Compartmental modeling approach for prediction of unreported cases of COVID-19 with awareness through effective testing program

Abhishekh Singh; Vikash Rana; Vijai Shanker Verma

doi:10.1515/cmb-2024-0014

Article Open Access

Compartmental modeling approach for prediction of unreported cases of COVID-19 with awareness through effective testing program

Abhishekh Singh , Vikash Rana and Vijai Shanker Verma

Published/Copyright: November 23, 2024

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From the journal Computational and Mathematical Biophysics Volume 12 Issue 1

Abstract

The objective of this article is to study the compartmental modeling approach for the prediction of unreported cases of coronavirus disease 2019 by considering six compartments. Our model is described by a system of six ordinary differential equations with initial conditions. The basic properties of solution of the model are established. The model is shown to have two equilibrium points, i.e., the disease-free and endemic equilibrium points. The basic reproduction number R 0 is derived by the next-generation matrix method. Stability analysis is carried out in the study. Furthermore, sensitivity analysis is also performed to identify the impact of important parameters that significantly affect R 0 . Numerical simulations provide a good approximation model for COVID-19, which will be utilized to investigate future pandemic with similar nature of spread as COVID-19 and estimate the number of unreported cases worldwide.

Keywords: COVID-19; basic reproduction number; stability analysis; sensitivity; numerical simulation

MSC 2010: 92D30; 93C15; 34D08

1 Introduction

At the end of the second decade of the twentieth century discovery of the origin of the coronavirus disease 2019 (COVID-19) outbreaks and the subsequent declaration of the pandemic in March 2020, efforts in research and development have led to an expanded comprehension of the epidemiology of COVID-19.

The coronavirus is transmitted when an infected individual speaks, sneezes, sings, or coughs from their mouth or nose. In the initial phase of coronavirus disease eruption, it is critical to comprehend the mechanics of the infection’s propagation [14,19]. Analyzing variations in transmission over time can shed light on the epidemiological scenario [9] and show whether outbreak management strategies are working as intended [7,12]. Millions of people died in the COVID-19 epidemic [16]. Governments, research laboratories, and world health organization have reported only cases that result in positive tests [15]; however, a sizable number of cases are infected with the virus but are either not found or not diagnosed for a variety of reasons, such as the infected person’s lack of awareness of their symptoms, inability to seek medical attention, or belief that their symptoms are normal and do not require testing [5,9]. There are many cases where the virus is present, but it is either not found or not diagnosed because the infected person was not affected by the virus’s symptoms, was unable to visit the hospital, or believed that the symptoms were normal and did not require testing. However, asymptomatic persons are more harmful, as a result, large number of people is being infected daily. The unreported cases are a matter of serious concern as they can cause severe problems to the patients with comorbidities [1].

Mathematical modeling plays a crucial role in understanding the disease dynamics and predicting the future patterns of the disease transmission [6]. For any pandemic/endemic of an infectious disease, it is crucial to run its affecting parameters with awareness through effective testing to take any further precautions [3]. To examine the momentum of disease outbreaks and guiding public healthcare initiatives, mathematical models estimate the course of diseases [11].

Several investigators have studied the transmission dynamics and control of COVID-19 using different epidemiological models. During an epidemic wave in Austria, Rippinger et al. [13] used the proportions of reported and unreported COVID-19 cases to calculate the influence of detection probability. Hamou et al. [9] have discussed the undetected cases of the novel coronavirus in Morocco. In a review article, AlArjani et al. [1] have summarized nine mathematical models that have been used in predicting the transmission of COVID-19. All the nine models followed by case studies have been thoroughly reviewed and characterized to study the intrinsic properties of each model in predicting disease transmission dynamics. Goudiaby et al. [8] have investigated optimal control strategies for COVID-19 and tuberculosis co-dynamics by incorporating five control measures.

Bhadauria et al. [4] have presented an susceptible, infected, quarantined, vaccinated mathematical model on COVID-19 with virus population in the environment which supported the fact that the ability of the virus to stay alive in the environment plays a crucial role in the fast and broad spread of infection among the population. Verma et al. [19] developed an susceptible, vaccinated, infected, quarantined, recovered epidemic model for COVID-19 and proposed that if control measures are used above a certain threshold for an extended length of time, the disease will finally go extinct. Kunwar and Verma [10] provided a mathematical analysis of the susceptible, vaccinated, exposed, infected, quarantined, recovered model for COVID-19, utilizing numerical simulation to determine the effects of varying control intervention intensities.

