Modeling the control of infectious diseases: Effects of TV and social media advertisements

Arvind Kumar Misra; Rajanish Kumar Rai; Yasuhiro Takeuchi

doi:10.3934/mbe.2018061

Article Contents

2018, Volume 15, Issue 6: 1315-1343. Doi: 10.3934/mbe.2018061

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Modeling the control of infectious diseases: Effects of TV and social media advertisements

1.
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi - 221005, India
2.
College of Science and Engineering, Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa 252-5258, Japan

* Corresponding author: Arvind Kumar Misra
* Corresponding author: Arvind Kumar Misra

Received: September 25, 2017

Revised: April 24, 2018

Published: November 2018

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Abstract

Public health information through media plays an important role to curb the spread of various infectious diseases as most of the populations rely on what media projects to them. Social media and TV advertisements are important mediums to communicate people regarding the spread of any infectious disease and methods to prevent its spread. Therefore, in this paper, we propose a mathematical model to see how TV and social media advertisements impact the dynamics of an infectious disease. The susceptible population is assumed vulnerable to infection as well as information (through TV and social media ads). It is also assumed that the growth rate of TV and social media ads is proportional to the number of infected individuals with decreasing function of aware individuals. The feasibility of possible equilibria and their stability properties are discussed. It is shown that the increment in growth rate of TV and social media ads destabilizes the system and periodic oscillations arise through Hopf-bifurcation. It is also found that the increase in dissemination rate of awareness among susceptible population also gives rise interesting dynamics about the stability of endemic equilibrium and causes stability switch. It is observed that TV and social media advertisements regarding the spread of infectious diseases have the potential to bring behavioral changes among the people and control the spread of diseases. Numerical simulations also support analytical findings.

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Mathematics Subject Classification: Primary: 34A34, 34D20, 92D30.

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Figure 1. Effect of changing the values of $\lambda$ and $M_0$ on $R_1$

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Figure 2. Non-linear stability behavior of model system (2) in $I-A-M$ space for $\lambda = 0.00003$, keeping rest of parameter values same as given in Table 1, which shows that all solution trajectory attains their equilibrium $E^*$ inside the region of attraction $\Omega$

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Figure 3. Variation of $S(t)$, $I(t)$, $A(t)$ and $M(t)$ with respect to time $t$ for $r = 0.005$, which shows that the equilibrium $E^*$ is stable and we have damped oscillation

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Figure 4. Phase portrait of model system (2) for $r = 0.005$ in $I-A-M$ space, which shows that the equilibrium $E^*$ is stable

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Figure 5. Variation of $S(t)$, $I(t)$, $A(t)$ and $M(t)$ with respect to time $t$ for $r = 0.011$, which shows that the equilibrium $E^*$ is unstable and we have undamped sustained oscillation

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Figure 6. Appearance of limit cycle of model system (2) for $r = 0.011$ in $I-A-M$ space, which shows that the equilibrium $E^*$ is unstable

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Figure 7. Bifurcation diagram of infected population $I(t)$, aware population $A(t)$ and the cumulative number of TV and social media ads $M(t)$ with respect to $r$, keeping rest of parameters same as given in Table 1

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Figure 8. Bifurcation diagram of infected population $I(t)$, aware population $A(t)$ and the cumulative number of TV and social media ads $M(t)$ with respect to $\lambda$ for $r = 0.05$, $\omega = 60$, keeping rest of parameters same as given in Table 1

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Table 1. Parameter values for the model system (2)

Parameter	Values	Parameter	Values
$\Lambda$	$5$ ${\rm{day}}^{-1}$	$\beta$	$0.0000030$ ${\rm{day}}^{-1}$
$\lambda$	$0.012$ ${\rm{day}}^{-1}$	$\lambda_0$	$0.008$ ${\rm{day}}^{-1}$
$\nu$	$0.2$ ${\rm{day}}^{-1}$	$\alpha$	$0.00001$ ${\rm{day}}^{-1}$
$d$	$0.00004$ ${\rm{day}}^{-1}$	$r$	$0.006$ ${\rm{day}}^{-1}$
$r_0$	$0.005$ ${\rm{day}}^{-1}$	$\theta$	$0.0005$
$\omega$	$6000$	$p$	$1200$
$M_0$	$500$

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