key: cord-296891-23xkaa19
authors: Sahu, Govind Prasad; Dhar, Joydip
title: Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity
date: 2015-01-15
journal: Journal of Mathematical Analysis and Applications
DOI: 10.1016/j.jmaa.2014.08.019
sha: 
doc_id: 296891
cord_uid: 23xkaa19

Abstract An autonomous deterministic non-linear epidemic model SEQIHRS is proposed for the transmission dynamics of an infectious disease with quarantine and isolation control strategies in a community with pre-existing immunity. The model exhibits two equilibria, namely, the disease-free and a unique endemic equilibrium. The existence and local stability of the disease free and endemic equilibria are explored in terms of the effective reproduction number R C . It is observed that media coverage does not affect the effective reproduction number, but it helps to mitigate disease burden by lowering the number of infectious individuals at the endemic steady state and also lowering the infection peak. A new approach is proposed to estimate the coefficient of media coverage. Using the results of central manifold theory, it is established that as R C passes through unity, transcritical bifurcation occurs in the system and the unique endemic equilibrium is asymptotically stable. It is observed that the population level impact of quarantine and isolation depend on the level of transmission by the isolated individuals. Moreover, the higher level of pre-existing immunity in the population decreases the infection peak and causes its early arrival. Theoretical findings are supported by numerical simulation. Sensitivity analysis is performed for R C and state variables at endemic steady state with respect to model parameters.

Mathematical modeling has become an important tool in analyzing the spread and control of infectious diseases taking into account the main factors governing development of a disease, such as transmission and recovery rates. Mathematical models are being used to predict how the disease will spread over a period of time. In recent years, many attempts have been made to develop realistic mathematical models for investigating the transmission dynamics of infectious diseases, and the asymptotic behaviors of these epidemic models are studied [10, 11, 42, 62] . Vaccination and antiviral drugs are the two most effective pharmaceutical interventions used for the control of an infectious disease. But due to strain's novelty most people may lack innate immunity to the disease, available vaccine may not provide protection against the pathogen and effective antiviral drug may not be available in sufficient amount. For example, Bird flu viruses H5N1 and H7N9 that have sporadically infected humans, could, with a few mutations to a key protein on their surface, become capable of infecting cells along the human upper airway and thereby take a step towards turning into pandemic-causing strains [50, 51] . In that scenario, the role of isolation, quarantine and other non-pharmaceutical interventions stimulated by media coverage becomes more significant as disease control strategies.

Despite initial concern that little protective immunity existed in the general population for pandemic H1N1 (2009), subsequent epidemiological data showed that morbidity in the elderly was lower than that in younger individuals, suggesting the existence of pre-existing immunity [16, 19, 27, 58] . The Centers for Disease Control and Prevention (Atlanta, GA, USA) reported that among persons > 60 years old, 33% have pre-existing, cross-reactive neutralizing antibodies against the new virus of pandemic (H1N1) 2009 [58] . Phylogenetic analyses on the HA of the 2009 pandemic H1N1 virus demonstrated its close relationship with the 1918-1919 Spanish H1N1 virus [15, 59] . Low virulence of the virus and pre-existing immune status are among the main factors that account for lower death rates in influenza outbreaks [58] . Pre-existing immunity is an important factor countering the pandemic potential of an emerging infectious disease. Thus, studying of pre-existing immunity will advance our understanding of the pathogenesis and disease dynamics of the emerging pathogen.

When an infectious disease breaks out in a population, people's response to the threat of the disease is dependent on their perception of risk, which is affected by public and private information disseminated widely by the media [28, 36, 49] . Media coverage about an epidemic gives a sense about the risk level and the relative need for precautions in risk areas and encourage the public to take precautionary measures against the disease such as wearing masks, avoiding public places, avoiding travel when sick, frequent hand washing, etc. Massive news coverage and fast information flow can generate a profound psychological impact on public health [55] . This is extremely important in the early stages of an epidemic, when pharmaceutical interventions are not often possible because treatment or vaccination options have not yet been developed [47] . Media coverage and education may reduce the contact rate of human beings and the use of NPIs may reduce the transmission probability. Many researchers investigated the impact of media awareness using mathematical modeling [2, 7, 8, 14, 22, 23, 26, 28, [30] [31] [32] 34, 36, 43, [47] [48] [49] 52, 53, 55, 57, 61] . One method is to form an information set summarized by a new state variable, mostly based on the publicly available information on both present and on the recent past spreading of the disease [2, [30] [31] [32] 43] . The population is broadly classified as educated and non-educated to reflect the awareness through media coverage [30] [31] [32] 43] 

