\displaystyle F_5}\\ {\displaystyle F_6}\\ {\displaystyle F_7}\end{array}\right)=\left(\begin{array}{c}{\displaystyle (1-u_1-u_2)\left[\frac{{\beta }_1{c}_1{X}_p+{\beta }_2{c}_2{I}_a+{\beta }_3{c}_3{I}_s+{\beta }_4{c}_4{S}_s+{\beta }_5{c}_5{H}_i}{N\left(t\right)}\right]S_p}\\ {\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 0}\end{array}\right)$$

At C-19FE, the linearization of $F_i$ gives

$$F_0=\left(\begin{array}{ccccccc}{\displaystyle \varphi {\beta }_1{c}_1\frac{S_p}{N}}& \varphi {\beta }_2{c}_2\frac{S_p}{N}& \varphi {\beta }_3{c}_3\frac{S_p}{N}& \varphi {\beta }_4{c}_4\frac{S_p}{N}& \varphi {\beta }_5{c}_5\frac{S_p}{N}& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\end{array}\right)$$

where $\varphi =(1-u_1-u_2)$.

Substituting Eq. (10), we have

$$F_0=\left(\begin{array}{ccccccc}{\displaystyle \varphi {\beta }_1{c}_1\frac{b_p}{\mu }}& \varphi {\beta }_2{c}_2\frac{b_p}{\mu }& \varphi {\beta }_3{c}_3\frac{b_p}{\mu }& \varphi {\beta }_4{c}_4\frac{b_p}{\mu }& \varphi {\beta }_5{c}_5\frac{b_p}{\mu }& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0\end{array}\right).$$ (11)

Computing for $V_i$, we have

$$V_{i=1,...,5}=\left(\begin{array}{c}{\displaystyle [(1-u_1)\lambda +m_1]X_p}\\ {\displaystyle [(1-u_2)k\theta +(1-a_1)(1-a_2){\phi }_1+m_2]A_u-(1-u_1)\lambda X_p}\\ {\displaystyle [(1-a_1)(1-a_2){\rho }_2+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)+m_3]I_a-(1-u_2)k\theta A_u}\\ {\displaystyle [(1-a_1)(1-a_2){\delta }_h+m_4]I_s-(a_1{\tau }_1{\rho }_1{I}_a+a_1{\tau }_2{\gamma }_s{S}_s)}\\ {\displaystyle (a_1{\tau }_2{\gamma }_s+m_5)S_s-[(1-{\rho }_1-{\rho }_2)I_a-{\phi }_2{A}_u]}\\ {\displaystyle m_6{H}_i-(1-a_1)(1-a_2)[{\phi }_1{A}_u+{\rho }_2{I}_a+{\delta }_h{I}_s]}\\ {\displaystyle m_7{R}_p-({\eta }_1{H}_i+{\eta }_2{I}_s+{\eta }_3{S}_s)}\end{array}\right).$$

Applying the linearization method, we have

$$V_0=\left(\begin{array}{ccccccc}{\displaystyle {\Omega }_1}& 0& 0& 0& 0& 0& 0\\ {\displaystyle -(1-u_1)\lambda }& {\Omega }_2& 0& 0& 0& 0& 0\\ {\displaystyle 0}& -(1-u_2)k\theta & {\Omega }_4& 0& 0& 0& 0\\ {\displaystyle 0}& 0& -a_1{\tau }_1{\rho }_1& {\Omega }_6& -a_1{\tau }_2{\gamma }_s& 0& 0\\ {\displaystyle 0}& {\phi }_2& -(1-{\rho }_1-{\rho }_2)& 0& {\Omega }_8& 0& 0\\ {\displaystyle 0}& -{\Omega }_3& -{\Omega }_5& -{\Omega }_7& 0& m_6& 0\\ {\displaystyle 0}& 0& 0& -{\eta }_2& -{\eta }_3& -{\eta }_1& m_7\end{array}\right),$$ (12)

where

$$\{\begin{array}{cc}{\displaystyle {\Omega }_1=(1-u_1)\lambda +m_1,}& {\Omega }_2=(1-u_2)k\theta +{\Omega }_3+m_2,\\ {\displaystyle {\Omega }_3=(1-a_1)(1-a_2){\phi }_1,}& {\Omega }_4=(1-a_1)(1-a_2){\rho }_1+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)+m_3,\\ {\displaystyle {\Omega }_5=(1-a_1)(1-a_2){\rho }_2,}& {\Omega }_6={\Omega }_7+m_4,\\ {\displaystyle {\Omega }_7=(1-a_1)(1-a_2){\delta }_h,}& {\Omega }_8=a_1{\tau }_2{\gamma }_s+m_5.\end{array}.$$ (13)

Thus, the COVID-19 reproduction number as applied by [26,27] corresponds to the spectral radius of $F_0{V}_0^{-1}$ is computed to give

$${\Re }_0=\rho \left(F_0{V}_0^{-1}\right)=\frac{\varphi b_p}{\mu }\left(\frac{{\beta }_1{c}_1}{{\Omega }_1}+\frac{{\beta }_2{c}_2}{{\Omega }_2}+\frac{{\beta }_3{c}_3}{{\Omega }_4}+\frac{{\beta }_4{c}_4}{{\Omega }_6}+\frac{{\beta }_5{c}_5}{{\Omega }_8}\right),$$

or

$${\Re }_0=\sum _{i=1}^5\left(\varphi R_j\right),$$ (14)

where $j=1,...,5$ represent the reproduction numbers for the infectious state variables defined in ${\beta }_i\left(\widehat{N}\right)$ with $\varphi =(1-u_1-u_2)$. For simplicity, if $u_i=0,a_i=0$, then we have the reproduction number for off-treatment scenario denoted by ${\Re }_{0\left(1\right)}$ and Eq. (14) becomes

$${\Re }_{0\left(1\right)}=\frac{b_p}{\mu }\left(\frac{{\beta }_1{c}_1}{{\Omega }_1}+\frac{{\beta }_2{c}_2}{{\Omega }_2}+\frac{{\beta }_3{c}_3}{{\Omega }_4}+\frac{{\beta }_4{c}_4}{{\Omega }_6}+\frac{{\beta }_5{c}_5}{{\Omega }_8}\right),$$ (14b)

with computed value of ${\Re }_{0\left(1\right)}=10.94$. If $u_i>0,a_i>.0$, then we have system basic reproduction number for onset-treatment scenario denoted by ${\Re }_{0\left(2\right)}$ derived as

$${\Re }_{0\left(2\right)}=\frac{\varphi b_p}{\mu }\left(\frac{{\beta }_1{c}_1}{{\Omega }_1}+\frac{{\beta }_2{c}_2}{{\Omega }_2}+\frac{{\beta }_3{c}_3}{{\Omega }_4}+\frac{{\beta }_4{c}_4}{{\Omega }_6}+\frac{{\beta }_5{c}_5}{{\Omega }_8}\right)$$ (14c)

with computed value of ${\Re }_{0\left(2\right)}=3.224$.

