Study of transmission pattern of COVID-19 among cardiac and noncardiac population using a nonlinear mathematical model

Gopishetty Shirisha; Saroj Vernekar

doi:10.1515/cmb-2025-0020

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Study of transmission pattern of COVID-19 among cardiac and noncardiac population using a nonlinear mathematical model

Gopishetty Shirisha
Gopishetty Shirisha
Department of Mathematics, Stanley College of Engineering and Technology for Women, Abids, Hyderabad-500001, Telangana, India

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and Saroj Vernekar
Saroj Vernekar
Department of Mathematics, BMS Institute of Technology and Management, Avalahalli, Yelahanka, Bengaluru-560064, Karnataka, India

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Published/Copyright: April 11, 2025

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From the journal Computational and Mathematical Biophysics Volume 13 Issue 1

Abstract

Recent studies have intensified the risk of post-COVID-19 cardiovascular diseases. In an effort to understand the intricate dynamics of COVID-19 infection, this article suggests a four-compartment model to portray the dynamic interplay between the susceptible population, the COVID-19-infected population detected without and with cardiovascular disease, and the recovered population. Basic properties such as nonnegativity and boundedness of solutions and the existence of disease-free and endemic equilibria are discussed. The model’s basic reproduction number is obtained. Sufficient conditions for local and global stability at the equilibrium point are established by restricting the functionals and parameters of the system. Numerical examples are illustrated to support the results. The relative significance of the model parameters to disease transmission is determined by performing a sensitivity analysis of the model. It is found that a rise in infection among the cardio population will drastically affect the overall infection rate compared to that of the noncardio population, supporting the real-time scenario. This model also emphasizes the importance of vaccination and treatment in controlling the spread of the virus.

Keywords: COVID-19; cardiovascular disease; equilibria; basic reproduction number; stability

MSC 2010: 92-10; 34D20; 34D23; 49K40

1 Introduction

In the twenty-first century, the COVID-19 pandemic has had the worst effects on humanity. In addition to the financial and political crises in the afflicted countries, it affected general human health issues. As per the COVID-19 epidemiological update of January 7, 2024, there have been over 774 million confirmed cases and over seven million deaths worldwide [35]. Reports on mortality may provide more insight into the true causes of death and the adverse outcomes associated with COVID-19 [23]. Based on data from retrospective case series from Italy, China, and New York, of the COVID-19 patients who were admitted to the intensive care unit, 56% had hypertension, 17% had cardiovascular disease, and 19% had diabetes [4]. Thus, patients with pre-existing comorbidities, such as hypertension, diabetes, and cardiovascular disease, were at higher risk for mortality with COVID-19 infection [14]. According to early findings from China and Italy, two countries where COVID-19 first spread during the pandemic, up to 1 in 5 COVID-19 individuals experience heart damage. Even in individuals without serious breathing issues like acute respiratory distress syndrome or ARDS, heart failure has been the cause of mortality in COVID-19 patients [20]. Hence, the impact of COVID-19 on infection persistence is notably high in individuals with pre-existing cardiovascular diseases. This correlation contributes to an elevated transmission rate of the virus. Individuals with cardiovascular conditions face an increased susceptibility to prolonged infection, thereby amplifying the overall transmission dynamics of COVID-19.

Following the July 2023 Heart Letter, even a slight infection with SARS-CoV-2 (the virus that causes COVID-19) may raise your chance of having a heart attack, stroke, or heart failure for up to a year after you recover from the illness [9]. It has been identified that multiple mechanisms for heart damage exist in COVID-19, which include low oxygen levels, myocarditis, a condition in which the heart is inflamed, stress cardiomyopathy, etc. [11,36,37]. As a result of the virus, many people who are not cardiac have reported experiencing heart problems, increasing the number of cardiac patients, which is alarming for the global health of humanity.

While COVID-19 is still prevailing and new variants like JN.1 are increasing, designing effective strategies to restrict the rise in the population of cardiovascular diseases by controlling the spread of the virus using treatment and vaccination is the need of the hour. This requires an understanding of the dynamics of the spread of the infection. As mathematics has a long history of having different tools to be used in modern infectious disease epidemiology, both for prediction and for deeper understanding, we will use some of its tools for our study.

Many research groups like pharmacists, mathematicians, chemists, biologists, and epidemiologists in different parts of the world have put much effort into developing models [1,5,7,8,18] that help in the analysis of effective control and preventive measures for different communicable diseases such as COVID-19. Among them are, to name a few: A dynamic compartmental mathematical model describing the transmissibility of the mers-cov virus in public is introduced in [32]. In a partially comorbid community, mathematical modelling of COVID-19 transmission dynamics was proposed in [31]. A nonlinear mathematical model on the COVID-19 transmission pattern among diabetic and nondiabetic populations was modelled in [2]. A fractional order control model for diabetes and COVID-19 co-dynamics with Mittag-Leffler function was studied in [22]. Mathematical modelling and analysis of COVID-19 and tuberculosis transmission dynamics were examined in [29]. The transmission dynamics of SARS-CoV-2 among diabetic patients were addressed in [21]. In [30], the effects of COVID-19 on the cardio-circulatory system were investigated using clinical data and statistical analysis by means of a mathematical model. The influence of COVID-19 disease on heart function was examined computationally in [26]. Various optimal control strategies for COVID-19 transmission dynamics with or without comorbidities have been studied in the literature, but none have worked on the dynamics of a mathematical model exhibiting the transmission of COVID-19 infection among the community of people with and without cardiovascular disease. Also, there is a need to emphasize the effect of vaccination on susceptible individuals and the treatment of infected individuals [6,10,17,28].

Owing to the aforementioned points, it is necessary to investigate the dynamics of the COVID infection in cardiac and noncardiac patients. To obtain the required dynamics, the total population is divided into four groups: the susceptible population, the infected population with no-cardiovascular disease (N-CVD), the infected population with preexisting cardiovascular disease (CVD), and the recovered population. As we are proposing a new model, we examine the behavior of the model by studying the basic properties such as the existence, positivity and boundedness of solutions, the existence of equilibria, basic reproduction number, and local and global stability of the model. To assess the significance of the parameters of the model, we will be using sensitivity analysis, by which one can explore which parameters have the most or the slightest influence on the basic reproduction number $( R 0 )$ of the model [12,25,27].