Possible multiple outbreaks and sustained periodic oscillations of the infection were observed. Cui et al. used similar transmission coefficient and established that multiple positive equilibria are possible when the media effect is sufficiently strong [7] . Reduction function of the negative exponential form to describe the reduction factor either by large number of infectious cases or by significant change in the number of infectious cases is used to model the media coverage [48, 57] . Cui et al. [8] proposed a general contact rate, β(I) = c 1 − c 2 f (I), to reflect some intrinsic characters of media coverage. Using the same contact rate proposed by Liu and Cui [26] , Tchuenche et al. [49] studied the impact of media coverage on the spread and control of an influenza strain. Sun et al. [47] used similar non-linear contact rate as in [8] to study media-induced social distancing in a two patch setting. Li and Cui [23] studied the effect of constant and pulse vaccination on SIS epidemic models incorporating media-induced incidence function similar as in [26, 49] .

Emerging infectious diseases have devastating impacts on public health and impose great financial burden on the community, which attracts a major concern to public health agency. So it is of great importance to evaluate optimal methods for controlling these diseases [46] . Historically, quarantine (of individuals feared exposed to a communicable disease) is one of the oldest public health control measures for the spread of communicable diseases. These measures have been successfully applied dating back to the plague epidemic of the 13th century, to the influenza epidemics of the 20th century. More recently, this measure was successfully used to combat the spread of some emerging and re-emerging human and animal diseases, such as the severe acute respiratory syndrome (SARS) [18, 25, 29, 33, 54, 60] , foot-and-mouth disease [21] and the 2009 swine influenza pandemic [17] . The SARS outbreaks of 2003 provided an important example of a novel disease that was effectively controlled using quarantine and isolation [18, 25, 29] . Implementing these measures, however, can inflict significant socio-economic and psychological costs. Public health officials need to be able to present comprehensive, understandable assessments of the options to other government officials in a timely manner [37] . For the purpose of this study, quarantine means the removal of individuals suspected of being infected (yet exhibiting no clinical symptoms) from the general population. Thus, these individuals could be asymptomatically-infected or susceptible. On the other hand, isolation refers to the removal of infected individuals exhibiting clinical symptoms of the disease. Although isolation is probably always a desirable public health measure, quarantine is more controversial. Mass quarantine can inflict significant social, psychological, and economic costs without resulting in the detection of many infected individuals [9] . Isolation is primarily used for controlling the disease when it suddenly emerges or reemerges [5] . A successful example is the isolation of those infected with SARS during 2003-2004. However, the disadvantages of this strategy are the difficulty of detecting infected individuals and the cost of isolation. In general, to achieve perfect isolation is difficult at large scale resulting in a leaky isolation causing nosocomial infections.

Numerous mathematical modeling works have been carried out to assess the impact of quarantine and isolation in controlling the spread of communicable diseases in human and animal populations [9, 18, 20, 25, 29, 33, 35, [38] [39] [40] [41] 54, 56, 60] . Further, the emergence of SARS in 2003 led to the formulation of numerous quarantine and isolation models for curtailing its spread [9, 18, 25, 29, 54] . Most of the disease modeling studies, published in the literature, provided quantitative evaluation of the control measures (quarantine and isolation) by simulating the models with available epidemiological and demographic data [9, 18, 25, 29, 54] .