4.3. Local stability in terms of ${\Re }_0$

Here, we explore the eigenvalues of the linearized Jacobian matrix, which is evaluated in terms of the negative trace and a positive determinant or as having negative eigenvalues [28]. Of note, Eq. (14) is significant in the sense that it affirm the existence of Eq. (10) for a COVID-19 free population provided ${\Re }_0<1$. Moreso, the endemicity of COVID-19 can be effectively control, if $\left({\beta }_i{c}_i\right)<1$ and under significant $\varphi$. A situation that will lead to reduction in ${\Re }_0$ and subsequently, the elimination of COVID-19 from the population. The following theorem justify the role of ${\Re }_0$ in COVID-19 transmission.

Theorem 3

Whenever the C-19FE for model (1) exists, it is locally asymptomatically stable provided ${\Re }_0<1$ and unstable if ${\Re }_0>1$.

Proof

Results of existing theorem by [29] is invoke for this prove. Let $J$ be the Jacobian matrix for the system (1).

Then, $J$ at C-19FE point $\left(E^0\right)$ is derive as:

$$J\left(E^0\right)=\left(\begin{array}{cccccccc}{\displaystyle -\mu }& {\sigma }_1-\varphi {\beta }_1{c}_1& -\varphi {\beta }_2{c}_2& -\varphi {\beta }_3{c}_3& 0& -\varphi {\beta }_4{c}_4& -\varphi {\beta }_5{c}_5& {\sigma }_2\\ {\displaystyle 0}& -({\Omega }_1+\varphi {\beta }_1{c}_1)& \varphi {\beta }_2{c}_2& \varphi {\beta }_3{c}_3& 0& \varphi {\beta }_4{c}_4& \varphi {\beta }_5{c}_5& 0\\ {\displaystyle 0}& (1-u_1)\lambda & -{\Omega }_2& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& (1-u_2)k\theta & -{\Omega }_4& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& a_1{\tau }_1{\rho }_1& -{\Omega }_6& a_1{\tau }_2{\gamma }_s& 0& 0\\ {\displaystyle 0}& 0& {\phi }_2& (1-{\rho }_1-{\rho }_2)& 0& -{\Omega }_8& 0& 0\\ {\displaystyle 0}& 0& {\Omega }_3& {\Omega }_5& {\Omega }_7& 0& -m_6& 0\\ {\displaystyle 0}& 0& 0& 0& {\eta }_2& {\eta }_3& {\eta }_1& -m_7\end{array}\right).$$ (17)

Taking the eigenvalues of $J\left(E^0\right)$, we have $-\mu$, $-({\Omega }_1+\varphi {\beta }_1{c}_1)$, $-{\Omega }_2$, $-{\Omega }_4$, $-{\Omega }_6$, $-{\Omega }_8$, $-m_6$ and $-m_7$. Then, the eigenvalues of $J\left(E^0\right)$ is all negative. Hence, localization of disease infection is locally asymptomatically stable for all ${\Re }_0<1$. This complete the result. □

4.4. COVID-19 endemic equilibrium (C-19EE)

Theorem 4

The system (1) exhibits C-19EE in the population if and only if ${\Re }_0>1$.

Proof

From Eq. (4), if $\widehat{\alpha }=0$, then infection persist i.e. $X_p=A_u=I_a=S_s=H_i\ne 0$. That is, system (1) has an equilibrium point call endemic equilibrium point. If $E^{*}$ define the endemic equilibrium point, then with Eq. (8), the differential sum of the system (1) at C-19EE, is derived as: [24]

$$b_p-\mu N-\widehat{\alpha }\widehat{N}=0,$$

or

$$N^{*}=\frac{b_p-\widehat{\alpha }\widehat{N}}{\mu },$$ (15)

which corresponds to the fact that at equilibrium, ${\beta }_i^{*}=0$. If ${\beta }_i^{*}>0$, then there exists disease endemicity, which in terms of ${\Re }_0$, we have

$${\beta }_i^{*}(1+{\beta }_i^{*}\widehat{\Omega }-{\Re }_0)=0,$$

or

$${\beta }_i^{*}=\frac{{\Re }_0-1}{\widehat{\Omega }},$$ (16)

provided ${\Re }_0>1$ with $\widehat{\Omega }$ as the disease constant derived from Eqs (9) to (13). Hence, proof completed. □

Remark 1

The Eq. (16) further affirm the incorporation of the system force of infection in terms of model reproduction number.

4.5. Global stability analysis

We shall consider in this section the global stability of C-19FE and that of C-19EE. The following notations are necessary.

Notation 1

Lemma 1

Let $A>0(<0)$ be a $n\times n$ real matrix. If $A$ is symmetric positive definite (or symmetric negative definite), then all the eigenvalues of $A$ have negative (positive) real parts if and only if there exists a matrix $H>0$ such that

$$HA+ATHT<0(>0).$$

Lemma 2

Consider a disease model system written in the form:

$$\{\begin{array}{cc}{\displaystyle \frac{d{X}_1}{dt}=}\hfill & F(X_1,X_2)\hfill \\ {\displaystyle \frac{d{X}_2}{dt}=}\hfill & G(X_1,X_2)\hfill \end{array}$$ (17)

with $G(X_1,X_2)=0$, where $X_1\in {\Re }^m$ denotes the uninfected population and $X_2\in {\Re }^n$ denotes the infection population; $X_0=(X_1^E,0)$ denotes the C-19EE of the system. Also, assume that the following conditions holds:

• (C₁) For $\frac{d{X}_1}{dt}=F(X_1,0)$, $X_1^E$ is globally asymptomatically stable.

• (C₂) $G(X_1,X_2)=A{X}_2-\widehat{G}(X_1,X_2)$, with $\widehat{G}(X_1,X_2)\ge 0$ for all $(X_1,X_2)\in \Omega$, where the Jacobian matrix

  $$A=\frac{\partial G}{\partial x_2}(X_1^E,0),$$

has all non-negative off-diagonal elements and $X$ is the region where the model makes biological meaning. Then, the C-19EE, $X_0=(X_1^E,0)$ is globally asymptomatically stable provided that ${\Re }_0<1$ and unstable otherwise.