The following is the structure of this article. Section 2 describes a model that depicts the dynamics of the COVID-19 virus in the N-CVD and CVD populations, and a result of basic analysis of nonnegative and bounded solutions has been examined. Section 3 discusses the stability analysis at equilibria, wherein the existence of disease-free and endemic equilibria, calculation of BRN and results on local stability at disease-free and endemic equilibria have been focused. Global stability results are presented in Section 4. A description of the sensitivity analysis of $R 0$ is given in Section 5. We illustrate our assertion with numerical examples in Section 6 and also determine the sensitive index for specific key parameters in the examples of the model. In Section 7, the conclusion is given.

2 Model description and analysis

In the present work, the examination of epidemic outbreaks is conducted from an interdisciplinary perspective by extending a susceptible-infectious-recovered model, which is a mathematical compartmental model based on the normative behavior of a population under study. Our proposed model subdivides the infected population into N-CVD and CVD. Thus, the model subdivides the total human population size at time $t$ , into susceptible (as $u ( t )$ ), infected N-CVD (as $v ( t )$ ), infected CVD (as $w ( t )$ ), and recovered (as $r ( t )$ ). The followings are the assumptions on the dynamical interaction between the different populations

Both CVD and N-CVD are infectious populations, and they can spread the infection through contact.
Susceptible can obtain infected either from infected N-CVD or infected CVD.
The newly infected individuals are classified as either N-CVD or CVD based on the diagnosis of pre-existing cardiac diseases.
It is presumed that the vaccinated population will not contract the virus because of the immune response induced by vaccination. As such, they are included in the population that has recovered.
Due to COVID-19 infection, there is a possibility that a person without pre-cardiac issues may develop cardiac problems.
Here, natural death, COVID-19 infection-related fatalities, and CVD-related deaths are all taken into consideration.
The treated individuals are no longer contagious and are regarded as having recovered.

The schematic illustration of our model is shown in Figure 1.

Figure 1

The flow dynamics of transmission and recovery of COVID-19 among the community with and without cardiovascular disease. Source: Created by the authors using MATLAB software.

In view of the aforementioned assumptions, the dynamics of the model are proposed as follows:

(1)

$u ′ = k − a f 1 ( u ) v − b f 2 ( u ) w − d u − γ u , v ′ = a ( 1 − α ) f 1 ( u ) v + b ( 1 − α ) f 2 ( u ) w − ( d + d 1 ) v − p 1 v − η v , w ′ = a α f 1 ( u ) v + b α f 2 ( u ) w + η v − ( d + d 2 ) w − p 2 w , r ′ = γ u + p 1 v + p 2 w − d r ,$

with initial conditions

$u ( t = 0 ) = u 0 , v ( t = 0 ) = v 0 , w ( t = 0 ) = w 0 , r ( t = 0 ) = r 0 .$

Description of the parameters of system (1) are provided in Table 1.

Table 1

Parameters of system (1)

Variables	Biological significance
$u$	Susceptible population
$v$	Infected population with no-cardiovascular disease (N-CVD)
$w$	Infected population with cardiovascular disease (CVD)
$r$	Recovered population
Parameters	Biological significance
$k$	Recruitment rate of $u$
$a$	Tansmission rate of $u$ to $v$
$b$	Transmission rate of $u$ to $w$
$d$	Natural death rate in all the compartments
$γ$	Vaccination rate of $u$
$d 1$	Rate of mortality of $v$ because of infection
$d 2$	Rate of mortality of $w$ because of infection or CVD
$α$	Fraction of infected population having pre-cardiac issues
$η$	Rate of conversion of $v$ to $w$ during infection period
$p 1$	Recovered rate (by treatment) of $v$
$p 2$	Recovered rate (by treatment) of $w$
$f 1$	Interaction function of $u$ and $v$
$f 2$	Interaction function of $u$ and $w$

The parameter $k ≥ 0$ is the growth rate of the susceptible population, $f 1$ and $f 2$ denotes the nonlinear and nonnegative interaction functions showing how the susceptible $u$ exposed to N-CVD population $v$ and CVD population $w$ respectively. $a$ and $b$ are transmission rates of $u$ to $v$ and $u$ to $w$ , respectively. The natural death rate is $d > 0$ . The parameter $d 1 ≥ 0$ is the death rate of the $v$ due to infection and $d 2 ≥ 0$ is the death rate of $w$ either due to infection or due to cardiovascular diseases. The parameter $α > 0$ is the fraction of the infected population having pre-history of cardiovascular disease, whereas $( 1 − α ) > 0$ are the rest of the infected population. As there is a possibility of N-CVD becoming CVD during the infection period, the rate of conversion of $v$ to $w$ during the infection period is taken as $η$ . $p 1$ and $p 2$ are the rates of recovery (by treatment) of the infected populations $v$ and $w$ , respectively. The rate at which the susceptible population $u$ gets vaccinated is the parameter $γ > 0$ .

2.1 Basic properties

As it is a biological model, the solutions of the system should be nonnegativity, and to predict the real-time conditions, it is necessary to have bounded solutions. So, in this section, we will study the basic properties of the model, such as the existence of solutions, conditions for nonnegative and bounded solutions, and the existence and uniqueness of the equilibrium point.

Next, we are going to check for what conditions the system has nonnegative and bounded solutions

Theorem 2.1

For every $t > 0$ , the solutions of system (1) with nonnegative initial conditions remain nonnegative and bounded.

Proof

From the given system (1), we have

$[ u ′ ] u = 0 = k ≥ 0 , [ v ′ ] v = 0 = b ( 1 − α ) f 2 ( u ) w ≥ 0 , ∀ f 2 ( u ) , w ≥ 0 , [ w ′ ] w = 0 = a α f 1 ( u ) v + η v ≥ 0 , ∀ f 1 ( u ) , v ≥ 0 , [ r ′ ] r = 0 = γ u + p 1 v + p 2 w ≥ 0 , ∀ u , v , w , p 1 , p 2 ≥ 0 .$

Thus, the solutions are nonnegative.

