Mathematical Model for Impact of Media on Cleanliness Drive in India

Published online: September 6, 2018 A mathematical model on cleanliness drive in India is analysed for active cleaners and passive cleaners. Cleanliness and endemic equilibrium points are found. Local and global stability of these equilibrium points are discussed using Routh-Hurwitz criteria and Lyapunov function respectively. Impact of media (as a control) is studied on passive cleaners to become active. Numerical simulation of the model is carried out which indicates that with the help of media transfer rate to active cleaners from passive cleaners is higher.


Introduction
Maintaining a clean environment plays an important role in the health of every human being, because their health is fully based on the surrounding environment. The importance of cleanliness cannot be ignored by any society. Every religion and civilization emphasis the significance of cleanliness. It is a virtue of God. Woefully, this virtue of cleanliness not reflected in our society. People usually throw garbage in the roads, streets and gardens etc. Spitting anywhere is also common practice. Students of schools and colleges also throw waste in the classroom and the other premises. pitiful conditions about cleanliness are observed in our society. We must develop positive attitude to tackle this situation. In fact, there is a need to inspire people about importance of cleanliness in our life [4].
Taking in to the account the urgent need of cleanliness in India, the Prime Minister Narendra Modi had launched cleanliness drive as "Swachh Bharat Abhiyan" on 2 nd October 2014, at Rajghat, New Delhi [3]. The objectives of the cleanliness drive in India are to clean up houses, streets, roads and infrastructure of India's cities, smaller towns, and as well as eliminating open defecation in rural areas [4]. The cleanliness drive aims to achieve the vision of 'Clean India' by 2 nd October 2019. Each citizen of India should join in this cleanliness drive as 'cleaner' to achieve the goal of "Swachh Bharat Abhiyan".
There are two types of cleaners, namely 'active cleaners' and 'passive cleaners'. Active cleaners are involved in cleanliness activity with their own interest, while passive cleaners need an external force to engage in cleanliness related activity. It is observed that media plays vital role as motive force for passive cleaners to join in cleanliness drive and become active media happens to be the most readily available and potentially most economical means of spreading information about cleanliness drive. Media can effectively lift public awareness and concern about sanitation. Moreover, every day public see a lot of advertisements in electronic and print media, where they tell people to support this drive [7]. Eventually, the media works as stimulant in cleanliness drive. Shah et al. [9] developed a mathematical model for cleanliness drive and has discussed its stability through graph theory.
In this paper, we will analyse how passive cleaners turns to be active cleaners under impact of media. In section 2, a mathematical model is discussed with schematic diagram along with notations, description and parametric values as described in Table 1 global stability of the equilibrium points are discussed in section 3.1 and 3.2 respectively. Section 4 studies optimal control on the system. In section 5, numerical simulation of the model is carried out and interpretations are work out.

Mathematical Model
Here, a mathematical model has been formulated to study the impact of media on cleanliness drive in India as an application of SEIR model. The notations along with its description and parametric values are given in Table 1. The transmission diagram for cleanliness drive is shown in figure 1.
Every susceptible individuals (S) either act as passive cleaners (P) or active cleaners (A). β and η are the rates described for the passive and active cleaners respectively who are indulge in cleanliness related activities. Gradually, passive cleaners (P) tries to become active cleaners (A) which is mentioned by the rate γ . To boost up this activity of cleanliness among individuals media campaign is essential which is described as the rate u 1 . Many times, it happens that this active cleaners (A) are forced to lose their interest and thus becomes a passive cleaners (P) which is described by the rate σ . B represents new recruitment rate and µ represents escape rate in the model. Now, from the figure 1 a set of nonlinear ordinary differential equations for cleanliness has been obtained.
Adding all above equations, we have The feasible region of solution of (1) is Now, using next generation matrix method [1], a basic reproduction number is to be calculated. For this, let ℑ( ) X denotes rate of new passive and active cleaners and ν X ( ) denotes rate of transfer of cleanliness activity among individuals.
Now, the derivative of ℑ and ν at cleanliness equilibrium point E 0 gives matrices F and V of order 3 × 3 defined as, Thus, the basic reproduction number (R 0 ) is given by Now, on equating the set of equation (1) to zero, an endemic equilibrium point defined as E * among individuals is ; where, X 2 = + σ µ

Stability Analysis
In this section, the local and global stability of E 0 and E * are to be studied.