The primary goal of this article is to theoretically study the impact of use of NPIs stimulated by media coverage, quarantine and isolation for an infectious disease in a community with pre-existing immunity. The rest of the paper is organized as follows: In Section 2, the proposed model in formulated. In Section 3, existence and local behavior of disease free (DFE) and endemic equilibria are explored. Global stability of DFE and uniform persistence is established. In Section 4, important thresholds are calculated. Numerical simulation is performed in Section 5. Sensitivity analysis is performed for effective reproduction number and steady states at endemic level with respect to model parameters in Section 6. Finally, results are discussed in Section 7.

In this section, we will formulate an epidemic model incorporating quarantine, isolation, use of nonpharmaceutical interventions stimulated by media coverage in presence of pre-existing cross-protective immunity. The total population at time t , denoted by Ñ (t), is sub-divided into six mutually exclusive compartments of susceptible (S(t)), exposed (Ẽ(t)), quarantined (Q(t)), infectious (Ĩ(t)), hospitalized (H(t)) and recovered (R(t)) individuals, so that Ñ (t) =S(t) +Ẽ(t) +Q(t) +Ĩ(t) +H(t) +R(t). In the modeling of infectious diseases, the incidence function plays a very important role, it can determine the rise and fall of epidemics [42] . In many epidemic models, the bilinear incidence rate βSĨ and the standard incidence rate βSĨ /Ñ are frequently used, where β measures the effect of both the infectiousness of the disease and the contact transmission rates. However, these incidence functions do not consider the impact of media coverage to the spread and control of infectious diseases. The media induced transmission rate β (Ĩ) =βe −mĨ , used in [7, 28] , has two limitations. First, β e −mĨ → 0 as Ĩ → ∞, independent of the value of m. It is not reasonable since the media coverage is not the intrinsic deterministic factor responsible for the transmission and hence the transmission rate cannot be reduced below a certain level merely through media awareness and alert. Second, even for a fixed m, the minimum transmission rate differs for different population sizes, regardless the similarity in social structure (i.e., education and awareness level) and climatic condition. We propose media induced transmission rate as β (Ĩ) =βe −mĨ N , which is more reasonable than that used in [7, 28] overcoming the aforementioned limitations (see Fig. 1 ).

Keeping in view the above, in our proposed model we choose the media induced transmission rate of the form β e −m 1Ĩ N −m 2H N . The parameters m 1 and m 2 represent the coefficients of media coverage using nonpharmaceutical interventions corresponding to infectious (Ĩ) and isolated (H) individuals, respectively. Now, the next important question is how to measure the media coefficient m. One innovative way of estimation of the media awareness coefficient m will be discussed in numerical simulation Section 5. Other modeling assumptions are as follows: The population grows at constant rate Λ. Population in all the compartments decreases at rate μ due to natural death. Susceptible individuals acquire infection through effective contact with infectious individuals at rateβ

The parameter β is the effective contact rate (that is, contact capable of leading to infection), while the modification parameter, 0 < η < 1, accounts for the assumed reduction in disease transmission by isolated individuals in comparison to non-hospitalized infectious individuals in the Ĩ class. Thus, η measures the efficacy of isolation or treatment given to hospitalized individuals (isolation is perfect if η = 0; leaky if 0 < η < 1 and completely ineffective if η = 1). The schematic flow diagram of the proposed model is shown in Fig. 2 . Based on the aforementioned modeling assumptions, the proposed model is govern by the following system of ordinary differential equations: Table 1 Description of parameters for the system (2)- (7).

Contact rate (in absence of NPIs through media coverage) days −1 η

Modification parameter for reduction in infectiousness of hospitalized individuals m 1 , m 2 Coefficients of media coverage corresponding toĨ andH σ

Progression rate from exposed to infectious class days −1 ξ

Recovery rate due to pre-existing cross-protective immunity days −1 k

Quarantine rate for exposed individuals days −1 α

Hospitalization rate for quarantined individuals days −1 φ

Hospitalization rate for infectious individuals days −1 γ 1

Recovery rate for non-hospitalized infectious individuals days −1 γ 2

Recovery rate for hospitalized infectious individuals days −1 1/θ Average waning period of disease-induced immunity days δ 1 Disease-induced death rate for non-hospitalized infectious individuals days −1 δ 2 Disease-induced death rate for hospitalized individuals

with initial conditions:

Descriptions of all the parameters are summarized in Table 1 . Note that

We consider only solutions with initial conditions inside the biologically feasible region Γ = (S,Ẽ,Q,Ĩ,H,R) ∈ R 6 + : 0 ≤S,Ẽ,Q,Ĩ,H,R,S +Ẽ +Ĩ +Q +H +R ≤ Λ μ in which the usual existence, uniqueness of solutions and continuation results hold. We study the system (2)- (7) and claim that the region Γ is bounded and positively invariant with respect to the proposed system (2)- (7).