Now, to prove the boundedness of the solutions, we define $ϰ$ as follows:

$ϰ ( t ) = u ( t ) + v ( t ) + w ( t ) + r ( t ) .$

Then derivative of $ϰ$ after simplifying is given as follows:

$d ϰ d t = k − d u − v ( d + d 1 ) − w ( d + d 2 ) − d r ≤ k − τ ϰ ( t ) , where τ = min { d , d + d 1 , d + d 2 } = d .$

After integrating and simplifying, we obtain $lim t → ∞ sup ϰ ( t ) ≤ k τ .$ Conclusion follows [2,13].□

3 Stability analysis at equilibria

It is essential to examine the model’s behavior to apply it to real-time data. A better approach would be to examine the model’s stability at the equilibrium point. For biological models, there are two types of equilibria–disease free and endemic. Stability at the disease-free equilibrium here suggests that the system will eventually reach an environment devoid of endemics, but stability at an endemic equilibrium indicates that the endemic will always be present. So, to analyze it, first, we investigate the existence of equilibrium solutions of model (1).

3.1 Equilibria

3.1.1 Disease-free equilibria

As disease-free equilibrium solution is a steady-state solution of the model in the absence of disease, we take the equilibrium solution as $( u * , 0 , 0 , r * )$ . As this is one of the solutions of (1), we obtain

(2)

$k − d u * − γ u * = 0 , γ u * − d r * = 0 .$

Solving (2), we obtain, $u * = k d + γ$ and $r * = γ k d ( d + γ )$ .

Thus, $k d + γ , 0 , 0 , γ k d ( d + γ )$ is disease-free equilibrium solution of system (1).

3.1.2 Endemic equilibria

Let the endemic equilibrium solution be given by $( u * , v * , w * , r * )$ .

As $( u * , v * , w * , r * )$ is the solution of the system, we obtain

(3)

$k − a f 1 ( u * ) v * − b f 2 ( u * ) w * − d u * − γ u * = 0 ,$

(4)

$a ( 1 − α ) f 1 ( u * ) v * + b ( 1 − α ) f 2 ( u * ) w * − ( d + d 1 ) v * − p 1 v * − η v * = 0 ,$

(5)

$a α f 1 ( u * ) v * + b α f 2 ( u * ) w * + η v * − ( d + d 2 ) w * − p 2 w * = 0 ,$

(6)

$γ u * + p 1 v * + p 2 w * − d r * = 0 .$

As equations (3)–(6) cannot be solved explicitly, we solve them implicitly as follows: from equation (4), we obtain

(7)

$w * = − [ a ( 1 − α ) f 1 ( u * ) − ( d + d 1 + p 1 + η ) ] v * b ( 1 − α ) f 2 ( u * ) .$

Similarly from equation (5), we obtain

(8)

$w * = − [ a α f 1 ( u * ) + η ] v * b α f 2 ( u * ) − ( d + d 2 + p 2 ) .$

From Eqs. (7) and (8), we arrive at

(9)

$( d + d 1 + p 1 + η ) − a ( 1 − α ) f 1 ( u * ) b ( 1 − α ) f 2 ( u * ) = − [ a α f 1 ( u * ) + η ] b α f 2 ( u * ) − ( d + d 2 + p 2 ) = φ .$

From Eqs. (7) and (8), for $v * > 0$ and $w * > 0$ , we require

(10)

$f 1 ( u * ) < d + d 1 + p 1 + η a ( 1 − α ) & f 2 ( u * ) < d + d 2 + p 2 b α .$

By assuming $f 1$ and $f 2$ to be continuous functions on defined intervals and $f 1 ( 0 ) = 0 = f 2 ( 0 )$ , it leads to the existence of $u *$ , which satisfies conditions (10).

Although solving (9) for $u *$ is difficult, in Section 6, we will demonstrate that for a chosen interaction function $f 1$ and $f 2$ , $u *$ exists and satisfies (9).

To find $v *$ and $w *$ , we substitute equation (7) into equation (3) to obtain

(11)

$v * = [ k − ( d + γ ) u * ] ( 1 − α ) d + d 1 + p 1 + η .$

Similarly, we obtain $w *$ .

From equation (6), $r * = u * + p 1 v * + p 2 w * d$ .

Thus, we obtain an endemic equilibrium point $( u * , v * , w * , r * )$ , where $u *$ , $v *$ , $w *$ , $r *$ are nonzero if $f 1$ and $f 2$ satisfy conditions (10).

From the aforementioned discussion, we conclude that an endemic and disease-free equilibrium exists for the system (1).

3.1.3 Interdependence of infected populations

If $a ( 1 − α ) f 1 ( u * ) − ( d + d 1 + p 1 + η ) = 0$ , then from equation (3), we obtain $b ( 1 − α ) f 2 ( u * ) w * = 0$ , which implies $f 2 ( u * ) = 0$ or $w * = 0$ .

We assume that $f 1 ( u * ) > 0 ∀ u * > 0$ and $f 2 ( u * ) > 0 ∀ u * > 0$ . Otherwise, $v *$ and $w *$ cannot infect any $u *$ , and infection does not prevail.

Thus, as $f 2 ( u * ) > 0$ , we obtain $w * = 0$ if $a ( 1 − α ) f 1 ( u * ) − ( d + d 1 + p 1 + η ) = 0$ .

Note that $w * = 0$ implies $v * = 0$ from equation (6).

Likewise, $b α f 2 ( u * ) − ( d + d 2 + p 2 ) = 0 ⇒ v * = 0$ and $w * = 0$ .

Remark 3.1

Thus, if $a ( 1 − α ) f 1 ( u * ) − ( d + d 1 + p 1 + η ) = 0$ or $b α f 2 ( u * ) − ( d + d 2 + p 2 ) = 0$ , we obtain $v * = 0$ and $w * = 0$ . This seems to be reasonable as distinct infected populations are interdependent, and it is typically not feasible for one infected group to become extinct from the other.

In the next section, we will discuss an important component that epidemiologists typically consider to reflect the dynamics of an epidemic and also study the stability analysis based on it.

3.2 Basic reproduction number

The basic reproduction number $R 0$ serves as a gauge for the transmissibility and contagiousness of viral and infectious diseases [3,16,33]. An epidemic does not occur during a stable disease outbreak (if $R 0 = 1$ ), and in the epidemic environment, the spread of the disease is regarded as weak (if $R 0 < 1$ ) or vigorous (if $R 0 > 1$ ), respectively. $R 0$ is obtained by calculating the spectral radius of the next-generation matrix $G = F V − 1$ [34].