Lemma 3

Let $D=|\begin{array}{cc}{\displaystyle d_{11}}& d_{12}\\ {\displaystyle d_{21}}& d_{22}\end{array}|$ be a $2\times 2$ matrix. Then, $D$ is stable $iff$ $d_{11}<0,d_{22}<0$ and $\det \left(D\right)=d_{11}{d}_{22}-d_{12}{d}_{21}>0$.

Definition 1

A non-singular $n\times n$ matrix $A$ is diagonally stable (or positive stable) if there exists a positive diagonal $n\times n$ matrix $M$ such that $MA+A^T{M}^T>0$.

4.5.1. Global stability of C-19FE

We prove our global stability by invoking the theorem from [30].

Theorem 5

The fixed point $E^0=(\frac{b_p}{\mu },0,0,0,0,0,0,0)$ is globally asymptotically stable equilibrium for the system (1) provided ${\Re }_0<1$ with Lemma 2 satisfied.

Proof

Using Lemma 3 in model (1), then

$$X_1=\left[\begin{array}{c}{\displaystyle S_p}\\ {\displaystyle I_s}\\ {\displaystyle R_p}\end{array}\right],X_1=\left[\begin{array}{c}{\displaystyle X_p}\\ {\displaystyle A_u}\\ {\displaystyle I_a}\\ {\displaystyle S_s}\\ {\displaystyle H_i}\end{array}\right].$$

When $X_p=A_u=I_a=S_s=H_i=0$, the uninfected sub-systems (i.e., $S_p,I_s,R_p$) becomes

$$\{\begin{array}{ccc}{\displaystyle \frac{d{S}_p}{dt}}& =& b_p-\mu S_p\\ {\displaystyle \frac{d{I}_s}{dt}}& =& 0\\ {\displaystyle \frac{d{R}_p}{dt}}& =& 0\end{array}$$

or

$$X_1=\frac{d{S}_p}{dt}=b_p-\mu S_p,$$ (18)

which has the solution

$$S_p\left(t\right)\le \frac{b_p}{\mu }+\left(S_p\left(0\right)-\frac{b_p}{\mu }\right)e^{-\mu t}.$$ (19)

Clearly, $S_p\left(t\right)\to \frac{b_p}{\mu }$ as $t\to \infty$ regardless of the initial value $S_P\left(0\right)$. Therefore, it shows that condition C₁ (of Lemma 2) holds for our model.

Taking on the infectious sub-systems and using Eq. (13) with condition C₂ (of Lemma 2), we have

$$\frac{d{X}_2}{dt}=\left(\begin{array}{c}{\displaystyle \begin{array}{c}{\displaystyle -{\Omega }_1{X}_p+\frac{\varphi {\beta }_1{c}_1{b}_p}{\mu N}{X}_p+\frac{\varphi {\beta }_2{c}_2{b}_p}{\mu N}{A}_u}\hfill \\ {\displaystyle +\frac{\varphi {\beta }_3{c}_3{b}_p}{\mu N}{I}_a+\frac{\varphi {\beta }_4{c}_4{b}_p}{\mu N}{S}_s+\frac{\varphi {\beta }_5{c}_5{b}_p}{\mu N}{H}_i}\hfill \end{array}}\\ {\displaystyle (1-u_1)\lambda X_p-{\Omega }_2{A}_u}\\ {\displaystyle (1-u_2)k\theta A_u-{\Omega }_4{I}_a}\\ {\displaystyle (1-{\rho }_1-{\rho }_2)I_a+{\phi }_2{A}_u-{\Omega }_8{S}_s}\\ {\displaystyle (1-a_1)(1-a_2)[{\phi }_1{A}_u+{\rho }_2{I}_a+{\delta }_h{I}_s]-m_6}\end{array}\right).$$

Now, using condition C₂ i.e.,

$$\frac{d{X}_2}{dt}=G(X_1,X_2)$$

$$\frac{d{X}_2}{dt}=\left(\begin{array}{ccccc}{\displaystyle -{\Omega }_1+{\omega }_1}& {\omega }_2& {\omega }_3& {\omega }_4& {\omega }_5\\ {\displaystyle (1-u_1)\lambda }& -{\Omega }_2& 0& 0& 0\\ {\displaystyle 0}& (1-u_2)k\theta & -{\Omega }_4& 0& 0\\ {\displaystyle 0}& {\phi }_2& (1-{\rho }_1-{\rho }_2)& -{\Omega }_8& 0\\ {\displaystyle 0}& (1-u_1)(1-u_2){\phi }_1& (1-u_1)(1-u_2){\rho }_1& 0& -m_6\end{array}\right)\left(\begin{array}{c}{\displaystyle X_p}\\ {\displaystyle A_u}\\ {\displaystyle I_a}\\ {\displaystyle S_s}\\ {\displaystyle H_i}\end{array}\right)-\left(\begin{array}{c}{\displaystyle {\beta }_i\frac{b_p}{\mu }-{\beta }_i{S}_p}\\ {\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 0}\end{array}\right),$$

where ${\omega }_1=\frac{\varphi {\beta }_1{c}_1{b}_p}{\mu N}$, ${\omega }_2=\frac{\varphi {\beta }_2{c}_2{b}_p}{\mu N}$, ${\omega }_3=\frac{\varphi {\beta }_3{c}_3{b}_p}{\mu N}$, ${\omega }_4=\frac{\varphi {\beta }_4{c}_4{b}_p}{\mu N}$ and ${\omega }_5=\frac{\varphi {\beta }_5{c}_5{b}_p}{\mu N}$ with ${\Omega }_1$, ${\Omega }_2$, ${\Omega }_4$, ${\Omega }_8$ and $m_6$ defined by Eqs (13) and (9) respectively. This implies that

$$\frac{d{X}_2}{dt}=A{X}_2-\widehat{G}(X_1,X_2),$$

i.e., $G(X_1,X_2)=A{X}_2-\widehat{G}(X_1,X_2)$ for all $\widehat{G}(X_1,X_2)\ge 0$, which satisfies condition C₂. Then, we see that $S_p\le \frac{b_p}{\mu }$, implying that $G(X,Y)\ge 0$ for all $(X,Y)\in {\Re }_{+}^8$. We also notice that the matrix $A$ is an $M$-matrix since its off-diagonal elements are non-negative. Hence, this proves the global stability of C-19FE ($E^0$). □

4.5.2. Global stability of C-19EE

Here, with the incorporation of the LaSalle's invariant principle, we shall adopt the Lyapunov function for the investigation of the global stability of system endemic equilibrium (C-19EE). This approach, which had been explored by [24,31] is found useful in compartmental epidemic models. Then, the following theorem establishes the system global stability of C-19EE.

