The Impact of Media Awareness in Controlling the Spread of Infectious Diseases in Terms of SIR Model

Infectious diseases, caused by microorganisms (viruses, fungi, bacteria, parasites and arthropods) which are pathogenic, are very detrimental and life threatening to human health since such diseases can spread indirectly or directly from individuals to individuals, may transmit from contaminated water and food or from animals (zoonotic) or insects [1]. Although the symptoms and signs of infectious diseases may vary or depend on specific types, general indications such as fever, coughing, muscle aches, runny nose, rashes, fatigue and diarrhea are very common for most of the diseases. Diagnosis of contagious diseases, that mainly identifies infectious agents, is done by microbial cultures, microscopy, PCR diagnostics, biochemical test, metagenomic sequence and symptomatic diagnostics [2]. Some of the common infectious diseases are AIDS, Tuberculosis, Hepatitis B, Measles, Chickenpox, Ebola, Influenza, Malaria, Dengue, Chikungunya, Nipah virus infection, MERS, SARS etc. [3]. Recently Coronavirus Disease (COVID-19), caused by a new virus: primarily named as 2019-nCoV; later known as SARS-CoV-2, has been declared as a worldwide pandemic which was initially identified at the city of Wuhan in China on December 31, 2019 when severe pneumonia cases with unidentified explanations were confirmed among a group of individuals and within June 23, 2020, around nine million infective cases with above 469,587 deaths have been confirmed worldwide due to COVID-19 [4]. To slow down or prevent the spread of critical diseases, several characteristics related to the diseases are needed to be understood and recognized [5]. Mathematical modelling in this regard may help predict the forthcoming growth characteristics, project the progress of the disease and choose which interventions should be used as a trial or avoided. Models are typically formulated by considering simple assumptions with specific parameters which are meant for certain diseases and the theoretical methods are applied to understand the special effects of designed interventions, for example, vaccination program [6].

A model on HIV epidemic presented the effects of timely treatment and predicted that early treatment is worthy enough to increase the immunity and considerably lessen new transmissions [7] whereas another work based on testing and treatment revealed that HIV new infections may be minimized to 69.1% within twenty years [8].

A study showed that testing and treatment have the potentiality in minimizing the HIV prevalence to below 1% in next 50 years [9]. A paper focusing on the important issues regarding AIDS epidemic and MDG 2015 investigated that any effective vaccine is necessary since approximately 7400 individuals become infected daily due to HIV. It also presented that the vaccine RV144 was 31% successful during 2009 in preventing HIV infection [10].

A model on tuberculosis established that the disease may be controlled when the efficacy of treatment and vaccination reaches to a convinced threshold [11] and this model was extended later with control schemes so that the intercession cost and the infection burden can be minimized [12]. Another study [13] presented that a particular vaccine which can function both as pre and post exposure is essential to attain the universal control over the deadly TB.

A model representing the influence of reinfection and relapse for Ebola dynamics exposed that the new infective cases will decrease with high level control interventions [14] while another study emphasized that education, quarantine and tracing can significantly decrease the total dimension of Ebola epidemic [15].

An analysis highlighted that Nipah virus infection (NiV) is possible to reduce quickly if quarantined cases are maximized and safety hygiene is maintained [16]. Based on the propagation of NiV [17], it has been anticipated that social distancing and mass awareness may be effective in controlling the situation of NiV [18].

Some works on the dynamics of Dengue [19], Chikungunya [20], Influenza [21], Measles [22], Hepatitis B [23] etc. similarly represent the applications of modelling, from where important insights and strategies were seen to be developed which are sufficiently needed for controlling the spread.

Media as a form of social communication facilitates people by not only sharing information but also providing data to the health administrators during any outbreak situation so that forecasting of the outbreaks can be made possible [24]. A study emphasized that education and media, in Bangladesh, have significance in preventing married people from the deadly HIV. The study also found that the couples who watch television regularly are nearly 8.6 times conscious about HIV than those of who do not watch television [25]. It has been understood that the effects of information transfer play an important role in minimizing the risk of infection [26] while the flow of awareness has the potentiality to decrease the percentage of diseased population [27]. The reports made by mass media for the general public during a pandemic or an epidemic deliver significant information which encourage people to practice healthy and positive behaviors such as maintaining social distance, hand washing etc. These positive practices can reduce the possibility of disease transmission [28]. Activities performed by the media have already shown the potentiality in predicting the evolution and development of infectious diseases. As a result, it is now possible to detect and easily analyze several disease behaviors [29]. Media coverage and its impact on contagious diseases have been explored and investigated through a model by using variable contact rate [30]. Another analysis based on SIS model (which actually motivated us to do this work) focused that awareness campaigns can effectively control the transmission of a spreadable disease but for continuous immigration, the disease may be endemic [31].

In this paper, we introduce a general model, which is based on SIR type [32], to study the possible control and preventive strategy, more specifically, the impact of media awareness programs during the period of any pandemic or epidemic from a common point of view. The purpose of this paper is to analyze the properties of the model, investigate the effects of the publicity rate of awareness programs and the news collection rate which function for making people conscious about the harmful consequences of short or long term infectious diseases.  