Matrices $F$ and $V$ for model (1) are as follows:

$F = a ( 1 − α ) f 1 k d + γ b ( 1 − α ) f 2 k d + γ a α f 1 k d + γ b α f 2 k d + γ , V = d + d 1 + p 1 + η 0 − η d + d 2 + p 2 , G = F V − 1 = a ( 1 − α ) f 1 k d + γ d + d 1 + p 1 + η + η b ( 1 − α ) f 2 k d + γ ( d + d 1 + p 1 + η ) ( d + d 2 + p 2 ) b ( 1 − α ) f 2 k d + γ d + d 2 + p 2 a α f 1 k d + γ d + d 1 + p 1 + η + η b α f 2 k d + γ ( d + d 1 + p 1 + η ) ( d + d 2 + p 2 ) b α f 2 k d + γ d + d 2 + p 2$

Characteristic equation of $G$ is $λ 2 − A λ + B = 0$ , where

(12)

$A = a ( 1 − α ) f 1 k d + γ d + d 1 + p 1 + η + η b ( 1 − α ) f 2 k d + γ ( d + d 1 + p 1 + η ) ( d + d 2 + p 2 ) + b α f 2 k d + γ d + d 2 + p 2 , B = 0 .$

Therefore, $λ = A 2 ± A 2 − 4 B 2 .$

As $R 0$ is the spectral radius of a square matrix G, i.e., it is the eigenvalue with the largest absolute real value. So it is given by $R 0 = A$ .

Next, we will focus on the stability analysis of the model at equilibria, which helps us comprehend the impact of endemics in the environment of cardiovascular diseases.

3.3 Local stability analysis for disease-free equilibrium point

Local stability involves analyzing the behavior of a system or model in the vicinity of an equilibrium point or a particular state. As $R 0$ is a mathematical threshold that indicates the stability of a disease-free equilibrium. Firstly, we will determine the conditions in terms of $R 0$ under which system (1) is locally stable for disease-free equilibria. We state the following:

Theorem 3.2

System (1) is locally asymptotically stable at disease-free equilibrium point $( u * , 0 , 0 , r * ) = k d + γ , 0 , 0 , γ k d ( d + γ )$ , if

$R 0 < 1 ,$
$a ( 1 − α ) f 1 k d + γ + b α f 2 k d + γ < 2 d + d 1 + d 2 + p 1 + p 2 + η$ .

Proof

The Jacobian matrix $J 1$ of the system at disease-free equilibrium point $( u * , 0 , 0 , r * ) = k d + γ , 0 , 0 , γ k d ( d + γ )$ , can be written as follows:

$J 1 = − d − ν − a f 1 k d + γ − b f 2 k d + γ 0 0 a ( 1 − α ) f 1 k d + γ − ( d + d 1 + p 1 + η ) b ( 1 − α ) f 2 k d + γ 0 0 a α f 1 k d + γ + η b α f 2 k d + γ − ( d + d 2 + p 2 ) 0 γ p 1 p 2 − d$

The characteristics equation of $J 1$ at the disease free equilibrium point $k d + γ , 0 , 0 , γ k d ( d + γ )$ is given by

(13)

$( λ + d ) ( λ + ( d + γ ) ) ( λ 2 + A 1 λ + A 2 ) = 0 ,$

where

$A 1 = − a ( 1 − α ) f 1 k d + γ + b α f 2 k d + γ − ( d + d 1 + p 1 + η ) − ( d + d 2 + p 2 ) , A 2 = a ( 1 − α ) f 1 k d + γ − ( d + d 1 + p 1 + η ) b α f 2 k d + γ − ( d + d 2 + p 2 ) − b ( 1 − α ) f 2 k d + γ a α f 1 k d + γ + η .$

As $λ = − d < 0$ and $λ = − d − γ < 0$ , equation (13) has negative real roots if $A 1 > 0$ and $A 2 > 0$ .

$A 1 > 0$ only if $a ( 1 − α ) f 1 k d + γ + b α f 2 k d + γ < 2 d + d 1 + d 2 + p 1 + p 2 + η$ .

$A 2 > 0$ only if $R 0 = A < 1$ ( $A$ is taken from (12)).□

3.4 Local stability analysis for endemic equilibrium point

Now, we shall proceed to obtain sufficient conditions for local stability at endemic equilibrium point $( u * , v * , w * , r * )$ . We state the Jacobian system as $J 2 = u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4 w 1 w 2 w 3 w 4 r 1 r 2 r 3 r 4$ ,

where $u 1 = − a f 1 ′ ( u * ) v * − b f 2 ′ ( u * ) w * − d − γ$ , $u 2 = − a f 1 ( u * )$ , $u 3 = − b f 2 ( u * )$ , $u 4 = 0$ ,

$v 1 = a ( 1 − α ) f 1 ′ ( u * ) v * + b ( 1 − α ) f 2 ′ ( u * ) w *$ , $v 2 = a ( 1 − α ) f 1 ( u * ) − ( d + d 1 + p 1 + η )$ , $v 3 = b ( 1 − α ) f 2 ( u * )$ , $v 4 = 0$ ,

$w 1 = a α f 1 ′ ( u * ) v * + b α f 2 ′ ( u * ) w *$ , $w 2 = a α f 1 ( u * ) + η$ , $w 3 = b α f 2 ( u * ) − ( d + d 2 + p 2 )$ , $w 4 = 0$ ,

$r 1 = γ$ , $r 2 = p 1$ , $r 3 = p 2$ , and $r 4 = − d$ .

The characteristics equation of the model around the endemic equilibrium point $( u * , v * , w * , r * )$ is given by $∣ J 2 − β I ∣ = 0 ,$ which reduces to $β 4 + P β 3 + Q β 2 + R β + S = 0$ , where

$P = − ( u 1 + v 2 + w 3 + r 4 ) , Q = r 4 ( u 1 + v 2 + w 3 ) + w 3 ( u 1 + v 2 ) + u 1 v 2 − u 2 v 1 − u 3 w 1 − v 3 w 2 , R = r 4 ( − u 1 ( v 2 + w 3 ) + u 2 v 1 + u 3 w 1 − v 2 w 3 + v 3 w 2 ) − u 1 v 2 w 3 + u 1 v 3 w 2 + u 2 v 1 w 3 − u 2 v 3 w 1 − u 3 v 1 w 2 + u 3 v 2 w 1 , S = r 4 ( u 2 v 3 w 1 + u 3 v 1 w 2 + u 1 v 2 w 3 − u 1 v 3 w 2 − u 2 v 1 w 3 − u 3 v 2 w 1 ) .$

By Routh-Hurwitz stability criteria [24], all the roots of the aforementioned equation have negative real parts lie in the open left half plane if and only if $P > 0$ , $S > 0$ , and $P Q > R$ hold true. It is a sufficient condition for the local stability of the model at the endemic equilibrium point.

