A Study of Shigellosis Bacteria disease Model with Awareness Effects

In this paper, a mathematical model is proposed and studied to describe the spread of shigellosis disease in the population community. We consider it divided into four classes namely: the 1 class consists of unaware susceptible individuals, 2 class of infected individuals, 3 class of aware susceptible individuals and 4 class are people carrying bacteria. The solution existence, uniqueness as well as bounded-ness are discussed for the shigellosis model proposed. Also, the stability analysis has been conducted for all possible equilibrium points. Finally the proposed model is studied numerically to prove the analytic results and discussing the effects of the external sources for disease and media coverage on the dynamical behaviors of shigellosis disease.


Introduction
In fact, the infectious diseases always have an impact on people's health, so it is necessary to study the mechanism by which the disease spread and the conditions of minor and major infections and learn how to control diseases. Recently, a global pandemic have spread in most countries of the world, which is the Corona epidemic (nCovid19), as it appeared in the State of in Wuhan and moved to most countries of the world, where the number of cases of this disease reached more than 6,048,844 million injuries and the number of deaths was more than 367,227 thousand.
Throughout history, infectious diseases have had a major impact on the population. The effects of epidemics are the most obvious and exciting. In the last decade, that was, from 2010 to 2020, many epidemics have spread, most notably new types of influenza such as the Middle East corona and swine flu, and preceded them in the previous decade. For the World Health Organization, swine flu was considered one of the most dangerous viruses. In June 2012, a report was published for a study of a group of doctors, researchers and agencies, announcing the death of 280,000 people, while in 2010, the World Health Organization announced that 18,000 people died as a result of the epidemic. An epidemic has played a major role of infectious diseases in the formation of the invasion of the New World and trolls these epidemics are epidemiologists in coordination with others in the medical field and researchers.
We well know, the many infectious disease are spread by virus as COVID-19 or bacteria as Cholera it is caused by the bacterium Vibrio cholera and there were many researchers studied the Cholera epidemic. Ridha and Muhseen [1] proposed the epidemic disease model with general recovery function. Muhseen and Zhou [2] studied spread Cholera disease with nonlinear incidence rate. M. Al-arydah, et al [3] studied cholera disease with education and chlorination. F.J. Luquero, et al [4] used vibrio cholera vaccine in an outbreak in Guinea. We depend here on one from the basic modeling to study and analyze the infection disease which is the SIR model of  The mathematical modeling is an important interdisciplinary activity in the study of some aspects of various disciplines. Bailey [8] and several authors provided many models for the spread and control of infectious diseases [9][10][11][12]. Methods used to control epidemics are outreach programs driven by the media that can modify the behavior human towards the disease. These awareness campaigns may differ between the various groups at risk that help limit the spread of infection. We would like to mention those who have a clear evaluation of the impact of awareness programs [13][14][15]. There are studies on the effect of media and the effectiveness of using face masks to reduce the spread of the influenza epidemic [16][17]. Misra [18] presented a non-linear mathematical model that evaluated awareness programs and showed their control that the outbreak of infectious diseases can be reduced through media coverage. In this work, we have proposed the mathematical model of the shigella disease involving media programs effect to control the spread the disease. We displayed the full details of the mathematical modeling of the shigella disease in section 2. We have also discussed some basic properties equilibrium points in section 3. In section 4 the local stability analysis was studied with the support of Gresgorin theorem. Furthermore, we studied the using of Lyapunov function to show the global stability of the proposed model at all equilibrium points in section 5. Finally, the effect of media coverage have been done to awareness for the shigella disease and also the risk of direct and indirect contact with carrier individuals on the out breaking of the shigella disease in the population. This was done through a numerical simulation. A discussion of the results effects and limitations involved were concluded in this paper.

Model formulation:
In this section, the population in this work can divided into to four classes, namely unaware susceptible class denoted by ( ), infected class denoted by ( ), aware susceptible denoted by ( ), people carrying bacteria denoted by ( ) and the number of media campaigns in that region at time t denoted by ( ). The model is given by the following differential equation.
The natural birth into susceptible class by > 0. 1 > 0 is the incidence of the disease by direct contact with the carriers of the bacteria. 2 > 0 is the awareness rate. 3 > 0 is the contact rate between susceptible and infected. 4 ≥ 0 represents the number of cases of the disease due to the external sources such as (food, water,…etc).
> 0 is the natural death rate of the in population . > 0 is the disease related death. > 0 is the removal rate. and are respectively, the growth rate carrying capacity of shigella disease.
> 0 is the increasing of the shigella disease bacteria due to infected class. > 0 represents the implementation rate of awareness programs which is proportional to the number of unaware susceptible and infective individuals in the population. Finally the depletion rate of awareness programs due to ineffectiveness, social problems in the population and similar factors is represented by > 0.