Theorem 6

Let $V$ be the Lyapunov function defined for system (1), then the global stability of the system endemic equilibrium (C-19EE) holds if its time derivative $\frac{dL}{dt}\le 0$.

Proof

Recall system (1) with biological feasible domain

$${\Re }_D=\left\{(S_p,X_p,A_u,I_a,I_s,S_s,H_i,R_p)\in {\Re }_{+}^8:N\le \frac{b_p}{\mu }:S_p+X_p+A_u+I_a+I_s+S_s+H_i+R_p\le \frac{b_p}{\mu }\right\},$$

which is clearly positively invariant in ${\Re }_{+}^8$. We also noted that at $\widehat{\alpha }=0$, $N=N^{*}=\frac{b_p}{\mu }$ as $t\to \infty$. Then, to prove for the global stability result, we construct the following Lyapunov function

$$V=\sum _{i=1}^8{w}_i{(N_i-N^{*})}^2,$$ (20)

where $w_i>0$ is a Lyapunov constant, $N_i$ is the population of $i^{th}$ compartment, $N_i^{*}$ is the equilibrium value of $N_i$ and $V$, a continuous and differentiable Lyapunov function. Computing the time derivative of $V$, along the trajectories of the system (1), we obtain

$$\dot{V}=\{\begin{array}{c}{\displaystyle 2w_1(S_p-S_p^{*})\frac{d{S}_p}{dt}+2w_2(X_p-X_p^{*})\frac{d{X}_p}{dt}+2w_3(A_p-A_p^{*})\frac{d{A}_u}{dt}+2w_4(I_a-I_a^{*})\frac{d{I}_a}{dt}}\\ {\displaystyle +2w_5(I_s-I_s^{*})\frac{d{I}_s}{dt}+2w_6(S_s-S_s^{*})\frac{d{S}_s}{dt}+2w_7(H_i-H_i^{*})\frac{d{H}_i}{dt}+2w_8(R_p-R_p^{*})\frac{d{R}_p}{dt}}\end{array}$$ (21)

Substituting the derivative of system (1) into Eq. (21) and accounting for Eq. (9), we have

$$\dot{V}=2w_1(S_p-S_p^{*})\left[{\sigma }_1(X_p-X_p^{*})+{\sigma }_2(R_p-R_p^{*})-\frac{1-u_1-u_2}{N^{*}}\left(\begin{array}{c}{\displaystyle {\beta }_1{c}_1(X_p{S}_p-X_p{S}_p^{*})+{\beta }_2{c}_2(A_u{S}_p-A_u{S}_p^{*})}\hfill \\ {\displaystyle +{\beta }_3{c}_3(I_a{S}_p-I_a{S}_p^{*})+{\beta }_4{c}_4(S_s{S}_p-S_s{S}_p^{*})}\hfill \\ {\displaystyle +{\beta }_5{c}_5(H_i{S}_p-H_i{S}_p^{*})}\hfill \end{array}\right)-\mu (S_p-S_p^{*})\right]$$

$$+2w_2(X_p-X_p^{*})\left[\begin{array}{c}{\displaystyle \frac{1-u_1-u_2}{N^{*}}\left(\begin{array}{c}{\displaystyle {\beta }_1{c}_1(X_p{S}_p-X_p{S}_p^{*})+{\beta }_2{c}_2(A_u{S}_p-A_u{S}_p^{*})}\hfill \\ {\displaystyle +{\beta }_3{c}_3(I_a{S}_p-I_a{S}_p^{*})+{\beta }_4{c}_4(S_s{S}_p-S_s{S}_p^{*})}\hfill \\ {\displaystyle +{\beta }_5{c}_5(H_i{S}_p-H_i{S}_p^{*})}\hfill \end{array}\right)}\hfill \\ {\displaystyle -(1-u_1)\lambda (X_p-X_p^{*})-m_1(X_p-X_p^{*})}\hfill \end{array}\right]$$

$$+2w_3(A_u-A_u^{*})\left[\begin{array}{c}{\displaystyle (1-u_1)\lambda (X_p-X_p^{*})-k\theta (1-u_2)(A_u-A_u^{*})}\hfill \\ {\displaystyle -(1-a_1)(1-a_2){\phi }_1(A_u-A_u^{*})-m_2{\phi }_1(A_u-A_u^{*})}\hfill \end{array}\right]$$

$$+2w_4(I_a-I_a^{*})\left[\begin{array}{c}{\displaystyle (1-u_2)k\theta (A_u-A_u^{*})-[k\theta (1-a_1)(1-a_2){\rho }_2+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)(I_a-I_a^{*})}\hfill \\ {\displaystyle -m_3(I_a-I_a^{*})}\hfill \end{array}\right]$$

$$+2w_5(I_s-I_s^{*})\left[a_1{\tau }_1{\rho }_1(I_a-I_a^{*})+a_1{\tau }_2{\gamma }_s(S_s-S_s^{*})-(1-a_1)(1-a_2){\delta }_h(I_s-I_s^{*})-m_4(I_s-I_s^{*})\right]$$

$$+2w_6(S_s-S_s^{*})\left[(1-{\rho }_1-{\rho }_1)(I_a-I_a^{*})+{\phi }_2(A_u-A_u^{*})-a_1{\tau }_2{\gamma }_s(S_s-S_s^{*})-m_5(S_s-S_s^{*})\right]$$

$$+2w_7(H_i-H_i^{*})\left[(1-a_1)(1-a_2)[{\phi }_1(A_u-A_u^{*})+{\rho }_2(I_a-I_a^{*})+{\delta }_h(I_s-I_s^{*})-m_6(H_i-H_i^{*})\right]$$

$$+2w_8(R_p-R_p^{*})\left[{\eta }_1(H_i-H_i^{*})+{\eta }_2(I_s-I_s^{*})+{\eta }_3(S_s-S_s^{*})-m_7(R_p-R_p^{*})\right].$$ (22)

Then, we add the expression ${\beta }_i{c}_i\widehat{N}{S}_p^{*}$ for all $\widehat{N}=(X_p,A_u,I_a,S_s,H_i)$ into the first and second square brackets and we obtain