After simplification, we obtain $P > 0$ if $a f 1 ′ ( u * ) v * + b f 2 ′ ( u * ) w * + a ( 1 − α ) f 1 ( u * ) v * + 3 d + d 2 + p 2 >$ $b ( 1 − α ) f 2 ′ ( u * ) w * + b α f 2 ( u * )$ .

Now, $S > 0$ if $u 2 v 3 w 1 + u 3 v 1 w 2 + u 1 v 2 w 3 > u 1 v 3 w 2 + u 2 v 1 w 3 + u 3 v 2 w 1$ .

Finally, $P Q > R$ if

$( r 4 + u 1 + v 2 + w 3 ) ( u 1 v 2 − u 3 w 1 + u 1 w 3 − u 2 v 1 + v 2 w 3 − v 3 w 2 + r 4 ( u 1 + v 2 + w 3 ) ) > u 3 v 2 w 1 + u 1 v 3 w 2 + u 2 v 1 w 3 + r 4 ( u 2 v 1 + u 3 w 1 + v 3 w 2 ) − r 4 ( v 2 w 3 + u 1 ( v 2 + w 3 ) ) − u 2 v 3 w 1 − u 3 v 1 w 2 − u 1 v 2 w 3 .$

Thus, we state the following.

Theorem 3.3

The endemic equilibrium point $( u * , v * , w * , r * )$ is locally asymptotically stable if

(i) $P > 0$ , (ii) $S > 0$ , and (iii) $P Q > R$ ,

where $P = − ( u 1 + v 2 + w 3 + r 4 )$ ,

$Q = r 4 ( u 1 + v 2 + w 3 ) + w 3 ( u 1 + v 2 ) + u 1 v 2 − u 2 v 1 − u 3 w 1 − v 3 w 2 , R = r 4 ( − u 1 ( v 2 + w 3 ) + u 2 v 1 + u 3 w 1 − v 2 w 3 + v 3 w 2 ) − u 1 v 2 w 3 + u 1 v 3 w 2 + u 2 v 1 w 3 − u 2 v 3 w 1 − u 3 v 1 w 2 + u 3 v 2 w 1 , S = r 4 ( u 2 v 3 w 1 + u 3 v 1 w 2 + u 1 v 2 w 3 − u 1 v 3 w 2 − u 2 v 1 w 3 − u 3 v 2 w 1 ) .$

Remark 3.4

If all eigenvalues of an associated Jacobian matrix have negative real parts, the disease-free equilibrium is locally stable, indicating that small perturbations decay over time, and the disease is unlikely to establish in the population. If any eigenvalue has a positive real part, it suggests instability and the disease may persist in the population. Local stability at an endemic point, which is a fixed point, shows that the system reaches a stable equilibrium with a nonzero level of the infected population.

Next, we examine the global stability of the system at both disease-free and endemic equilibrium points.

4 Global stability

Global stability analysis is a mathematical approach used to study the long-term behavior of a mathematical model representing an epidemic. It helps determine whether a particular equilibrium point in the model is stable or unstable, providing insights into the overall dynamics of the epidemic [19].

4.1 Global stability at disease-free equilibria

Theorem 4.1

The disease-free equilibrium point $k d + γ , 0 , 0 , γ k d ( d + γ )$ is globally asymptotically stable if $A = max { a f 1 ( u ) − d − d 1 − p 1 , b f 2 ( u ) − d − d 2 − p 2 } < 0$ .

Proof

By (1), we have

$v ′ + w ′ = ( a f 1 ( u ) − d − d 1 − p 1 ) v + ( b f 2 ( u ) − d − d 2 − p 2 ) w ≤ A ( v + w ) , where A = max { a f 1 ( u ) − d − d 1 − p 1 , b f 2 ( u ) − d − d 2 − p 2 } .$

Therefore, we obtain $( v + w ) ≤ e A t ( v 0 + w 0 ) ⟶ 0$ , if $A < 0$ for sufficiently large $t$ .

By using this result for sufficiently large $t$ , from (1), we obtain $( u + r ) ′ = k − d ( u + r )$ .

Solving it we obtain $u + r ⟶ k d + γ$ , for sufficiently large $t$ .

$( u * + r * = k d + γ + γ k d ( d + γ ) = k d + γ )$ .

Thus, $u + r ⟶ u * + r *$ .□

4.2 Global stability at endemic equilibria

The endemic equilibrium point $( u * , v * , w * , r * )$ will satisfy equation (1), and thus, we obtain

(14)

$k − a f 1 ( u * ) v * − b f 2 ( u * ) w * − d u * − γ u * = 0 , a ( 1 − α ) f 1 ( u * ) v * + b ( 1 − α ) f 2 ( u * ) w * − ( d + d 1 ) v * − p 1 v * − η v * = 0 , a α f 1 ( u * ) v * + b α f 2 ( u * ) w * + η v * − ( d + d 2 ) w * − p 2 w * = 0 , γ u * + p 1 v * + p 2 w * − d r * = 0 .$

By using (1) and (14), we can write the system as follows:

(15)

$( u − u * ) ′ = − a f 1 ( u ) ( v − v * ) − a v * ( f 1 ( u ) − f 1 ( u * ) ) − b f 2 ( u ) ( w − w * ) − b w * ( f 2 ( u ) − f 2 ( u * ) ) − ( d + γ ) ( u − u * ) , ( v − v * ) ′ = a ( 1 − α ) f 1 ( u ) ( v − v * ) + a ( 1 − α ) v * ( f 1 ( u ) − f 1 ( u * ) ) + b ( 1 − α ) f 2 ( u ) ( w − w * ) + b ( 1 − α ) w * ( f 2 ( u ) − f 2 ( u * ) ) − ( d + d 1 + p 1 + η ) ( v − v * ) , ( w − w * ) ′ = a α f 1 ( u ) ( v − v * ) + a α v * ( f 1 ( u ) − f 1 ( u * ) ) + b α f 2 ( u ) ( w − w * ) + b α w * ( f 2 ( u ) − f 2 ( u * ) ) + η ( v − v * ) − ( d + d 2 + p 2 ) ( w − w * ) , ( r − r * ) ′ = γ ( u − u * ) + p 1 ( v − v * ) + p 2 ( w − w * ) − d ( r − r * ) .$

We state the result on global stability as follows:

Theorem 4.2

The endemic equilibrium point $( u * , v * , w * , r * )$ of system (1) is globally asymptotic stable.