In view of the above, we have considered a deterministic susceptible, exposed, infected, recovered, testing (SEIRT) compartmental model to describe the transmission dynamics of COVID-19. The main aim of this model is to study the disease dynamics of COVID-19 and is to predict the number of reported and unreported patients [5]. Therefore, we provide a comprehensive model to calculate the number of unreported infected and recovered patients. In this model, we try to provide a comprehensive model to calculate the number of unreported infected and recovered patients. We have also incorporated effective testing compartments in our model so that maximum number of reported cases can be diagnosed.

2 Mathematical model

Here, the total population under consideration denoted by N ( t ) at any time t has been divided into five compartments, namely, susceptible S ( t ) , exposed E ( t ) , reported infective I R ( t ) , unreported infective I U ( t ) , and recovered R ( t ) . Therefore, at any time t , we can write N ( t ) = S ( t ) + E ( t ) + I R ( t ) + I U ( t ) + R ( t ) . An effective testing implementation compartment T ( t ) is also introduced in the model due to the involvement of social and psychological factors with the population in which the susceptible population enters with testing adoption rate ϕ and leaves with testing breakdown rate ϕ 0 . Thus, the model is named as the SEIRT model.

To construct the model, the following assumptions are made:

(i) Susceptible class

The susceptible population is assumed to be generated by the recruitment rate A . It is further assumed that the susceptible population decreases following the effective contact at a rate β S ( E + I U ) . Here, it is also assumed that the reported individuals quarantined themselves, that is why, they did not come into the susceptible class. Also, in the susceptible class, the population decreases due to natural death μ S . It is further assumed that due to loss of immunity, susceptible population increases by θ R . All these can be summarized in the following equation:

d S d t = A − β S ( E + I U ) − μ S + θ R .

(ii) Exposed class

The exposed population is assumed to be generated and increased by the β S ( E + I U ) . It is further assumed that the exposed cases decrease by η R E . It also decreases by η U E . Also, in the exposed class, the population decreases due to natural death μ E . All these can be summarized in the following equation:

d E d t = β S ( E + I U ) − ( η R + η U + μ ) E .

(iii) Reported infective class

The reported infective class is assumed to be generated and increased by the progression rate from asymptomatic to reported symptomatic with rate η R . Furthermore, it is also assumed that the number of reported infective decreases after recovery from the COVID-19 disease by the recovery rate γ R . Also, the number of reported infective decreases due to the natural death by μ I R as well as death due to disease by α R I R . It is further assumed that the reported class is increased by ψ T I U such that unreported infected population goes for testing and being diagnosed positive. All these can be summarized in the following equation:

d I R d t = η R E − ( γ R + α R + μ ) I R + ψ T I U .

(iv) Unreported infective class

The unreported infective class is assumed to be generated and increased by the progression rate from asymptomatic to unreported symptomatic with rate η U . Furthermore, it is also assumed that the number of unreported infective decreases after recovery from the COVID-19 disease by the recovery rate γ U . Also, the number of unreported infective decreases due to the natural death by μ I U as well as death due to disease by α U I U . It is further assumed that the unreported class decreases by ψ T I U because the unreported class after being tested positive moves to the reported class. All these can be summarized in the following equation:

d I U d T = η U E − ( γ U + α U + μ ) I U − ψ T I U .

(v) Recovered class

The recovered class is assumed to be generated and increased by the recovery of the reported class γ R I R . Similarly, it is also increased by the recovery of the unreported class γ U I U . The recovered class is also assumed to be decreased by the natural death rate μ R. It is also supposed that the class decreases by θ R because the recovered population moves to the susceptible class due to the loss of immunity after recovery by the rate θ . All these can be summarized in the following equation:

d R d t = γ R I R + γ U I U − μ R − θ R .

(vi) Effective testing class

An effective testing compartment incorporated with testing adoption rate ϕ to increase the susceptible population by ϕ S. The testing decreases with testing breakdown rate ϕ 0 by ϕ 0 T , because of the lack of percentage of testing adoption rate. Also, it increases after the baseline number of testing is done by ϕ 0 T 0 . All these can be summarized in the following equation:

d T d t = ϕ S − ϕ 0 ( T − T 0 ) .