Simplifying Eq. (22), we have

$$\dot{V}=2w_1(S_p-S_p^{*})\left[{\sigma }_1(X_p-X_p^{*})+{\sigma }_2(R_p-R_p^{*})-\frac{1-u_1-u_2}{N^{*}}\left(\begin{array}{c}{\displaystyle {\beta }_1{c}_1{X}_p(S_p-S_p^{*})-{\beta }_1{c}_1{S}_p^{*}(X_p-X_p^{*})}\hfill \\ {\displaystyle -{\beta }_2{c}_2{A}_u(S_p-S_p^{*})-{\beta }_2{c}_2{S}_p^{*}(A_u-A_u^{*})}\hfill \\ {\displaystyle -{\beta }_3{c}_3{I}_a(S_p-S_p^{*})-{\beta }_3{c}_3{S}_p^{*}(I_a-I_a^{*})}\hfill \\ {\displaystyle -{\beta }_4{c}_4{S}_s(S_p-S_p^{*})-{\beta }_4{c}_4{S}_p^{*}(S_s-S_s^{*})}\hfill \\ {\displaystyle -{\beta }_5{c}_5{H}_i(S_p-S_p^{*})-{\beta }_5{c}_5{S}_p^{*}(H_i-H_i^{*})}\hfill \end{array}\right)-\mu (S_p-S_p^{*})\right]$$

$$+2w_2(X_p-X_p^{*})\left[\begin{array}{c}{\displaystyle \frac{1-u_1-u_2}{N^{*}}\left(\begin{array}{c}{\displaystyle {\beta }_1{c}_1{X}_p(S_p-S_p^{*})-{\beta }_1{c}_1{S}_p^{*}(X_p-X_p^{*})}\hfill \\ {\displaystyle -{\beta }_2{c}_2{A}_u(S_p-S_p^{*})-{\beta }_2{c}_2{S}_p^{*}(A_u-A_u^{*})}\hfill \\ {\displaystyle -{\beta }_3{c}_3{I}_a(S_p-S_p^{*})-{\beta }_3{c}_3{S}_p^{*}(I_a-I_a^{*})}\hfill \\ {\displaystyle -{\beta }_4{c}_4{S}_s(S_p-S_p^{*})-{\beta }_4{c}_4{S}_p^{*}(S_s-S_s^{*})}\hfill \\ {\displaystyle -{\beta }_5{c}_5{H}_i(S_p-S_p^{*})-{\beta }_5{c}_5{S}_p^{*}(H_i-H_i^{*})}\hfill \end{array}\right)}\hfill \\ {\displaystyle -(1-u_1)\lambda (X_p-X_p^{*})-m_1(X_p-X_p^{*})}\hfill \end{array}\right]$$

$$+2w_3(A_u-A_u^{*})\left[\begin{array}{c}{\displaystyle (1-u_1)\lambda (X_p-X_p^{*})-k\theta (1-u_2)(A_u-A_u^{*})}\hfill \\ {\displaystyle -(1-a_1)(1-a_2){\phi }_1(A_u-A_u^{*})-m_2{\phi }_1(A_u-A_u^{*})}\hfill \end{array}\right]$$

$$+2w_4(I_a-I_a^{*})\left[\begin{array}{c}{\displaystyle (1-u_2)k\theta (A_u-A_u^{*})-[k\theta (1-a_1)(1-a_2){\rho }_2+a_1{\tau }_1{\rho }_1+(1-{\rho }_1-{\rho }_2)(I_a-I_a^{*})}\hfill \\ {\displaystyle -m_3(I_a-I_a^{*})}\hfill \end{array}\right]$$

$$+2w_5(I_s-I_s^{*})\left[a_1{\tau }_1{\rho }_1(I_a-I_a^{*})+a_1{\tau }_2{\gamma }_s(S_s-S_s^{*})-(1-a_1)(1-a_2){\delta }_h(I_s-I_s^{*})-m_4(I_s-I_s^{*})\right]$$

$$+2w_6(S_s-S_s^{*})\left[(1-{\rho }_1-{\rho }_1)(I_a-I_a^{*})+{\phi }_2(A_u-A_u^{*})-a_1{\tau }_2{\gamma }_s(S_s-S_s^{*})-m_5(S_s-S_s^{*})\right]$$

$$+2w_7(H_i-H_i^{*})\left[(1-a_1)(1-a_2)[{\phi }_1(A_u-A_u^{*})+{\rho }_2(I_a-I_a^{*})+{\delta }_h(I_s-I_s^{*})-m_6(H_i-H_i^{*})\right]$$

$$+2w_8(R_p-R_p^{*})\left[{\eta }_1(H_i-H_i^{*})+{\eta }_2(I_s-I_s^{*})+{\eta }_3(S_s-S_s^{*})-m_7(R_p-R_p^{*})\right].$$ (23)

Or

Then, Eq. (23) can be written in compact form as:

$$\dot{V}=Q\left(ZY+Y^T{Z}^T\right)Q^T$$ (24)

with $Q=[S_p-S_p^{*},X_p-X_p^{*},A_u-A_u^{*},I_a-I_a^{*},I_s-I_s^{*},S_s-S_s^{*},H_i-H_i^{*},R_p-R_p^{*}]$, $Z=diag(w_1,w_2,...,w_8)$ and

$$Y=\left(\begin{array}{cccccccc}{\displaystyle {\lambda }_1}& -{\lambda }_3& -{\lambda }_5& -{\lambda }_7& 0& -{\lambda }_{11}& -{\lambda }_{14}& 0\\ {\displaystyle {\lambda }_2}& {\lambda }_4& {\lambda }_5& {\lambda }_8& 0& {\lambda }_{11}& {\lambda }_{14}& 0\\ {\displaystyle 0}& 0& (1-u_1)\lambda -{\lambda }_6& 0& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& {\lambda }_9-{\lambda }_{10}& 0& 0& 0& 0\\ {\displaystyle 0}& 0& 0& 0& a_1{\tau }_1{\rho }_1+a_1{\tau }_2{\gamma }_s& -{\lambda }_{12}& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& {\lambda }_{13}& 0& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& {\lambda }_{15}& 0\\ {\displaystyle 0}& 0& 0& 0& 0& 0& 0& {\lambda }_{16}\end{array}\right),$$ (25)