Proof

Consider the Lyapunov function $V = ∣ u − u * ∣ + ∣ v − v * ∣ + ∣ w − w * ∣$ .

Taking Dini derivative along the solutions of (15) and simplifying, we obtain

$D + V ≤ − d ∣ u − u * ∣ − ( d + d 1 ) ∣ v − v * ∣ − ( d + d 2 ) ∣ w − w * ∣ − d ∣ r − r * ∣ ≤ − A V , where A = min { d , d + d 1 , d + d 2 } = d > 0 .$

The conclusion follows from the standard argument [2,15].□

Remark 4.3

For a specific initial state, the local stability conditions at disease-free equilibria guarantee that the disease will eventually disappear, whereas, at endemic equilibria, local stability conditions ensure that the disease will persist over time. In contrast, despite of the initial state (high or low levels of infection and spread), the global stability criteria at the disease-free equilibrium are essential to ensure that the diseases eventually disappear. On the other hand, global stability at the endemic equilibrium point aids in determining the conditions in which the disease is most likely to continue infecting people. Overall, this analysis is valuable for public health planning, intervention strategies, and understanding the impact of various factors on the endemic state of the disease.

An overview of sensitivity analysis, which we will utilize to assess the significance of the model’s parameter, is given in the next section.

5 Sensitivity analysis of $R 0$

Sensitivity analysis is a valuable technique in the field of modelling and simulation, helping researchers and analysts understand the impact of variations in input parameters on the model’s output. This analysis is particularly crucial when dealing with complex systems or models where uncertainty exists in input parameters. As the reproduction number $R 0$ is a key metric in epidemiological models, sensitivity analysis can be applied to identify which parameters have the most significant influence on $R 0$ . The normalized forward sensitivity index is used in sensitivity analysis to quantify the impact of changes in a parameter on a model variable. Mathematically, the normalized forward sensitivity index for a variable $X$ with respect to a parameter $P$ is given by:

(16)

$X R 0 P = ∂ R 0 ∂ P P R 0 .$

This information can guide efforts to prioritize data collection or parameter estimation for improving model predictions.

Sensitivity analysis is a metric that reflects the ability of a surveillance system to accurately detect and report cases of a particular disease. In the context of infectious diseases, such as a pandemic, monitoring the sensitivity index helps public health officials gauge the effectiveness of vaccination and treatment efforts.

In the context of vaccination, this means that the system is likely to capture and document a large portion of individuals who have received the vaccine, providing an accurate representation of the vaccination rate. Similarly, for treatment rates, a high sensitivity index suggests that the system is effective at identifying and recording individuals who have undergone the prescribed treatment.

6 Numerical example

In this section, we provide some numerical examples to support the theoretical results from the previous sections. For assumed values of parameters, as our model forms a system of differential equations, we solve them by using MATLAB software using the ode45 solver, which applies the Runge-Kutta method. A few parameters like infection transmission rates $a$ and $b$ , transmission rate $η$ from N-CVD to CVD, treatment rates $p 1$ and $p 2$ , and vaccination rate $γ$ are considered as the sensitive parameters of the model, which govern the model. So, we determine the sensitivity index of these parameters using (16) and examine their impact on the model.

Example 6.1

Let the interaction functions be $f 1 ( u ) = u u + 1$ and $f 2 ( u ) = u u + 1$ , and the values of the parameters be

Parameters	$k$	$a$	$b$	$d$	$γ$	$d 1$	$d 2$	$η$	$p 1$	$p 2$
Values	100	2.1	2.65	1.5	0.5	1.2	1.33	0.25	0.12	0.133

(i) For the aforementioned values of the parameters, $R 0 = 0.828581$ , $a ( 1 − α ) f 1 k d + γ + b α f 2 k d + γ$ = 2.38235

$2 d + d 1 + d 2 + p 1 + p 2 + η = 6.003$ .

Clearly, $R 0 < 1$ and $a ( 1 − α ) f 1 k d + γ + b 1 α f 2 k d + γ < 2 d + d 1 + d 2 + p 1 + p 2 + η$ .

Thus, the system with the aforementioned parameters satisfies the conditions of Theorem 3.2. By virtue of Theorem 3.2, the disease-free equilibria $( 50 , 0 , 0 , 17 )$ is locally asymptotically stable.

(ii) Also, as $max { a 1 f 1 ( u ) − d − d 1 − p 1 , b 1 f 2 − d − d 2 − p 2 } = − 0.3350 < 0$ , as this condition is independent of $α$ , so by virtue of Theorem 4.1, the disease-free equilibria $( 50 , 0 , 0 , 17 )$ is globally stable.

For a given set of values, the sensitivity index using (16) for the sensitive parameters $a$ , $b$ , $η p 1$ , $p 2$ , and $γ$ is shown in Table 2, and Figure 2 depicts the bar graph of it.

Table 2

Sensitivity indices of $R 0$

Parameters	Sensitivity index
$a$	0.323747
$b$	0.676253
$η$	0.00562305
$p 1$	$‒ 0.0140157$
$p 2$	$‒ 0.0306654$
$γ$	$‒ 0.00490196$

Figure 2

(a) and (b) shows bar plot of the sensitive indices of the model parameters Example 6.1. Source: Created by the authors using MATLAB software.

Numerical simulations of the behaviour of the infected (N-CVD and CVD) populations in relation to the different values of the sensitive parameters, with initial data $( 50 , 20 , 25 , 12 )$ , are illustrated in Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.

Figure 3

Infected NCVD population for different values of $a$ . Source: Created by the authors using MATLAB software.

Figure 4

Infected CVD population for different values of $a$ . Source: Created by the authors using MATLAB software.

Figure 5

Infected NCVD population for different values of $b$ . Source: Created by the authors using MATLAB software.

Figure 6

Infected CVD population for different values of $b$ . Source: Created by the authors using MATLAB software.

$Figure 7 Infected NCVD population for different values of η \eta . Source: Created by the authors using MATLAB software.$

Figure 7

Infected NCVD population for different values of $η$ . Source: Created by the authors using MATLAB software.

$Figure 8 Infected CVD population for different values of η \eta . Source: Created by the authors using MATLAB software.$

Figure 8

Infected CVD population for different values of $η$ . Source: Created by the authors using MATLAB software.