In Section 2, the framework of the model has been represented whereas the analytical part of the model has been discussed in Section 3. Numerical results are shown graphically in Section 4. Finally, the overall summary with findings is discussed in Section 5.

The analysis required for model (2) is discussed in this section. We need to observe the boundedness criterion of (2), determine its possible equilibria with basic reproduction number and prove the stability of equilibria.

3.1 Boundedness

Following [7], we obtain from model (2): $\frac{d N}{d t} \leq b-\mu N$ which implies that $N(t) \leq \frac{b}{\mu}+\left(N(0)-\frac{b}{\mu}\right) \exp (-\mu t) .$ When $t \rightarrow \infty, N(t)$ approaches to a certain threshold $\frac{b}{u}$ which suggests that N(t) is actually bounded on [0, t), as a result S, I, R and C are also bounded. Therefore the biological feasible region of model (2) is:

$\Omega =\left\{ (S(t),\,I(t),\,R(t),\,C(t))\in \Re _{+}^{4}\,\,:\,\,S,I,R,C\ge 0\,\,;\,\,0\le N\le \frac{b}{\mu } \right\}$

3.2 Equilibria

Apparently, model (2) possesses two equilibrium points, a disease free equilibrium point $(\mathrm{DFE}): \varepsilon_{1}\left(S^{\circ}, I^{o}, R^{o}, M^{o}, C^{o}\right)$ and an endemic equilibrium point $(\mathrm{EE}): \varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right) .$

Clearly $\varepsilon_{1}\left(S^{o}, I^{o}, R^{o}, M^{o}, C^{o}\right) \equiv\left(\frac{b}{\mu}, 0,0,0,0\right)$.

In order to determine EE which satisfies the equations:

$\frac{dS}{dt}=0,\frac{dI}{dt}=0,\frac{dR}{dt}=0,\frac{dM}{dt}=0,\,\frac{dC}{dt}=0$

i.e., $b-(\beta I+mM+\mu )\,S=0$,$\beta \,SI-(\mu +\gamma +d)\,I=0$,$\gamma I-\mu R=0$, $\sigma I-qM=0$and $mSM-\mu C=0$     (3)

we solve Eq. (3) and thus obtain $\varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right)$ where,

${{S}^{*}}=\frac{d+\gamma +\mu }{\beta }$, ${{I}^{*}}=\frac{bq\beta -q{{\mu }^{2}}-dq\mu -q\gamma \mu }{(d+\gamma +\mu )\,(m\sigma +q\beta )}$

${{R}^{*}}=\frac{\gamma \,(bq\beta -q{{\mu }^{2}}-dq\mu -q\gamma \mu )}{\mu \,(d+\gamma +\mu )\,(m\sigma +q\beta )}$, ${{M}^{*}}=\frac{\sigma \,(b\beta -d\mu -\gamma \mu -{{\mu }^{2}})}{(d+\gamma +\mu )\,(m\sigma +q\beta )}$

and ${{C}^{*}}=\frac{m\sigma \,(b\beta -d\mu -\gamma \mu -{{\mu }^{2}})}{\beta \mu \,(m\sigma +q\beta )}$.

3.3 Basic reproduction number ($\mathfrak{R}_{0}$)

A precise method termed as next generation matrix approach [33] is applied in this subsection to obtain the basic reproduction number (${{\Re }_{0}}$) for model (2).

In model (2), I(t) is the only infection component and therefore the new infection matrix is $F=(\,\beta \,{{S}^{o}}\,){{\,}_{1\times 1}}$ and the transfer matrix is $V=(\,d+\gamma +\mu \,){{\,}_{1\times 1}}$. Consequently, the next generation matrix becomes$F{{V}^{-1}}=(\,\beta \,{{S}^{o}}\,)\,\,\left( \frac{1}{d+\gamma +\mu \,} \right)\,$.

Hence, the spectral radius of $F{{V}^{-1}}$, i.e., $\rho \,(F{{V}^{-1}})$ is defined to be the basic reproduction number (${{\Re }_{0}}$) which can be written as: ${{\Re }_{0}}=\frac{\beta \,{{S}^{o}}}{d+\gamma +\mu \,}=\frac{b\beta }{\mu \left( d+\gamma +\mu  \right)}$.

3.4 Stability

The Jacobian matrix (J) associated to the model (2) is essentially required to establish the stability [34] of DFE and EE and therefore is defined as follows:

$J=\left( \,\begin{matrix}   -(\beta \,I+\mu +mM) & -\beta S & 0 & -mS & 0  \\   \beta \,I & \beta S-\mu -\gamma -d & 0 & 0 & 0  \\   0 & \gamma  & -\mu  & 0 & 0  \\   0 & \sigma  & 0 & -q & 0  \\   mM & 0 & 0 & mS & -\mu   \\\end{matrix}\, \right)$     (4)

Theorem 1. The DFE: $\varepsilon_{1}\left(S^{o}, I^{o}, R^{o}, M^{o}, C^{o}\right)$ is asymptotically stable when $\Re_{0}<1$ and unstable when ${{\Re }_{0}}>1$.