Theorem (1):
The uniformly bounded of the any solutions are discussed in the following. By a similar way we get: We obtain that, the solution of system (1) is confined in the following region Ω = {( , , , , ) ∈ + 5 : ≤ , 0 ≤ ≤ } Thus, these solutions are uniformly bounded and the proof is complete. ∎

The number of equilibrium points:
It is easy that aware susceptible is related with variable ( ) and ( ) only. Hence for fixed values of, ( ) and ( ), the calculate value of can be found simply by solving the system (1). In fact, we can determine the value of by the following equation Consequently, we can reduce system (1) and rewrite it to the following system Clearly, there are only two equilibrium points of system (3) While is a positive real root of the following quadratic equation While ( * , * ) represents a positive intersection point of the following two isoclines: ( , ) = 1 2 + 2 2 + 3 + 4 + 5 = 0 (7) ( , ) = 1 2 + 2 2 + 3 + 4 + 5 = 0 Consequently, these two isoclines (7) and (8)  Provided that the following conditions are satisfied Therefore, we have the endemic equilibrium point 1 = ( * , * , * , * ) if the above conditions (9a)-(9b) hold and the following condition is satisfied too

Local stability analysis:
In this section, the local stability conditions of system (3)  Now, by applying the condition in the Gersgorin theorem [19].
We get all the eigenvalues of above Jacobian exist in the region Now, by applying the condition in the Gersgorin theorem [19].
We get all the eigenvalues of above Jacobian exist in the region Then all the eigenvalues of ( 1 ) exist in the disc centered at . Thus if the diagonal elements are negative and the conditions (13a) and (13b) are held, all the eigenvalues will exist in the left half plane and the 1 is locally asymptotically stable. ∎

Global stability analysis:
In this section, we discuss the global stability conditions and determine the basin of attraction of these equilibrium points of system (3) that is presented as shown in the following theorems.  Clearly, 1 < 0, and then 1 is a Lyapunov function provided that the given condition (16a) and (16g) held . Therefore, 1 is globally asymptotically stable. ∎

Numerical Simulation:
In this present section, the spread and control of shigellosis disease are investigated by numerically simulation for many sets of initial values and different sets of parameters values. The objectives of this section are determined by the effect of contact rate, media rate and external sources as well confirm our obtained results. It is observed that through choosing the following data  Clearly Figure 1. confirms our obtained analytical results regarding the existence of globally asymptotically stable endemic equilibrium point.
However, for the same data by equation (17) with 4 = 0, = 0.01 the solution of system (1) converge to the disease free equilibrium point is shown in the following Figure 2.   Figure 3, it is clear that the dynamical behavior of system (1), transmission from endemic point to disease free point that is mean the endemic point became unstable. Now, the dynamical behavior of system (1) under the effect of varying the contact infected rate value 3 is investigated. System (1) is solved numerically by choosing the parameters values given by equation (17)   Clearly, from figure 4. We see that the solution of system (1) is still a converge to the endemic equilibrium point. In addition it is observed that the number of asymptomatic susceptible unaware and aware population decreases while the number of infected population increases.

According to
Finally, the dynamical behavior of system (1) under the effect of varying the incidence rate of the disease by direct contact with the carriers of the bacteria. So, we can chose the same set of parameters values given by equation (17) with 1 = 5.5,20.5,35.5 respectively and then the trajectories of the system (1) are shown in the Figure 5. Clearly, system (1) has an asymptotically stable to the endemic equilibrium point. In addition it is observed that there is a slight change in the system.

Conclusion and Discussion:
Awareness programs that control the spread of the epidemic strong steps should be taken regarding their implementation of diseases such as smallpox, measles, influenza, and others. Behavioral changes caused by awareness programs have the ability to control the size of the epidemic and then predict the future course of the outbreak to guide public health policy. This is a different thing because outbreak of the disease involves the environment, and (direct and indirect) multiple ways of transmission how to cover the media, the awareness programs will be affected the dynamics of shigellosis. In this work, we proposed and analyzed a model to study the effect of awareness programs on shigellosis disease. The model included four ordinary differential equations describing four different class: unaware susceptible individuals , infected individuals , aware susceptible individuals and people carrying bacteria . System (1) has only two equilibrium points. The conditions for existence, stability for each equilibrium points are obtained. Further, it is observed that the disease free equilibrium point ( 0 ) exists when 4 = 0 and = = 0 and locally stable if the conditions (11a-11c) are held, and then it is globally stable if the conditions (15a-15b) are held. The endemic equilibrium point ( 1 ) exists if the conditions