$Figure 9 Infected NCVD population for different values of p 1 {p}_{1} . Source: Created by the authors using MATLAB software.$

Figure 9

Infected NCVD population for different values of $p 1$ . Source: Created by the authors using MATLAB software.

$Figure 10 Infected CVD population for different values of p 1 {p}_{1} . Source: Created by the authors using MATLAB software.$

Figure 10

Infected CVD population for different values of $p 1$ . Source: Created by the authors using MATLAB software.

$Figure 11 Infected NCVD population for different values of p 2 {p}_{2} . Source: Created by the authors using MATLAB software.$

Figure 11

Infected NCVD population for different values of $p 2$ . Source: Created by the authors using MATLAB software.

$Figure 12 Infected CVD population for different values of p 2 {p}_{2} . Source: Created by the authors using MATLAB software.$

Figure 12

Infected CVD population for different values of $p 2$ . Source: Created by the authors using MATLAB software.

$Figure 13 Infected NCVD population for different values of γ \gamma . Source: Created by the authors using MATLAB software.$

Figure 13

Infected NCVD population for different values of $γ$ . Source: Created by the authors using MATLAB software.

$Figure 14 Infected CVD population for different values of γ \gamma . Source: Created by the authors using MATLAB software.$

Figure 14

Infected CVD population for different values of $γ$ . Source: Created by the authors using MATLAB software.

Figures 15, 16, 17 depict the behaviour of the recovered population when there is a change in treatment rates $p 1$ and $p 2$ and vaccination rate $γ$ .

$Figure 15 Recovered population for different values of p 1 {p}_{1} . Source: Created by the authors using MATLAB software.$

Figure 15

Recovered population for different values of $p 1$ . Source: Created by the authors using MATLAB software.

$Figure 16 Recovered population for different values of p 2 {p}_{2} . Source: Created by the authors using MATLAB software.$

Figure 16

Recovered population for different values of $p 2$ . Source: Created by the authors using MATLAB software.

$Figure 17 Recovered population for different values of γ \gamma . Source: Created by the authors using MATLAB software.$

Figure 17

Recovered population for different values of $γ$ . Source: Created by the authors using MATLAB software.

Example 6.2

Let the parameters take the value

Parameters	$k$	$a$	$b$	$d$	$d 1$	$d 2$	$γ$	$η$	$p 1$	$p 2$
Values	200	1.95	2.187	1.8	0.07	0.08	0.5	0.15	0.11	0.09

and functions be $f 1 = u u + 2$ and $f 2 = u u + 3$ .

For the aforementioned values of the parameters, $R 0 = 1.03995 > 1$ .

Calculating the parameters $P$ , $Q$ , $R$ , and $S$ of Theorem 3.3, we obtain $P = 6.31869$ , $Q = 13.3923$ , $R = 9.82846$ , and $S = 0.6532$ . Clearly, $P > 0$ , $S > 0$ , and $P Q > R$ . Thus, the system with the aforementioned parameters satisfies the conditions of Theorem 3.3. Therefore, the endemic equilibria $( 49.3156 , 20.3226 , 23.5206 , 16.1167 )$ is locally asymptotically stable.
From Theorem 4.2, this system is globally asymptotically stable also.

Table 3 gives the sensitive index of the sensitive parameters

$a$ ,

$b$ ,

$η , p 1$ ,

$p 2$ , and

$γ$ , for a given set of values. Figure 18 gives the bar graph of the sensitive index.

Table 3

Sensitivity indices of $R 0$

Parameters	Sensitivity index
$a$	0.435156
$b$	0.564844
$η$	0.00389899
$p 1$	$‒ 0.024392$
$p 2$	$‒ 0.025805$
$γ$	$‒ 0.00247158$

Figure 18

(a) and (b) shows the bar plot of the sensitive indices of the model parameters of Example 6.2. Source: Created by the authors using MATLAB software.

Simulations of the infected population (N-CVD and CVD) with initial data $( 50 , 10 , 12 , 6 )$ in response to sensitive parameters at distinct values are shown in Figures 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

Figure 19

Infected NCVD population for different values of $a$ . Source: Created by the authors using MATLAB software.

Figure 20

Infected CVD population for different values of $a$ . Source: Created by the authors using MATLAB software.

Figure 21

Infected NCVD population for different values of $b$ . Source: Created by the authors using MATLAB software.

Figure 22

Infected CVD population for different values of $b$ . Source: Created by the authors using MATLAB software.

$Figure 23 Infected NCVD population for different values of η \eta . Source: Created by the authors using MATLAB software.$

Figure 23

Infected NCVD population for different values of $η$ . Source: Created by the authors using MATLAB software.

$Figure 24 Infected CVD population for different values of η \eta . Source: Created by the authors using MATLAB software.$

Figure 24

Infected CVD population for different values of $η$ . Source: Created by the authors using MATLAB software.

$Figure 25 Infected NCVD population for different values of p 1 {p}_{1} . Source: Created by the authors using MATLAB software.$

Figure 25

Infected NCVD population for different values of $p 1$ . Source: Created by the authors using MATLAB software.

$Figure 26 Infected CVD population for different values of p 1 {p}_{1} . Source: Created by the authors using MATLAB software.$

Figure 26

Infected CVD population for different values of $p 1$ . Source: Created by the authors using MATLAB software.

$Figure 27 Infected NCVD population for different values of p 2 {p}_{2} . Source: Created by the authors using MATLAB software.$

Figure 27

Infected NCVD population for different values of $p 2$ . Source: Created by the authors using MATLAB software.

$Figure 28 Infected CVD population for different values of p 2 {p}_{2} . Source: Created by the authors using MATLAB software.$

Figure 28

Infected CVD population for different values of $p 2$ . Source: Created by the authors using MATLAB software.

$Figure 29 Infected NCVD population for different values of γ \gamma . Source: Created by the authors using MATLAB software.$

Figure 29

Infected NCVD population for different values of $γ$ . Source: Created by the authors using MATLAB software.

$Figure 30 Infected CVD population for different values of γ \gamma . Source: Created by the authors using MATLAB software.$

Figure 30

Infected CVD population for different values of $γ$ . Source: Created by the authors using MATLAB software.

Simulations of the recovered population when there is a rise in treatment rates and vaccination rates can be visualized in Figures 31, 32, 33.