The compartmental flow diagram of the proposed SEIRT model is shown in Figure 1. In the model, we have not considered the vital dynamics related to birth death during the epidemic. All parameters used in the model are listed with their description in Table 1. The mathematical formulation of the model is given as:

(1) d S d t = A − β S ( E + I U ) − μ S + θ R , d E d t = β S ( E + I U ) − ( η R + η U + μ ) E , d I R d t = η R E − ( γ R + α R + μ ) I R + ψ T I U , d I U d T = η U E − ( γ U + α U + μ ) I U − ψ T I U , d R d t = γ R I R + γ U I U − μ R − θ R , d T d t = ϕ S − ϕ 0 ( T − T 0 ) ,

with the initial conditions (Figure 2):

(2) S ( 0 ) = S 0 > 0 , E ( 0 ) = E 0 ≥ 0 , I U ( 0 ) = I U 0 ≥ 0 , I R ( 0 ) = I R 0 ≥ 0 , R ( 0 ) = R 0 > 0 , T ( 0 ) = T 0 ≥ 0 ,

Figure 1

Compartmental diagram of the SEIRT model.

Table 1

Description of model parameters

Parameters	Description
A	Recruitment rate
β	Effective contact rate of transmission of COVID-19
θ	Loss of immunity after recovery
μ	Natural death rate of population
η R	Progression rate from asymptomatic to reported symptomatic
η U	Progression rate from asymptomatic to unreported symptomatic
γ R	Recovery rate of reported COVID-19-infected individuals
γ U	Recovery rate of unreported COVID-19-infected individuals
α R	Death rate of reported COVID-19-infected individuals
α U	Death rate of unreported COVID-19-infected individuals
ψ	Rate at which unreported infected population enter into reported class after testing and being diagnosed as positive
ϕ	Testing adoption rate
T 0	Initial assessment of testing
ϕ 0	Testing breakdown rate

Figure 2

Trend of reported and unreported infective population with time.

3 Basic properties of the model

The basic properties of model, i.e., non-negativity and boundedness, are stated as follows:

3.1 Non-negativity of the model

In order to show that all state variables remain positive for t ≥ 0 , non-negativity requirements must be satisfied. Thus, we have the following theorem to show the non-negativity of solutions of the model:

Theorem 1

All the solutions ( S , E , I U , I R , R , T ) of system (1) under the initial conditions given by (2) are non-negative for all t ≥ 0 .

Proof

From the system of equations (1) and (2), we obtain

(3) d S d t S = 0 = A + θ R ≥ 0 ,

(4) d E d t E = 0 = β S I U ≥ 0 ,

(5) d I R d t I R = 0 = η R E + π T I U ≥ 0 ,

(6) d I U d t I U = 0 = η U E ≥ 0 ,

(7) d I R d t I R = 0 = γ R I R + γ U I U ≥ 0 ,

(8) d T d t T = 0 = ϕ S + ϕ 0 T 0 ≥ 0 .

From the above, we conclude that all the state variables of the system are non-negative for t ≥ 0 .□

3.2 Boundedness

We are going to show that system (1) possesses bounded solutions. The term boundedness refers to the natural constraints on the unchecked expansion of the infected population as a result of a variety of factors, including environmental factors or protective behaviors developed by individuals in order to prevent the infection. Now, we prove the following theorem, for boundedness of solution of the model:

Theorem 2

The feasible region of system (1) is given by a set Ω , which is defined as follows:

Ω = ( S , E , I R , I U , R , T ) : 0 ≤ S + E + I R + I U + R ≤ A μ , T = ϕ A ϕ 0 μ + T 0 ,

and is closed for system (1) with non-negative initial condition (2) for all solutions.

Proof

Adding the first five equations of system (1), and also using the relation N = S + E + I R + I U + R , we obtain

d N d t = A − μ N − α R I R − α U I U ,

∴ d N d t ≤ A − μ N .

Now, integrating it by the method of separation of variables, we obtain

(9) N ( t ) ≤ A μ + N 0 − A μ e − μ t .

Hence, if 0 ⩽ N ( 0 ) ⩽ A μ , then limsup t → ∞ N ( t ) ⩽ A μ .

Again, the boundedness for testing done is given by the following equation:

d T d t = ϕ S − ϕ 0 ( T − T 0 ) ,

which gives

(10) 0 < T ≤ ϕ A μ ϕ 0 + T 0 .

Thus, Ω = ( S , E , I R , I U , R , T ) : 0 ≤ S + E + I R + I U + R ≤ A μ , T = ϕ A ϕ 0 μ + T 0 is closed for system (1). Consequently, from a biological and mathematical point of view, the proposed model is good and well defined.□

4 Model analysis

The model analysis is done as given in the following.