Local stability
The characteristic polynomial of above matrix is where, a P S X X X a X P S X P X S X X X X

Global stability
Theorem 3: (Stability of E 0 ) Cleanliness equilibrium point E 0 ( )is globally asymptotically stable.
Theorem 4: (Stability of E * ) Endemic equilibrium point E * is globally asymptotically stable.

Proof: Consider a Lyapunov function
∴ E * is globally asymptotically stable.

Optimal Control Model
Our aim is to involve maximum number of individuals in cleanliness drive. Media is effective medium to motivate all individuals in broad sense to join the drive actively. In this section, a control function is introduced to get maximum transfer rate from passive cleaners to active cleaners. The objective function with of the control variable is given by where, Ω is the set of all compartmental variables,

J u A S A P A A w u dt
, , and are non-negative weight constants for state variables S P A , , and control variable u 1 respectively.
Existence of optimal control variable is guaranteed since Ω is nonempty. Control set is convex and closed. One can easily see that right hand side of is bounded by linear function in state and control variables. The integrand of the objective function is convex and bounded below by w u 1 1 2 1 − . Now, we will calculate the value of control variable u 1 from t = 0 to t T = such that J u t J u u The necessary condition for Lagrangian function to be optimal for control is By solving equation (4) we have, u P So, the required optimal control condition is obtained as where a 1 = lower bound and b 1 = upper bound.

Numerical Simulation
In this section, we will study numerical results of all compartments. Effect of media as control are studied on all compartments.  Figure 2 shows that in first month of cleanliness drive, susceptible individuals become passive cleaners as well as active cleaners. But, there are more passive cleaners then active cleaners in first month. It also indicates that it takes more than one month to have more active cleaners then passive cleaners in cleanliness drive. It reflects laziness of individuals in cleanliness drive. Figure 3 shows that under influence of media as control, susceptible individuals quickly join cleanliness activities as active and passive cleaners. It indicates that almost all individuals become active cleaners in a month and remains active for longer time.
It is observed from the figure 4 that with control, transmission rate of passive cleaners into active is higher compared to without control. It indicates that passive cleaners are inspired by media and they get actively involved in the drive. Media campaign increase passive cleaners by 12% as shown in figure 5.      Figure 6 indicates increment of active cleaners with control compare to without control. It shows that active cleaner increased by 100% in mid days of the first month of the drive with media campaign. Figure 7 shows that effect of media plays a vital role as it increases the active cleaners from 39% to 61%. One must put there step forward in making our country clean. Figure 8 shows the significance of media in the cleanliness drive. It can be seen from the figure that 27% awareness is required among individuals through media campaigning. It also indicates that after awareness about cleanliness among individuals, control can be gradually decreased.

Conclusion
In this paper, a mathematical model of cleanliness drive in India is studied. One can conclude that media is essential for the awareness of cleanliness among individuals since passive cleaners become active cleaners in cleanliness related activities comparatively at a higher rate when media impact is observed. Our first contribution for the clean India is to keep our house clean by proper wastage management of our house as per the instruction of the corporation. We should also actively participate in the cleanliness drive to clean our streets, school, college, office, public place etc. to achieve vision of 'Clean India'.
This model can be extended to various social and health related issues. Infectious diseases usually spread in unhygienic environment. Cleanliness play vital role to control it. Hence, the model can be extended to study the effect of cleanliness in controlling infectious diseases.