All the solution trajectories of system (2)- (7) initiating inside Γ approach enter or stay within the interior of Γ .

denote the non-negative cone in six-dimensional Euclidean space. From the system (2)-(7), we observe that

, R (t) are continuous functions oft. Thus the vector field on each bounding hyperplane of R 6 + is pointing inward direction of R 6 + . Hence all the solution trajectories initiating in R 6 + will remain inside R 6 + for all the time. This establishes the fact that R 6 + is positively invariant for the system (2)- (7) . Also, the total population Ñ (t) satisfies dÑ dt = Λ −μÑ −δ 1Ĩ −δ 2H . Then, dÑ dt < Λ −μÑ , applying Birkhoff's and Rota's theorems on differential inequality [1, 45] , as t → ∞, we have 0 ≤Ñ (t) ≤ Λ μ =Ñ 0 . Therefore the solution of system (2)-(7) is bounded and hence any solution of the system originated from Γ remains in Γ . 2

We reduce the above system into non-dimensional form using

the equivalent non-dimensional system is given by:

where β =β μ ,

and the initial conditions:

In the following sections, we will study the dynamical behavior of the system (8)- (13) with initial condition (14) .

In this section, we calculate all feasible steady states and the basic reproduction number for the system. Observe that the biologically feasible region for the non-dimensional system is

which is positively invariant for the system (8)-(13). We consider only solutions with initial conditions inside the region Ω.

The system (8)-(13) always has the disease-free equilibrium (DFE) (E 0 = (0, 0, 0, 0, 0, 1)). The local stability of DFE E 0 will be explored using the effective reproduction number R C . The non-negative matrix F, of the new infection terms, and the matrix V, of the remaining terms are given, respectively, by

The corresponding linearized matrices evaluated at the DFE E 0 are

respectively. It follows that

The effective reproduction number R C = ρ(FV −1 ), where ρ is the spectral radius, is given by

It is worth mentioning that in the absence of a combined quarantine and isolation program (k = 0, α = 0, φ = 0, δ 2 = 0, γ 2 = 0), the effective reproduction number reduces to the basic reproduction number

.

Using Theorem 2 in [12] , one can establish the following result.

Theorem 3.1. The disease-free equilibrium of the system (8)- (13) is locally-asymptotically stable if R C < 1, and unstable if R C > 1.

Biologically speaking, R C represents the average number of secondary infections produced by a typical infected individual in a community that adopts isolation and quarantine programs. The epidemiological implication of this result is that if R C < 1, then the influx of a few infected individuals will not generate large outbreaks (and the disease will die out). Disease outbreak will occur if R C > 1.

The possible endemic equilibria of proposed model are derived by solving the system of non-linear equations obtained from the system (8)-(13) equating the derivatives to zeroes. The endemic equilibrium Ē = (E * , Q * , I * , H * , R * , N * ) of the model (8)-(13) is given by

The value of I * is given by the solution of the equation In case, there is no media effect, i.e., m = 0, we get

. It follows that I * exists at positive level (and so as the unique endemic equilibrium Ē ) if and only if R C > 1. Otherwise, the value of I * is given from Eq. (15) . Now we establish the existence of I * for R C > 1 though graphical approach. In Figs. 3 and 4, we plot the curve e mI * R C and straight line 1 − b 5 I * against I * in the range [0-1]. Note that since b 5 > 1, we have 1/b 5 < 1. From Fig. 3 , it follows that there is no point of intersection when R C ≤ 1, resulting the non-existence of the endemic equilibrium, but as R C > 1, I * exists uniquely at positive level and hence the unique endemic exists in this case (see Fig. 4 ).