4.1 Existence of equilibrium points

Here, system (1) has two equilibrium points: one the disease-free equilibrium (DFE) point E 0 ( A μ , 0 , 0 , 0 , 0 , T 0 ) and the other endemic equilibrium (EE) point E * , which is obtained using the isocline method as follows.

4.2 Isocline method

The EE point E * is obtained using the isocline method as follows:

We compute the value of E and R in terms of S and I U using the second, third, and fifth equations of system (1) with α 1 > β S . Then, using these values in equation (1), we obtain

(11) ℋ ( S , I U ) = A − β 2 S 2 I U α 1 − β S − β S I U − μ S + θ γ R η R β S I U α 2 ( μ + θ ) ( α 1 − β S ) + θ γ R ψ ϕ S I U ϕ 0 α 2 ( μ + θ ) + θ γ R ψ ϕ 0 T 0 I U ϕ 0 α 2 ( μ + θ ) + θ γ U I U ( μ + θ ) .

Now, from the sixth equation of system (1), we compute the value of T in terms of S , and using this value in fourth equation, of system (1) we obtain

(12) G ( S , I U ) = η U β S I U α 1 − β S − α 3 I U − ψ ( ϕ S I U + ϕ 0 T 0 I U ) ϕ 0 .

Again, we find d S d I U as follows:

d S d I U = − ∂ ℋ ⁄ ∂ I U ∂ ℋ ⁄ ∂ S = − − α 1 β S α 1 − β S + θ γ R η R β S α 2 ( μ + θ ) ( α 1 − β S ) + θ γ R ψ ϕ S ϕ 0 α 2 ( μ + θ ) + θ γ R ψ T 0 α 2 ( μ + θ ) β 2 I U ( 2 α 1 S − β S 2 ) ( α 1 − β S ) 2 − β I U − μ + θ γ R η R β α 1 I U α 2 ( μ + θ ) ( α 1 − β S ) 2 + θ γ R ψ ϕ I U ϕ 0 α 2 ( μ + θ ) .

From the above, it can be easily obtained that d S d I U < 0 ,

if θ γ R η R β S α 2 ( μ + θ ) ( α 1 − β S ) + θ γ R ψ ϕ S ϕ 0 α 2 ( μ + θ ) + θ γ R ψ T 0 α 2 ( μ + θ ) + θ γ U ( μ + θ ) > α 1 β S α 1 − β S

and θ γ R η R β α 1 I U α 2 ( μ + θ ) ( α 1 − β S ) 2 + θ γ R ψ ϕ I U ϕ 0 α 2 ( μ + θ ) + β 2 I U ( 2 α 1 S − β S 2 ) ( α 1 − β S ) 2 > β I U + μ .

Again, we find d S d I U as follows:

d S d I U = − ∂ G ⁄ ∂ I U ∂ G ⁄ ∂ S = − η U β S ( α 1 − β S ) + α 3 + ψ ( ϕ S + ϕ 0 T 0 ) ϕ 0 η U β α 1 ( α 1 − β S ) 2 − ψ ϕ I U ϕ 0 .

From the above, it is noted that d S d I U > 0 if α 3 + ψ ( ϕ S + ϕ 0 T 0 ) ϕ 0 > η U β S ( α 1 − β S ) > ψ ϕ I U ϕ 0 .

Therefore, the function G ( S , I U ) is monotonically increasing and ℋ ( S , I U ) is monotonically decreasing if the aforementioned inequalities hold. From the aforementioned conditions, it is clear that isoclines (11) and (12) cut at a unique point ( S * , I U * ) . The point of intersection ( S * , I U * ) is shown in Figure 3.

Figure 3

Intersection of susceptible and infected unreported population.

4.3 Basic reproduction number

An essential component of any epidemiological model analysis is the basic reproduction number R 0 . Within a community, it is the total number of secondary illnesses that an infected individual spreads to all members of the community during the infectious period. The existence of the system’s DFE is ascertained analytically. We calculate the basic reproduction number using the next-generation matrix method, which was first presented by Driesseche and Watmough [18,20]. For system (1), the DFE point is E 0 ( A μ , 0 , 0 , 0 , 0 , T 0 ) .

To apply the next-generation matrix method, we take only the infected compartments from system (1), and thus, we have the following infective class sub-system:

(13) d E d t = β S ( E + I U ) − ( η R + η U + μ ) E ≡ F 1 − G 1 ,

(14) d I R d t = η R E − ( γ R + α R + μ ) I R + ψ T I U ≡ F 2 − G 2 ,

(15) d I U d T = η U E − ( γ U + α U + μ ) I U − ψ T I U ≡ F 3 − G 3 .