Proof. At DFE: $\varepsilon_{1}\left(S^{o}, I^{o}, R^{o}, M^{o}, C^{o}\right) \equiv\left(\frac{b}{\mu}, 0,0,0,0\right)$, Eq. (4) becomes

$\begin{align}  & J({{\varepsilon }_{1}})=\left( \,\begin{matrix}   -(\beta \,{{I}^{o}}+\mu +m{{M}^{o}}) & -\beta {{S}^{o}} & 0 & -m{{S}^{o}} & 0  \\   \beta \,{{I}^{o}} & \beta {{S}^{o}}-\mu -\gamma -d & 0 & 0 & 0  \\   0 & \gamma  & -\mu  & 0 & 0  \\   0 & \sigma  & 0 & -q & 0  \\   m{{M}^{o}} & 0 & 0 & m{{S}^{o}} & -\mu   \\\end{matrix}\, \right) \\ & \Rightarrow J\,({{\varepsilon }_{1}})=\left( \,\begin{matrix}   -\mu  & -\beta \frac{b}{\mu } & 0 & -m\frac{b}{\mu } & 0  \\   0 & \beta \frac{b}{\mu }-\mu -\gamma -d & 0 & 0 & 0  \\   0 & \gamma  & -\mu  & 0 & 0  \\   0 & \sigma  & 0 & -q & 0  \\   0 & 0 & 0 & m\frac{b}{\mu } & -\mu   \\\end{matrix}\, \right) \\\end{align}$

Considering λ as the eigen value, the characteristic equation becomes $\left| J\,({{\varepsilon }_{1}})-\lambda \,{{I}_{d}}\, \right|=\,\,0$, where I_dis an identity matrix of order 5×5.

Now by rearranging the characteristic equation, we obtain

$\left| \,\begin{matrix}   -\mu -\lambda  & -\beta \frac{b}{\mu } & 0 & -m\frac{b}{\mu } & 0  \\   0 & \beta \frac{b}{\mu }-\mu -\gamma -d-\lambda  & 0 & 0 & 0  \\   0 & \gamma  & -\mu -\lambda  & 0 & 0  \\   0 & \sigma  & 0 & -q-\lambda  & 0  \\   0 & 0 & 0 & m\frac{b}{\mu } & -\mu -\lambda   \\\end{matrix}\, \right|=0$     (5)

From Eq. (5), we have

$(\lambda +q)\,(\lambda +\mu )\,(\lambda +\mu )\,(\lambda +\mu )\,\left( \lambda -\frac{b\beta -d\mu -\gamma \mu -{{\mu }^{2}}}{\mu } \right)\,=0$

which implies that $\lambda=-q, \lambda=-\mu, \lambda=-\mu, \lambda=-\mu,$ and $\lambda=\frac{b \beta-d \mu-\gamma \mu-\mu^{2}}{\mu}$.

Clearly, all the eigen values, except $\lambda=\frac{b \beta-d \mu-\gamma \mu-\mu^{2}}{\mu}$, are negative.

Now $\lambda=\frac{b \beta-\mu(d+\gamma+\mu)}{\mu}$ can be written as $\lambda=\frac{\mu(d+\gamma+\mu)\left(\frac{b \beta}{\mu(d+\gamma+\mu)}-1\right)}{\mu}=(d+\gamma+\mu)\left(\Re_{0}-1\right)$, which is negative when $\Re_{0}<1$. Therefore, Theorem 1 holds.

Theorem 2. The EE: $\varepsilon_{2}\left(S^{*}, I^{*}, R^{*}, M^{*}, C^{*}\right)$ is asymptotically stable when $\Re_{0}>1$ and unstable when ${{\Re }_{0}}<1$.

Proof is provided in Appendix A.

A general model on infectious disease dynamics has been developed in this paper where media awareness is considered as a control variable function. The model is formulated with a purpose to study the impact of media awareness with important factors and its efficacy during any pandemic or epidemic situation. Media generally collects information about the severity of the ongoing disease, emphasizes on several health issues, as a result of which, mass population may become conscious and form a separate isolation group so that they may avoid the contagion. Since increased transmission rate is one of the main reasons for the persistence of a disease, consciousness regarding safety issues and disease characteristics is significantly needed to lessen the adverse scenario. It is observed that continuous publicity of awareness programs brings a tremendous change in human behavior which is very effective and substantial in preventing the disease transmission. Moreover, the news and information collected by media, despite some exhaustion, always reveal a positive impact which works as a pre-recovery option for mitigating the virus transmission. The present work suggests that in order to productively control the disease burden, especially at that time when there is scarcity of effective vaccines or proper treatments, early and continuous implementation of awareness programs may be one of the supportive interventions for any countries to meet the critical challenges against the new emerging diseases.