where

$$\begin{array}{ccc}\hfill {\lambda }_1& =& {\sigma }_1{X}_p+{\sigma }_2{R}_p-\frac{(1-u_1-u_2)}{N*}\left(({\beta }_1{c}_1{X}_p+{\beta }_2{c}_2{A}_u+{\beta }_3{c}_3{I}_a+{\beta }_4{c}_4{S}_s+{\beta }_5{c}_5{H}_i)-\mu ]\right),\hfill \\ \hfill {\lambda }_2& =& \left(\frac{1-u_1-u_2}{N*}\right)[{\beta }_1{c}_1{X}_p+{\beta }_2{c}_2{A}_u+{\beta }_3{c}_3{I}_a+{\beta }_4{c}_4{S}_s+{\beta }_5{c}_5{H}_i],{\lambda }_3=\left(\frac{1-u_1-u_2}{N*}\right){\beta }_1{c}_1{S}_p*,\hfill \\ \hfill {\lambda }_4& =& \frac{1-u_1-u_2}{N*}{\beta }_1{c}_1{S}_p*-(1-u_1)\lambda +m_1,{\lambda }_5=\left(\frac{1-u_1-u_2}{N*}\right){\beta }_2{c}_2{S}_p*,{\lambda }_6=k\theta (1-u_2)+(1-a_1)(1-a_2){\phi }_1+m_2\hfill \\ \hfill {\lambda }_7& =& \left(\frac{1-u_1-u_2}{N*}\right){\beta }_3{c}_3{S}_p*,{\lambda }_8=\left(\frac{1-u_1-u_2}{N*}\right){\beta }_3{c}_3{S}_p*,{\lambda }_9=(1-u_2)k\theta ,{\lambda }_{10}=(1-a_1)(1-a_2){\rho }_2-a_1{\tau }_1{\rho }_1-(1-{\rho }_1-{\rho }_2)+m_3,\hfill \\ \hfill {\lambda }_{11}& =& \left(\frac{1-u_1-u_2}{N*}\right){\beta }_4{c}_4{S}_p*,{\lambda }_{12}=(1-a_1)(1-a_2){\delta }_h+m_4,{\lambda }_{13}=(1-{\rho }_1-{\rho }_1)+{\phi }_2-a_1{\tau }_2{\gamma }_s+m_5,{\lambda }_{14}=\left(\frac{1-u_1-u_2}{N*}\right){\beta }_5{c}_5{S}_p*,\hfill \\ \hfill {\lambda }_{15}& =& (1-a_1)(1-a_2)[{\phi }_1+{\rho }_2+{\delta }_h-m_6\text{and}{\lambda }_{16}={\eta }_1+{\eta }_2+{\eta }_3-m_7.\hfill \end{array}$$

Now, we discuss the global asymptotic stability of $Q$ by showing that the matrix $Y$ of Eq. (25) is Lyapunov stable or $-Y$ is diagonally stable. The following lemmas and theorem yields the required proof.

Lemma 4

For the matrix $Y$ as defined by Eq. (25), let $D=-Y$ and then $D$ is diagonal stable.

Lemma 5

If $D$ is diagonal matrix, then the inverse denoted by $E$ i.e., $E=-Y^{-1}$ is also diagonal stable.

Proving Lemmas 4 and 5 follows from the next lemma.

Lemma 6

Let $D=\left[d_{ij}\right]$ be a non-singular $n\times n$ matrix $(n\ge 2)$ and $M=diag(m_1,....,m_n)$ be a positive diagonal $n\times n$ matrix. Let $E=D^{-1}$, then if $d_m>0$, $\tilde{M}\tilde{E}+{\left(\tilde{M}\tilde{E}\right)}^T>0$ and $\tilde{M}\tilde{D}+{\left(\tilde{M}\tilde{D}\right)}^T>0$, then it is possible to choose $m_n>0$ such that $\tilde{M}\tilde{D}+D^T{M}^T>0$. Hence, the following theorem holds.

Theorem 7

The matrix $Y$ defined by Eq. (25) is Lyapunov stable.

Proof

Based on Lemmas 5 and 6, and since $Y_{88}>0$ i.e., ${\lambda }_{16}$, then there exists a positive diagonal matrix

$Z(-Y)+{(-Y)}^T{Z}^T>0$. Thus, $ZY+Y^T{Z}^T<0$. □

Therefore, using the LaSalle's invariant principle, the proof for the global stability of the system endemic equilibrium is completed from the following theorem.

Theorem 8

The endemic equilibrium $E^{*}=(S_p^{*},X_p^{*},A_u^{*},I_a^{*},I_s^{*},S_s^{*},H_i^{*},R_p^{*})$ of (1) is globally asymptotically stable in $E^0$, provided ${\Re }_0>1$ and unstable otherwise.

Proof

Invoking Lemmas 5, 6 and Theorem 7, we obtain $\frac{dV}{dt}<0$ when $E^0\ne E^{*}$ and $E^0$ is not on $S-axis$ (a set of measure zero). Therefore, the largest invariant set in $E^0$ such that $\frac{dV}{dt}<0$ is a singleton $E^{*}$, which is our endemic equilibrium point. Then, by LaSalle's invariant principle [32], it follows that the endemic equilibrium of model (1), $E^{*}$ is globally asymptotically stable (GAS) in ${\Re }_D$, if $E^0\ne E^{*}$. This complete the proof. □

5. Numerical results

Here, attempt is made to verify the viability of derived theoretical predictions. That is, we numerically illustrate the system force of infection ${\beta }_i\left(\widehat{N}\right)$ under ${\Re }_{0\left(1\right)}=10.94$. Clearly, the inclusion of both system force of infection and the reproduction number in system (1) is an integral component of the model novelty. These two key components determine the dynamic behavior of the system (1) at $u_i=0,a_i=0$ for all $i=1,2$, which we shall simulate. Finally, we simulate the system stability endemic equilibrium following the application of control functions $u_i>0,a_i>0$ for all $i=1,2$ under reproduction number at onset of treatment computed to be ${\Re }_{0\left(2\right)}=3.224$. The entire simulations is feasible under in-built Runge-Kutta software in a Mathcad environment.

5.1. Simulation of model force of infection

Invoking Tables 1,2 and Eq. (2), we illustrate the system force of infection ${\beta }_i\left(\widehat{N}\right)$ against the susceptible

$S_p\left(t\right)$ and the exposed $X_p\left(t\right)$ components. That is, Fig. 2 (a)-(b) depicts the dynamics of the system force of infection for untreated COVID-19 scenario.

Fig. 2.

(a-b) Dynamics of model force of infection against $S_p\left(t\right)$, $X_p\left(t\right)$ with ${\Re }_{0\left(1\right)}=10.94$.