The right-hand side of the infective class sub-system can be written as ℱ ( x ) – V ( x ) , where

ℱ = β S E + β S I U 0 0 and V = α 1 E − η R E + α 2 I R + ψ T I U − η U E + α 3 I U + ψ T I U ,

where α 1 = η R + η U + μ , α 2 = γ R + α R + μ , and α 3 = γ U + α U + μ .

Let us define F = ∂ ( ℱ ) i ∂ E j and V = ∂ ( V ) i ∂ E j for i , j = 1 , 2 , 3 at E 0 .

Differentiating ℱ and V w.r.t. I U and I R at E 0 ( A μ , 0 , 0 , 0 , 0 , T 0 ), we have

F = β A μ 0 β A μ 0 0 0 0 0 0 and V = α 1 0 0 − η R α 2 − ψ T 0 − η U 0 α 3 + ψ T 0 .

The next-generation matrix of the model is given by F V − 1 , and the basic reproduction number R 0 is given by the spectral radius ρ (i.e., the largest eigenvalue) of F V − 1 . Thus, we find

(16) F V − 1 = β A μ α 1 + β A η U μ α 1 ( α 3 + ψ T 0 ) 0 β A μ ( α 3 + ψ T 0 ) 0 0 0 0 0 0 .

The eigenvalues of F V − 1 are given by ∣ F V − 1 − λ I ∣ = 0 , i.e.,

(17) β A μ α 1 + β A η U μ α 1 ( α 3 + ψ T 0 ) − λ 0 β A μ ( α 3 + ψ T 0 ) 0 − λ 0 0 0 − λ = 0 .

From equation (17), the eigenvalues are found as λ = 0 , 0 and β A μ α 1 + β A η U μ α 1 ( α 3 + ψ T 0 ) ; the third one is the largest eigenvalue.

Thus, the spectral radius ρ of F V − 1 is given by ρ ( F V − 1 ) = ( β A μ α 1 + β A η U μ α 1 ( α 3 + ψ T 0 ) ) .

Now, the basic reproduction number R 0 is given by R 0 = β A μ α 1 1 + η U ( α 3 + ψ T 0 ) ,

(18) i.e. R 0 = β A ( γ U + α U + ψ T 0 + η U + μ ) μ ( η R + η U + μ ) ( γ U + α U + ψ T 0 + μ ) .

5 Stability analysis

The Jacobian matrix of the system of equations (1) is given by

V ( E ) = − β ( E + I U ) − μ − β S 0 − β S θ 0 β ( E + I U ) β S − α 1 0 β S 0 0 0 η R − α 2 ψ T 0 ψ I U 0 η U 0 − α 3 − ψ T 0 − ψ I U 0 0 γ R γ U − μ − θ 0 ϕ 0 0 0 0 − ϕ 0 .

The eigenvalues of the Jacobian matrix V are given by ∣ V − λ I ∣ = 0 .

5.1 Local stability of DFE point

To discuss the stability of DFE point E 0 ( A μ , 0 , 0 , 0 , 0 , T 0 ) based on the basic reproduction number R 0 , we have the following theorem:

Theorem 3

If R 0 < 1 , then the DFE point E 0 ( A μ , 0 , 0 , 0 , 0 , T 0 ) is locally asymptotically stable, otherwise unstable.

Proof

The Jacobian matrix of system (1) at E 0 is given by

V ( E 0 ) = − μ − β A μ 0 − β A μ θ 0 0 β A μ − α 1 0 β A μ 0 0 0 η R − α 2 ψ T 0 0 0 0 η U 0 − α 3 − ψ T 0 0 0 0 0 γ R γ U − μ − θ 0 ϕ 0 0 0 0 − ϕ 0 .

Four eigenvalues of the aforementioned matrix are obtained as λ 1 = − ϕ 0 , λ 2 = − μ , λ 3 = − ( μ + θ ) , and λ 4 = − α 2 = − ( γ R + α R + μ ) , and the remaining two eigenvalues are determined by the following quadratic equation:

λ 2 + α 3 + ψ T 0 + α 1 − β A μ λ + α 1 ( α 3 + ψ T 0 ) [ 1 − R 0 ] = 0 .