From the above discussion we conclude that Theorem 3.2. The system (8)- (13) has no endemic equilibrium for R C ≤ 1, but has a unique endemic equilibrium Ē if R C > 1. Now we will state and prove the local stability of the endemic equilibrium in the following theorem:

The endemic equilibrium Ē is locally asymptotically stable for R C > 1, but close to 1.

Proof. The Jacobian matrix J 0 at DFE is given by

Here, we use the method based on the central manifold theory to establish the local stability of endemic equilibrium taking β as bifurcation parameter [4] . A critical value of bifurcation parameter β at R C = 1 is given as

It can be easily verified that the Jacobian J 0 at β = β c has a right eigenvector (corresponding to the zero eigenvalue) given by W = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) T , where

× γ 1 (1 + γ 2 + δ 2 )σ + γ 2 σφ + (1 + γ 2 + δ 2 )ξ(1 + γ 1 + δ 1 + φ) and

Furthermore, the components of the left eigenvector (corresponding to the zero eigenvalue), V = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ), must satisfy the equalities V.J 0 = 0 and V.W = 1, so that we obtain

Use the notations

Substituting the values of all the second order derivatives evaluated at DFE and β = β c , we get

and

Finally, substituting the values of V and W, we obtain

where

Since a < 0 and b > 0 at β = β c , therefore using Theorem 4.1 and Remark 1 stated in [4] , a transcritical bifurcation occurs at R C = 1 and the unique endemic equilibrium is locally asymptotically stable for R C > 1. 2

In this section, we analyze the global stability of the disease-free steady states for a special case. We state the following theorem: Theorem 3.4. Suppose R C < 1 and δ 1 = δ 2 = 0, then the disease-free equilibrium E 0 is globally asymptotically stable.

Proof. Here, we prove global stability of DFE applying the method used in [3] . When δ 1 = δ 2 = 0, we have dN dt = 1 − N . Then N → 1 as t → ∞. Take the N in the limiting case, i.e., N = 1, then the system (8)- (13) reduces to

Let X = (R) and Z = (E, Q, I, H), here U 0 = (X 0 , Z 0 ), where X 0 = (0) and Z 0 = (0, 0, 0, 0). We have

Clearly, B is an M-matrix. For I ≥ 0, H ≥ 0, we have 0 < e −m 1 I−m 2 H ≤ 1, therefore G(X, Z) ≥ 0 since 0 ≤ S ≤ 1. Thus both the conditions (H1) and (H2) are satisfied. Hence, the DFE E 0 is globally asymptotically stable if R C < 1. 2

Now, we explore the uniform persistence for the system (8)- (13) . Again, the system (8)-(13) is said to be uniformly-persistent if there exists a constant c such that any solution (E(t), Q(t), I(t), H(t), R(t)) satisfies

. Similar as in [41] , we can state the following theorem for persistent:

Theorem 3.5

Proof. From Theorem 3.1, the DFE of the model (8)-(13) is unstable whenever R C > 1. Apply the uniform persistence result stated in [13] , finally it can be proved in a similar manner as Proposition 3.3 of [24] . 2

The consequence of this result is that in limiting case, all the infected state variables E, Q, I and H of the model will remain above a certain positive threshold and the disease will persist in the population.

In this section, the effect of quarantine and isolation on the transmission dynamics of the disease is measured qualitatively. A threshold analysis on the parameters associated with the quarantine of exposed individuals (k) and the isolation of the infected individuals (φ) is performed by computing the partial derivatives of the effective reproduction number R C with respect to these parameters. We observe that

so that,

From the above analysis it is clear that if the relative infectiousness of the hospitalized individuals (η) does not exceed the threshold value η k , then quarantining of exposed individuals results in reduction of the effective reproduction number R C and therefore, reduction in disease burden (new infections, hospitalization etc.). On the other hand, if η < η k , then due to increase in the rate of quarantine, the effective reproduction number R C will increase and consequently, the disease burden also increases. Thus, the use of quarantine is detrimental in this case. The result is summarized as follows:

Theorem 4.1. For the model (8)- (13) , the use of quarantine of the exposed individuals will have positive

Similarly, the impact of isolation of infectious individuals is assessed by calculating the partial derivatives of R C with respect to the isolation parameter φ. Thus, we obtain ∂R C ∂φ = βσ(η(γ 1 + δ 1 + 1) − (1 + γ 2 + δ 2 )) (γ 2 + δ 2 + 1)(γ 1 + δ 1 + φ + 1) 2 (k + ξ + σ + 1) . Table 2 Parameter values used in the simulation for the system (8)- (13) .