$Figure 31 Recovered population for different values of p 1 {p}_{1} . Source: Created by the authors using MATLAB software.$

Figure 31

Recovered population for different values of $p 1$ . Source: Created by the authors using MATLAB software.

$Figure 32 Recovered population for different values of p 2 {p}_{2} . Source: Created by the authors using MATLAB software.$

Figure 32

Recovered population for different values of $p 2$ . Source: Created by the authors using MATLAB software.

$Figure 33 Recovered population for different values of γ \gamma . Source: Created by the authors using MATLAB software.$

Figure 33

Recovered population for different values of $γ$ . Source: Created by the authors using MATLAB software.

Observation:

Numerical examples, Examples 6.1 and 6.2, validate our Theorems.
Tables 2 and 3 gives sensitivity indices of $R 0$ with respect to sensitive parameters of Examples 6.1 and 6.2, respectively. It is noted from Tables 2 and 3 that the parameters $a$ and $b$ , and $η$ have a positive impact on $R 0$ ; that is, if the values of these parameters are increased, then there will be a rise in the value of $R 0$ and vice versa. However, the parameters $p 1$ , $p 2$ , and $γ$ have a negative impact; that is, if there is an increase in these parameter values, then there is a decrease in $R 0$ and vice versa. It is also observed that $b$ highly positively influences $R 0$ and $p 2$ highly negatively influences $R 0$ . The bar graphs shown in Figures 2 and 18 corroborate it.
Simulations in Figures 3–14 and Figures 19–30 support sensitivity analysis of Tables 2 and 3.
1. Figures 3–6 and 19–22 show that as the conversion rates $a$ (susceptible to N-CVD) and $b$ (susceptible to CVD) increase, there is an increase in both infected populations (N-CVD and CVD), showing that the transmissibility of virus is more. In particular, spread of the disease is more when there is a rise in the CVD population getting infected compared to that of the rise in the N-CVD population getting infected.
2. When there is an increase in the conversion rate $η$ (N-CVD to CVD), from Figures 7,8,23–24, it is observed that there is a decrease in the N-CVD population and an increase in CVD population, but overall, there will be a rise in the infection rate.
3. Figures 9–14 and 25–30 show that as the treatment rates $p 1$ and $p 2$ , and vaccination rate $γ$ increase, there is a reduction in both infected (N-CVD and CVD) populations, showing that the spread of infection is under control.
The effect of change of treatment rates $p 1$ and $p 2$ , and vaccination rate $γ$ on recovered population are shown in Figures 15–17 and 31–33. It is clear that as the treatment rate or vaccination rate increases, there is an increase in the recovered population.
In particular, as the susceptibles who are getting vaccinated are directly moving to the recovered class, so when there is an increase in the vaccination rate, there is a reduction in the number of susceptible who can get infected. Hence, vaccination rate has an indirect impact on the infection rate but has a direct impact on the recovery rate. This is shown in Figures 14, 17, 30, and 33.

7 Conclusion and future scope

Amid the COVID-19 scenario, individuals with cardiovascular illnesses were particularly affected, which consequently affected the infection rate. Even in the post-COVID-19 scenario, there has been a noticeable increase in the number of patients experiencing heart problems. Thus, research on the transmission of infection in people with and without cardiovascular disease is needed to address the aforementioned situations in the future. The present study proposes a mathematical model to accomplish this. Mathematical analysis of the model is examined by establishing the results on the existence of nonnegative and bounded solutions, the existence of disease-free and endemic equilibrium points, the calculation of BRN using a next-generation matrix approach, results on local stability in context with BRN, and global stability results. Numerical examples are provided to support the results. To know the impact of the key parameters on the spread of infection, sensitivity analysis of these parameters is calculated for the numerical examples. It is noted that the disease will spread more when there is an increase in conversion rates from susceptible to N-CVD, susceptible to CVD and N-CVD to CVD. It is also observed that the conversion rate of susceptibles to CVD has more impact on the spread of the virus compared to susceptibles to N-CVD. Also, the conversion rate of N-CVD to CVD has a significant effect on the disease spread, which cannot be ignored. The spread of the infection is less when there is a rise in vaccination rates and treatment rates for CVD and N-CVD populations. Here, it shows the need to put special efforts into the treatment of the CVD population to control the worst scenario of the disease. Even, the vaccination rate has an important role, where it indirectly influences the infected population and directly influences the recovered population.

The study opens up avenues for further research by introducing delays into the model and studying its bifurcation, refining the mathematical model with additional epidemiological data, exploring the impact of emerging variants of the virus, and investigating the long-term implications of COVID-19 on cardiovascular health.

Acknowledgments

The authors are grateful for the reviewer’s valuable comments, which improved the manuscript.

Funding information: The authors state no funding involved.
Author contributions: Gopishetty Shirisha: Conceptualization, methodology, derivation and interpretation of results and manuscript writing. Saroj Vernekar: Literature review, Simulations of numerical examples, manuscript editing, and formatting. All authors reviewed and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: This research did not involve any human participants or animals. Ethical approval was therefore not required.
Data availability statement: Data sharing is not applicable to this article as no new datasets were generated or analyzed during the current study.

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Received: 2024-04-24

Revised: 2024-12-13

Accepted: 2025-01-26

Published Online: 2025-04-11

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

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Keywords for this article

COVID-19; cardiovascular disease; equilibria; basic reproduction number; stability

Creative Commons

BY 4.0

Articles in the same Issue

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Mathematical model on influence of past experiences on present activities of human brain
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Simulation study on the impact of measurement errors in hierarchical Bayesian semi-parametric models
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Article

Abstract

1 Introduction

2 Model description and analysis

2.1 Basic properties

Theorem 2.1

Proof

3 Stability analysis at equilibria

3.1 Equilibria

3.1.1 Disease-free equilibria

3.1.2 Endemic equilibria

3.1.3 Interdependence of infected populations

Remark 3.1

3.2 Basic reproduction number

3.3 Local stability analysis for disease-free equilibrium point

Theorem 3.2

Proof

3.4 Local stability analysis for endemic equilibrium point

Theorem 3.3

Remark 3.4

4 Global stability

4.1 Global stability at disease-free equilibria

Theorem 4.1

Proof

4.2 Global stability at endemic equilibria

Theorem 4.2

Proof

Remark 4.3

5 Sensitivity analysis of R 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math>

6 Numerical example

Example 6.1

Example 6.2

Observation:

7 Conclusion and future scope

Acknowledgments

References

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5 Sensitivity analysis of $R 0$