Using the Routh-Hurwitz criterion if α 3 + ψ T 0 + α 1 > β A μ and R 0 < 1 , then α 3 + ψ T 0 + α 1 − β A μ and α 1 ( α 3 + ψ T 0 ) are positive. Hence, all the eigen values are negative, which means that, the DFE point E 0 ( A μ , 0 , 0 , 0 , 0 , T 0 ) is locally asymptotically stable, otherwise unstable.□

5.2 Local stability analysis of EE point

We linearize the system about the EE point E * to ascertain the local stability of the system by setting S = S 1 + S * , E = E 1 + E * , I R = I R 1 + I R * , I U = I U 1 + I U * , R = R 1 + R * , and T = T 1 + T * . After linearization, the system of equations (1) can be written as follows:

d S 1 d t = − β S 1 E * − β S * E 1 − β S 1 I U * − β S * I U 1 − μ S 1 + θ R 1 , d E 1 d t = β S 1 E * + β S * E 1 + β S 1 I U * + β S * I U 1 − α 1 E 1 , d I R 1 d t = η R E 1 − α 2 I R 1 + π T 1 I U * + ψ T * I U 1 , d I U 1 d t = η U E 1 − α 2 I U 1 + π T 1 I U * − ψ T * I U 1 , d R 1 d t = γ R I R 1 + γ U I U 1 − μ R 1 − θ R 1 , d T 1 d t = ϕ S 1 − ϕ 0 T 0 .

Now, let us consider the Lyapunov function

(19) V = 1 2 S 1 2 + 1 2 E 1 2 + 1 2 I R 1 2 + 1 2 I U 1 2 + 1 2 R 1 2 + 1 2 T 1 2 .

Differentiating equation (19) w.r.t. t, we have

V ˙ = − 1 4 a 11 S 1 2 + a 12 S 1 E 1 − 1 3 a 22 E 1 2 − 1 4 a 11 S 1 2 + a 14 S 1 I U 1 − 1 5 a 44 I U 1 2 − 1 4 a 11 S 1 2 + a 15 S 1 R 1 − 1 3 a 55 R 1 2 − 1 4 a 11 S 1 2 + a 16 S 1 T 1 − 1 3 a 66 T 1 2 − 1 3 a 22 E 1 2 + a 23 E 1 I R 1 − 1 4 a 33 I R 1 2 − 1 3 a 22 E 1 2 + a 24 E 1 I U 1 − 1 5 a 44 I U 1 2 − 1 4 a 33 I R 1 2 + a 34 I R 1 I U 1 − 1 5 a 44 I U 1 2 − 1 4 a 33 I R 1 2 + a 35 I R 1 R 1 − 1 3 a 55 R 1 2 − 1 4 a 33 I R 1 2 + a 36 I R 1 I 1 − 1 3 a 66 T 1 2 − 1 5 a 44 I U 1 2 + a 45 I U 1 R 1 − 1 3 a 55 R 1 2 − 1 5 a 44 I U 1 2 + a 46 I U 1 T 1 − 1 3 a 66 T 1 2 ,

where

(20) a 11 = − β E * − β I U * − μ , a 22 = β S * − α 1 , a 33 = − α 2 , a 44 = − α 3 − ψ T * , a 55 = − μ − θ , a 66 = − ϕ 0 , a 12 = − β S * + β E * + β I U * , a 13 = 0 , a 14 = − β S * , a 15 = θ , a 16 = ϕ , a 23 = η R , a 24 = β S * + η U , a 25 = 0 , a 26 = 0 , a 34 = ψ T * , a 35 = γ R , a 36 = ψ I U * , a 45 = γ U , a 46 = − ψ I U * , a 56 = 0 .

The Lyapunov function V ˙ is negative definite, if the following conditions hold:

( i ) ( − β S * + β E * + β I U * ) 2 < 1 3 ( β E * + β I U * + μ ) ( α 1 − β S * ) , ( i i ) ( − β S * ) 2 < 1 5 ( β E * + β I U * + μ ) ( α 3 + ψ T * ) , ( i i i ) ( θ ) 2 < 1 3 ( β E * + β I U * + μ ) ( + μ + θ ) , ( i v ) ( ϕ ) 2 < 1 3 ( β E * + β I U * + μ ) ( ϕ 0 ) , ( v ) ( η R ) 2 < 1 3 ( α 1 − β S * ) ( α 2 ) , ( v i ) ( β S * + η U ) 2 < 4 15 ( α 1 − β S * ) ( α 3 + ψ T * ) , ( v i i ) ( ψ T * ) 2 < 1 5 ( α 2 ) ( α 3 + ψ T * ) , ( v i i i ) ( γ R ) 2 < 1 3 ( α 2 ) ( μ + θ ) , ( i x ) ( ψ I U * ) 2 < 1 3 ( α 2 ) ( ϕ 0 ) ( x ) ( γ U ) 2 < 4 15 ( α 3 + ψ T * ) ( μ + θ ) , ( x i ) ( − ψ I U * ) 2 < 4 15 ( α 3 + ψ T * ) ( ϕ 0 ) .