Notably, Fig. 2(a)-(b) portrait the consequential rate of force of infection on the susceptible and the exposed by the infectious classes denoted by ${\beta }_i\left(\widehat{N}\right)$. Clearly, for an untreated COVID-19 dynamics, which is vindicated by high system reproduction number ${\Re }_{0\left(1\right)}=10.94$, the virulence ingress required to aggressively contaminate both the susceptible ($0.5\le S_p\left(t\right)\le 5.335\times {10}^{12}cell/m{l}^3$) and exposed sub-population ($0.3\le X_p\left(t\right)\le 3.201\times {10}^{12}cell/m{l}^3$) is computed to be in the range of $0.022\le {\beta }_i\left(\widehat{N}\right)\le 2.12\times {10}^{11}$ $m{l}^{3}vi{r}^{-1}{d}^{-1}$ for all $t_f\le 90$ $days$. This explains why the practical exponent spread of the virus is observed under off-treatment scenario world-wide. Furthermore, the consequential of system force of infection on varying subpopulations observed under off-treatment is seen in the next Section 5.2.

5.2. Simulation of system basic model (without control functions)

The impact of both the force of infection and system reproduction number under off-treatment protocol is explicitly illustrated as depicted by Fig. 3 (a)-(h) below:

Fig. 3.

(a-h) Simulation of COVID-19 infection dynamics under off-treatment scenario with ${\Re }_{0\left(1\right)}=10.94$ and ${\beta }_i\left(\widehat{N}\right)=0.022$.

Fig. 3(a)-(h) represent simulation of COVID-19 infection dynamics where zero control measures is experimented for all time interval $t_f\le 90$ days. We observe from Fig. 3(a) that within the first 18 days of onset COVID-19 viral load, the susceptible population is contaminated by the rapid spread of the virus resulting to population extinction with $S_p\left(t\right)\le -24.447cells/ml3$ for all $18\le t_{f}90$ days. This rapid decline is evident by the exponential increase in the rate at which population become exposed with value at $0.3\le X_p\left(t\right)\le 49.785cells/ml3$ for all $t_f\le 18$ days and then decline slightly to stability at $X_p\left(t\right)\le 41.024cells/ml3$ after $18\le t_f\le 90$ days (see Fig. 3(b)). Fig. 3(c) depicts rapid increase in the population that becomes unaware asymptomatic infectious class i.e., $0.1\le A_u\left(t\right)\le 11.675cells/ml3$ for the first 10days and then decline slightly to stability at $A_u\left(t\right)\le 9.8cells/ml3$ for all $20\le t_f\le 90$ days.

In Fig. 3(d), the screen aware infectives is simulated. We observe unsteady slow inclination curve of $0.15\le I_a\left(t\right)\le 1.312cells/m{l}^3$ but far lower compared to Fig. 3(c). From Fig. 3(e), following the non-availability of treatment measures, it is clear that isolation compartment, which constitute control/treatment measure remain at zero i.e., $I_s\left(t\right)=0.0$ for all $t_f\le 90$ days. Of interest, the system is bound to experience commensurate increase in the proportion of super spreaders under off-treatment scenario. Fig. 3(f) illustrate the dynamics of super spreaders with $0.05\le S_s\left(t\right)\le 4.174cells/m{l}^3$ for all $18\le t_f\le 90$ days. The increase in the number of hospitalized patients is observe in Fig. 3(g) with value at $0.0\le H_i\left(t\right)\le 7.463cells/m{l}^3$ in $18\le t_f\le 90$ days. None-the-less, possibility of recovery through any other means order than study designated preventive/control measures exists. Fig. 3(h) represent the amount of recovered proportion with $R_p\left(t\right)\le 6.176cells/m{l}^3$. This can be attributed to varying natural adaptive immune response.

5.3. Simulation of system endemic equilibria (with dual bilinear control functions)

Having theoretically investigated the endemicity for the spread of COVID-19 viral load and the methodological application of dual-bilinear control functions, we devote this sub-section to the illustration of the viability of our findings. Obviously, with the introduction of pair non-pharmaceutical control functions (face masks and social distancing) and pair pharmaceutical intervention functions (hydroxyl-chloroquine –HCQ and Azithromycin – AZT), we compute as represented by Fig. 4 (a)-(h) below, the varying dynamics of COVID-19 transmission under the aforementioned designated control measures.

Fig. 4.

(a-h) Stability dynamics for system endemic equilibria under dual-bilinear control functions, ${\Re }_{0\left(2\right)}=3.224$.

From Fig. 4(a)-(h), we see a set of undulating dynamic flow of COVID-19 infection under clinical application of both non-pharmaceutical ($u_{i(i-1,2)}>0$) and pharmaceutical therapies ($a_{i(i-1,2)}>0$) control measures. In Fig. 4(a), unlike Fig. 3(a), we observe a smooth resurging curve for the susceptible population at a significant range of $0.5\le S_p\left(t\right)\le 3.143cells/m{l}^3$ for all $18\le t_f\le 90$ days. Notably, the introduction of non-pharmaceutical preventive measures as in Fig. 4(b), shows decline in the rate of exposed class with value at $0.3\le X_p\left(t\right)\le 0.612cells/m{l}^3$. Fig. 4(c) portrait initial instantaneous decline of unaware asymptomatic population in the first $t_f\le 3$ days and then exhibit slight increase to stability value of $0.1\le A_u\left(t\right)\le 0.247cells/m{l}^3$ for all $3\le t_f\le 90$ days. Furthermore, from Fig. 4(d, e), we observe rapid decline in the spread of the virus as indicated by the aware infectives i.e., $0.15\ge I_a\left(t\right)\ge 0.01cells/m{l}^3$. The isolated infectious population exhibits undulating declining curve in the first $t_f\le 7$ days with attain stability range of $2\times {10}^{-3}\le I_a\left(t\right)\le 5.8\times {10}^{-3}cells/m{l}^3$ for all $20\le t_f\le 90$ days.

Considering Fig. 4(f), the dynamics of super spreaders is illustrated. The undulating smooth curve indicates drastic reduction in the rate of super spreaders with value range $0.05\le S_s\left(t\right)\le 0.061cells/m{l}^3$. This result is quite significant when compared to that off-treatment as portrait by Fig. 3(f). Moreso, we see from Fig. 4(g) the cases of hospitalization equally reducing drastically to value range of $0.0\le H_i\le 0031cells/m{l}^3$ as against $H_i\le 7.463cells/m{l}^3$ of Fig. 3(F). Finally, following the significant decrease in the rate of isolation, super spreaders and hospitalization, the proportion of recovery equally decline to stability $R_p\le 0.047cells/m{l}^3$ for all $40\le t_f\le 90$ days - see Fig. 4(h).