∂R C ∂φ < 0 (> 0) iff η < η φ (η > η φ ),

Nominal values (per day)

Therefore, the use of isolation of infected individuals will be helpful to control the disease in the community if the relative infectiousness of the hospitalized individuals (η) does not exceed the threshold η φ . The result is summarized below: Hence, we conclude that the combine use of quarantine of exposed individuals and isolation of individuals with symptoms will have positive population-level impact if and only if

The quarantine and isolation strategies will have negative population-level impact if

The effective reproduction number R C is a decreasing (non-decreasing) function of the quarantine and isolation parameters k and φ if condition (25) [ (26)] is satisfied (see Figs. 13 and 14 obtained from simulation of the model in which the results are consistent with the analytical findings discussed earlier).

In this section, we provide numerical simulations to illustrate previously established results with the biological feasible parametric values as shown in Table 2 case the disease-free equilibrium is LAS (see Fig. 5 ). When β = 0.40, the effective reproduction number R C = 4.1058 > 1, the unique endemic equilibrium exists and is locally asymptotically stable as shown in Fig. 6 . Whenever R C < 1, the system (8)- (13) has no endemic equilibrium and the DFE of the model with δ 1 =δ 2 = 0 is GAS in Ω, from Theorem 3.4. Fig. 8 depicts the numerical experimentation of the model (8)-(13) taking δ 1 =δ 2 = 0, with β = 0.05 and R C = 0.8952 < 1, for different initial infected population. From the figure it is clear that all solutions converge to the DFE (E 0 ). Similarly, for β = 0.4, we get R C = 4.1058 > 1, and all the solutions converge to the EE (Ē) as shown for infectious individuals I in Fig. 9 .

The coefficients of media coverage m 1 and m 2 should depend on the disease under consideration, the social structure (education, awareness, responsiveness, economy, etc.) of the population and the NPIs used in a particular region. Here, we use the formula m j = − log e (p + q j − pq j ), to quantify the coefficients m 1 and m 2 of media coverage, where q j quantifies the response of the population aware to media recommended NPIs with respect to the number of infective and the hospitalized individuals. If people are not responding to media alert, then q j = 1 and if all the people are adopting the recommended NPIs, then q j = 0. It is assumed that the disease transmission rate can be reduced by p fraction when all individuals follow the It is observed from the analysis that the coefficients of media coverage m 1 and m 2 do not affect R C and the qualitative features of the model remain unaltered. From (18) , we observe that a is always negative, which precludes the existence of backward bifurcation in the system and hence ensures transcritical (i.e., forward) bifurcation about R C = 1. Hence in this case, the classical requirement of R C < 1 is necessary and sufficient for disease control. Moreover, from (15) we observe that

One can easily observe that the use of NPIs stimulated by media coverage helps to mitigate the disease burden from the environment by lowering the level of infectious individuals at steady state. The effect of m 1 and m 2 on the fraction of infectious individuals (I) is shown in Figs. 10 and 11 taking β = 0.09, and β = 0.4, respectively and the rest of the parametric values are as in Table 2 . It is observed that the level of endemic equilibrium is significantly affected by media coefficients m 1 and m 2 .

The pre-existing immunity in the population has significant role in the disease outbreak as it lowers the basic as well as effective reproduction numbers. The effect of pre-existing immunity parameter ξ on the Fig. 12 . It is clear form the figure that higher level of pre-existing immunity helps to reduce disease burden.