Therefore, the EE point E * is locally asymptotically stable if all of the conditions (i)–(xi) are met.

6 Sensitivity analysis

Sensitivity analysis is an approach used in epidemiology to assess how an unmeasured quantity affects a sample’s random findings [2,17]. Here, we examine the model’s sensitive parameters. The partial derivatives show the sensitivity of R 0 w.r.t. the model parameters. For any parameter c , the sensitivity index with respect to c is defined as follows (Table 2):

K c = ∂ R 0 ∂ c × c R 0 , where R 0 = β A ( γ U + α U + ψ T 0 + η U + μ ) μ ( η R + η U + μ ) ( γ U + α U + ψ T 0 + μ ) .

The sensitivity indices for each of the parameters with respect to R 0 are computed and are given in the following:

K β = ∂ R 0 ∂ β β R 0 = 1 , K A = ∂ R 0 ∂ A A R 0 = 1 , K γ U = ∂ R 0 ∂ γ U γ U R 0 = − 0.6 × 1 0 − 4 , K η R = ∂ R 0 ∂ η R η R R 0 = − 0.39 , K η U = ∂ R 0 ∂ η U η U R 0 = − 0.294 × 1 0 − 3 , K α U = ∂ R 0 ∂ α U α U R 0 = − 0.2 × 1 0 − 3 , K ψ = ∂ R 0 ∂ ψ ψ R 0 = − 0.04 × 1 0 − 8 , K μ = ∂ R 0 ∂ μ μ R 0 = − 0.38675 × 1 0 − 10 , K T 0 = ∂ R 0 ∂ T 0 T 0 R 0 = − 0.2067 .

Table 2

Values and sources of model parameters

Parameters	Values	Sources
A	300	Assumed
β	0.000002	[18]
θ	0.28	Assumed
μ	0.014	Estimated
η R	0.013	Assumed
η U	0.006	Assumed
γ R	0.06	Estimated
γ U	0.01	Estimated
α R	0.005	[8]
α U	0.03	Estimated
ψ	0.0000001	Assumed
ϕ	0.177	[17]
T 0	0.600006	Estimated
ϕ 0	0.100089	Estimated

The various sensitivity indices of R 0 w.r.t. model parameters are shown in Table 3.

Table 3

Sensitivity indices of R 0 with model parameters

Model parameters	Sensitivity indices
A	+ 1
β	+ 1
μ	‒ 0.38675 × 1 0 ‒ 10
γ U	‒ 0.6 × 1 0 ‒ 4
η R	‒ 0.39
η U	‒ 0.294 × 1 0 ‒ 3
α U	‒ 0.2 × 1 0 ‒ 3
ψ	‒ 0.4 × 1 0 ‒ 9
T 0	‒ 0.2067

The sensitivity analysis shows that the recruitment rate A and contact rate β have a significant positive influence on the virus’s ability to spread. The analysis suggests that the impact magnitudes of A and β are the same. The remaining parameters μ , γ U , η R , η U , α U , ψ , and T 0 have an adverse effect. Figure 4 displays a graphical representation of the sensitivity indices.

Figure 4

Sensitivity analysis of parameters.

7 Numerical simulation

To support analytical results, numerical simulations are carried out using the parameter values taken in the model.

From Figure 5, it can be concluded that as we increase η R , the asymptomatic population moves to the reported symptomatic population, resulting in no further disease that would be transmitted to the susceptible population. The reason behind it is the reported symptomatic quarantines themselves and cannot come further in contact with susceptible population. From the figure, it can also be seen that by increasing η R , unreported population decreases and reported population increases.

$Figure 5 Variation of η R {\eta }_{R} with time for (i) unreported infectives (left panel) and (ii) reported infective (right panel).$

Figure 5

Variation of η R with time for (i) unreported infectives (left panel) and (ii) reported infective (right panel).

In Figure 6, we have plotted the graphs (left and right panels) for both the reported and unreported populations, respectively. In this figure, it has been observed that increasing η U shows that the asymptomatic population moves to the unreported symptomatic, i.e., the population have acquired the disease and are showing the symptoms. From this observation, it can be concluded that as soon as they found symptoms, they immediately moves for testing for no further transmission of the disease. Also, from figure, it can be clearly seen that by increasing η U , the reported class decreases, which is a matter of serious concern and should be taken seriously. This can only be reduced by increasing the number of testing for the susceptible population.