Of note, the choice of Lyapunov function in conjunction with LaSalle's invariant principle in analyzing the global stability conditions of design model have clearly given insight to COVID-19 infection dynamics and as well, portrait a novel methodological control approach in the quest to mitigate COVID-19 pandemic. These results provides unique treatment approach when compare with results of system motivating models [12,16], where the aforementioned technique were not considered. Moreso, the method implemented by this study have affirm the high effectiveness of the combination of designated non-pharmaceutical and pharmacotherapies in controlling the deadly disease.

6. Discussion

The study by [16] had been formulated as an adhoc compartmental COVID-19 model to account for the peculiarities of COVID-19 infection dynamics in Wuhan, China, while the study [12] considered contact parameter as its cardinal point in its mathematical formulation of COVID-19 transmission in Niger Republic. On account of earlier highlighted limitations of these two models, which borders on non-availability of mathematical model for the combination of medical pharmaceuticals and non-pharmaceuticals in the treatment protocols of spiking COVID-19 pandemic, the present study sought to determine and analyze the global stability conditions that could lead to predictions of COVID-19 infection dynamics and treatment control protocols. The study using nonlinear differential equations, formulated its model as an 8-Dimensional deterministic compartmental mathematical COVID-19 dynamic model involving the interactions between healthy and COVID-19 shrouded infectious population. Designated control functions are categorized into dual non-pharmaceuticals (face-masking and social distancing) and dual pharmacotherapies (hydroxylchloroquine and azithromycin).

The study adopted classical method of Lyapunov function in conjunction with LaSalle's invariant principle for the analysis of the system global stability conditions. The investigation was conducted for both untreated and onset treatment scenarios and numerical validations performed using in-built Runge-Kutta of order of precision 4 in a Mathcad surface. Results obtained indicated that under off-treatment scenario, the logistic dose response curve of heavy droplet of finer aerosol viral load of COVID-19 required to extensively contaminate the susceptible population of $S_p\left(t\right)\le 5.335\times {10}^{12}cells/m{l}^3$ is in the range of ${\beta }_i\left(\widehat{N}\right)\le 2.12\times {10}^{11}$ under exponential reproduction number of ${\Re }_{0\left(1\right)}=10.94$. This result clearly buttress why within short space of time, the entire world was highly ravaged by the deadly virus. Moreso, under off-treatment scenario as depicted by Fig. 3(a)-(h), we observed rapid extinction of the susceptible population leading to excruciating endemic spread of the virus for all $18\le t_f\le 90$ days.

Following the introduction of methodological control approach under dual-bilinear control functions, the system reproduction number declined drastically to ${\Re }_{0\left(2\right)}=3.224$. This result is in agreement with COVID-19 estimated reproduction number (2-3) by [1]. Moreso, we found from Fig. 4(a)-(h) that the application of only non-pharmaceuticals at onset of infection (mild) indicated immense reduction in the spread of the virus, while for severe cases, the application of non-pharmaceuticals enhanced by induced multi-therapies lead to significant reduction in the rate of isolations, super spreads and hospitalized population with attained stability of ($I_s\left(t\right)\le 5.8\times {10}^{-3}$, $S_s\left(t\right)\le 0.061$, $H_i\left(t\right)\le 0.031$) $cells/m{l}^3$.

Of note, the positive impact from dual-bilinear treatment protocols is vindicated by the enhanced reduction in the rate of recovery and the accelerated rejuvenation of the susceptible population as against population extinction under off-treatment scenario. That is, the application of the method of Lyapunov function incorporating LaSalle's invariant principle and methodological dual bilinear control protocols has a very desirable effect upon the susceptible population. Comparatively, the present results when compared with those of system motivating models [12,16], shows that our controls does behaves somewhat different from control functions used in motivating models not explicitly studied under dual-bilinear protocols.

7. Conclusion

Following the non-availability of mathematical model for the combination of medical pharmaceuticals and non- pharmaceuticals in the treatment protocols of spiking COVID-19 pandemic, the present study sought to determine and analyze the global stability conditions for the role of dual-bilinear control functions in the control and treatment dynamics of novel COVID-19 pandemic. The model adopted human-to-human transmission mode with population under-consideration partitioned into 8-Dimensional deterministic compartments: Susceptible population $S_p\left(t\right)$, Exposed class $X_p\left(t\right)$, Unaware asymptomatic infectious population $A_u\left(t\right)$ Aware infectives $I_a\left(t\right)$, isolated infectious population $I_s\left(t\right)$, Super spreaders $S_s\left(t\right)$, Hospitalized infectives $H_i\left(t\right)$ and the recovered population $R_p\left(t\right)$. First, we investigated the model state-space and established the system reproduction number for both off/on treatment protocols using modified generated data from certified models. Using classical method of Lyapunov functions in combination with the theory of LaSalle's invariant principle, we have discussed the global stability conditions of COVID-19 model. Obviously, the method of Lyapunov function has been widely applied to varying dynamical systems, but the essential part of this analysis is based on the incorporation of the theory of LaSalle's invariant principle under dual-bilinear control protocols. The analytical predictions of the global stability analysis are provided and numerically simulated. We found that under designated control functions, COVID-19 transmission was drastically reduced to insignificant threshold of ($0.1\le A_u\left(t\right)\le 0.247$, $0.15\ge I_a\left(t\right)\ge 0.01$, $0.05\le S_s\left(t\right)\le 0.061$ and $0.01\le H_i\left(t\right)\le 0.031$) $cells/m{l}^3$ for all $18\le t_f\le 90$ days, leading to tremendous rejuvenation of the susceptible population, $0.5\le S_p\left(t\right)\le 3.143cells/m{l}^3$. It is presumed that the insignificant persistence of the virus can be attributed to possible reinfection of recovered population upon integrated into the susceptible population. Furthermore, the present results in comparison with those of motivating models, have projected classical methodological and epidemiological concept under designated clinical conditions. Thus, for enhance optimal result, the study highly suggest possible application of optimal control theory to the existing model.

Authors' contribution

Bassey Echeng Bassey: Conceptualization, formulations, writing, investigation, editing, review, programing and analysis.

Jerimiah U. Atsu: Supervision, methodology, funding, editing and validation.

Declaration of Competing Interest

This undertaking certify that onbehalf of the authors, the corresponding author wish to state clearly that there exist no conflict of interest in the submission of this manuscript. I also would like to declare that the work described was an original research that has not been published previously and not under any consideration for publication elsewhere.