The effectiveness of quarantine and isolation depends on the size of the modification parameter (η) for the reduction in infectiousness of hospitalized individuals. For β = 0.04, the threshold value of η with respect to quarantine parameter k is η k = 0.91 and with respect to isolation parameter φ is η φ = 0.90. From Fig. 13 , it is clear that quarantine k has positive population-level impact (R C decreases with increase in k) for η < 0.91 and have negative population level impact for η > 0.91. Similarly for η < 0.90, isolation has positive level impact, whereas isolation has negative impact if η > 0.90 (see Fig. 14 ).

In this section, we perform sensitivity analysis of effective reproduction number R C and endemic equilibrium taking parametric values given in Table 2 . Sensitivity indices allow us to measure the relative change in a state variable/derived parameter when a model parameter changes. Table 3 The sensitivity indices, Υ R C y j = ∂R C ∂y j × y j R C , of the effective reproduction number R C to the parameters, y j , for parameter values given in Table 2 .

Definition. (See [6, 44] .) The normalized forward sensitivity index of a variable, u, that depends on a parameter, p, is defined as:

Estimation of highly sensitive parameter should be done very carefully, because a small variation in the parameter will lead to relatively large quantitative change. On the other hand, a less sensitive parameter does not require as much effort to estimate, since a small variation in that parameter will not produce large change to the quantity of interest.

The normalized sensitive indices of effective reproduction number R C with respect to parameters are shown in Table 3 . From Table 3 , we observe that β, η, σ, and α have positive impact on R C and the rest of the parameters have negative impact. For example, 10% increase (decrease) in σ, resulting in 0.304% increase (decrease) in R C , on the other hand 10% increase (decrease) in γ 1 , will decrease (increase) R C by 2.576%. Moreover, parameters β, δ 1 and γ 1 are most sensitive to R C , hence we observe significant change in R C by small changes in these parameters. Again, we perform sensitivity analysis of state variables at endemic steady state with respect to model parameters. Sensitivity indices of state variables at endemic equilibrium are shown in Table 4 using parametric values shown in Table 2 . From Table 4 , we observe that parameters β, η, δ 2 , α, θ and σ have positive impact on I * and the rest of the parameters have negative impact. Moreover, parameters φ, σ, β, γ 1 , θ and δ 1 are most sensitive parameters to I * , hence we observe significant change in I * by small changes in these parameters.

An SEQIHRS epidemic model for the transmission dynamics of an infectious disease is proposed and rigorous mathematical analysis is carried out to get insight into the qualitative dynamics in presence of pre-existing immunity and the use of NPIs stimulated by media coverage. The main mathematical and epidemiological findings of the proposed model presented in this article are as follows:

(i) The model (2)-(7) has a locally-asymptotically stable disease-free equilibrium whenever the associated effective reproduction number is less than unity (Theorem 3.1). Moreover, if the disease-induced death rates are neglected then the DFE is globally-asymptotically stable (Theorem 3.4). (ii) The model has a unique endemic equilibrium whenever the effective reproduction number exceeds unity and then EE is locally asymptotically stable (Theorems 3.2 and 3.3). (iii) The presence of pre-existing immunity (ξ) in the population has significant impact on the transmission dynamics of the disease. Higher level of pre-existing immunity in the population decreases the infection peak and causes its early arrival. (iv) The coefficients of media awareness m 1 and m 2 do not affect the effective reproduction number R C .

Hence, it does not change the qualitative behavior of the model, but it helps to mitigate disease burden by lowering the level of infection over time. (v) The use of quarantine and/or isolation could have positive, no or negative population level impact depending on the relative infectiousness of isolated individuals (η). (vi) Since disease transmission is directly related to the effective reproduction number, and the disease prevalence is directly related to the endemic equilibrium point Ē , specifically to the magnitude of I * , therefore normalized forward sensitivity indices of the effective reproduction number and I * will be helpful to determine decisive parameter(s) for containing the infectious disease. Sensitivity indices of the effective reproduction number and I * are calculated and highly sensitive parameters are discovered.

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The authors are grateful to the anonymous reviewers for their helpful suggestions and comments that have improved the quality and presentation of the manuscript.