$Figure 6 Variation of η U {\eta }_{U} with time for (i) unreported infectives (left panel) and (ii) reported infectives (right panel).$

Figure 6

Variation of η U with time for (i) unreported infectives (left panel) and (ii) reported infectives (right panel).

In Figure 7, we have shown the variation of ϕ for reported population with time. From this figure, we observe that as we increase testing adoption rate ϕ , the reported class increases, and hence, this observation reveals that the infected population go for testing and diagnosed positive and further isolate themselves; therefore, they will not transmit the disease anymore.

$Figure 7 Variation of ϕ \phi for reported infectives with time.$

Figure 7

Variation of ϕ for reported infectives with time.

In Figure 8, we have plotted the variation of ψ for reported population with time. From this figure, it is observed that an increase in ψ shows the increase in reported population, which explains that unreported infected population go for testing, and after testing, they are found to be positive and hence quarantine themselves. We have also drawn variation of ψ for unreported population with time. From the figure, it can be clearly seen that increasing ψ shows decreasing unreported population, which simply reveals that more and more unreported population go for testing are found positive and quarantine and no further transmission is done.

$Figure 8 Variation of ψ \psi with time for (i) unreported infectives (left panel) and (ii) reported infectives (right panel).$

Figure 8

Variation of ψ with time for (i) unreported infectives (left panel) and (ii) reported infectives (right panel).

The impact of various parameters on R 0 is further investigated and is shown in Figure 9. A shift from R 0 < 1 to R 0 > 1 can be seen, if we increase either of the parameters as shown in Figure 9(a)–(h).

Figure 9

Matrix plots showing the changing nature in basic reproduction number ( R 0 ) of SEIRT model under following parametric variations: (a) R 0 vs ( β , A) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [250, 340], (b) R 0 vs ( β , α U ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [0.01, 0.05], (c) R 0 vs ( β , η R ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [0.01, 0.015], (d) R 0 vs ( β , η U ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [ 1 × 1 0 ‒ 3 , 9 × 1 0 ‒ 3 ], (e) R 0 vs ( β , γ U ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [0.005, 0.03], (f) R 0 vs ( β , μ ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [0.011, 0.019], (g) R 0 vs ( β , ψ ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [ 0.9 × 1 0 ‒ 7 , 1.1 × 1 0 ‒ 7 ], (h) R 0 vs ( β , T 0 ) ∈ [ 1.376 × 1 0 ‒ 6 , 1.396 × 1 0 ‒ 6 ] × [0.1, 0.8].

8 Conclusion

In this article, we have suggested an epidemiological model with an effective testing procedure to predict the reported and unreported population class of COVID-19. This research demonstrates that the coronavirus does not spread only between susceptible and confirmed cases; there are multiple infected cases because of contact with unreported cases, who in turn are unaware that they are virus carriers. This is the cause of the difficulty in containing the virus’s spread. The results obtained are based on the constraints used, i.e., the values of parameters used in the model. The benefit of this model is that it is not only applicable to the COVID-19 disease but also it can be fitted to any communicable disease in which the symptomatic and asymptomatic populations are known. By demonstrating the existence, uniqueness, non-negativity, and boundedness of solutions in each region, the model’s biological significance is further demonstrated. The equilibrium points of the model are also computed. The basic reproduction number is also utilized in the stability analysis. This mathematical model reveals that if the basic reproduction number is less than unity, then the DFE point is stable; otherwise, it is unstable. For the existence of EE point, we have used the isocline method. Stability analysis of EE point is carried out by incorporating the Lyapunov function. To supplement the analytical results, numerical simulations are performed and the results are found to be in good agreement.

Acknowledgement

The authors are grateful to Editors-in-Chief and referees for their careful reading, valuable comments and making helpful suggestions to improve the paper.

Funding information: This research received no specific grant from any funding agency, commercial or nonprofit sectors.
Author contributions: All authors have equal contribution.
Conflict of interest: The authors have no conflicts of interest to disclose.
Ethical approval: This research did not require ethical approval.
Data availability statement: This manuscript has not any associated data in a data repository.

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Received: 2023-11-25

Revised: 2024-06-26

Accepted: 2024-10-03

Published Online: 2024-11-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cmb-2024-0014

Keywords for this article

COVID-19; basic reproduction number; stability analysis; sensitivity; numerical simulation

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