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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

 

  

 

 

A Study of Shigellosis Bacteria disease Model with Awareness Effects 

Watheq Ibrahim Jasim                                       Hassan Fadhil AL-Husseiny 

watheq.alasady75@gmail.com                              Hassan.fadhil.r@sc.uobaghdad.edu.iq 
 

Department of Mathematics, College of Sciences, University of Baghdad, Baghdad , Iraq. 

 

                                                                                                                                  

                                                                 

 

 

Abstract 

     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 1st class consists of  unaware susceptible individuals, 2nd class of infected 

individuals, 3rd class of aware susceptible individuals and 4th 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. 

 

Keywords: Shigellosis Disease, Awareness, Media Coverage, Stability, External Sources. 

 

1. 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, 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.2.2619 

Article history: Received 24, June, 2020, Accepted  20,July,2020, Published in April 2021 

 

mailto:watheq.alasady75@gmail.com
mailto:Hassan.fadhil.r@sc.uobaghdad.edu.iq


  

130 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

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 Kermack and Mckendrick [5-7. 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 [9-12]. 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 

[13-15]. 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. 

 

2. Model formulation: 

    In this section, the population in this work  can divided into to four classes, namely 

unaware susceptible class denoted by (𝑆𝑢), infected class denoted by (𝐼), aware susceptible 

denoted by (𝑆𝑎), people carrying bacteria denoted by (𝐵) and the number of media campaigns 

in that region at time t denoted by (𝑀). The model is given by the following differential 

equation. 



  

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𝑆�̇� = 𝜃 −𝑎1𝑆𝑢𝐵 −𝑎2𝑆𝑢𝑀 −𝑎3𝑆𝑢𝐼 −𝑎4𝑆𝑢 −𝜇𝑆𝑢
𝐼̇ = 𝑎1𝑆𝑢𝐵 +𝑎3𝑆𝑢𝐼 +𝑎4𝑆𝑢 −(𝜇 +𝑑 +𝜂)𝐼          

𝑆�̇� = 𝑎2𝑆𝑢𝑀 −𝜇𝑆𝑎                                                         

�̇� = 𝑟𝐵(1−
𝐵

𝐾
)−𝜇𝐵 +𝜖𝐼                                          

�̇� = 𝛼(𝑆𝑢 +𝐼)−  𝛾𝑀                                                    

                                       (1) 

With initial conditions 𝑆𝑢(0) > 0,𝐼(0) ≥ 0,𝑆𝑎(0) ≥ 0,𝐵(0) ≥ 0,𝑀(0) ≥ 0. The natural 

birth into susceptible class by 𝜃 > 0. 𝑎1 > 0 is the incidence of the disease by direct contact 

with the carriers of the bacteria.  𝑎2 > 0 is the awareness rate. 𝑎3 > 0 is the contact rate 

between susceptible and infected. 𝑎4 ≥ 0 represents the number of cases of the disease due to 

the external sources such as (food, water,…etc).                                               𝜇 > 0 is the 

natural death rate of the in population . 𝑑 > 0 is the disease related death. 𝜂 > 0 is the 

removal rate. 𝑟 and 𝑘 are respectively, the growth rate  carrying capacity of shigella disease. 

𝜖 > 0 is the increasing of the shigella disease bacteria due to infected class. 𝛼 > 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 𝛾 > 0. 

 

Theorem (1): The uniformly bounded of the any solutions are discussed in the following. 

Proof: Let (𝑆𝑢(𝑡),𝐼(𝑡),𝑆𝑎(𝑡),𝐵(𝑡),𝑀(𝑡)) is the solution of the system (1) with positive 

initial condition  (𝑆𝑢(0),𝐼(0),𝑆𝑎(0),𝐵(0),𝑀(0))  which defines the function 

𝑁(𝑡) = 𝑆𝑢(𝑡)+𝐼(𝑡)+ 𝑆𝑎(𝑡)+𝐵(𝑡) then take the time derivative of  𝑁(𝑡) along the 

solution of the system (1);this gives 

𝑑𝑁

𝑑𝑡
= Ө+𝑟𝐵(1−

𝐵

𝑁
)−𝑞(𝑆𝑢 +𝐼 +𝑆𝑎 +𝐵)                     

𝑑𝑁

𝑑𝑡
≤ 𝐻 −𝑞𝑁  ;  𝐻 =

𝑟𝐾

4
+Ө ,             𝑞 = 𝑚𝑖𝑛 {𝜇,𝜇 −𝜖}

𝑑𝑁

𝑑𝑡
+𝑞𝑁 ≤ 𝐻                                                                             

 

Clearly, by solving the above equation, we obtain  

𝑁(𝑡) ≤
𝐻

𝑞
+ (𝑁0 −

𝐻

𝑞
)𝑒−𝑞𝑡 

Therefore,  𝑁(𝑡) ≤
𝐻

𝑞
,         𝑎𝑠 𝑡 →  ∞ 

𝑑𝑀

𝑑𝑡
=  𝛼 ( 𝑆𝑢 + 𝐼 )−  𝛾𝑀                  

𝑑𝑀

𝑑𝑡
≤ �̃� −  𝛾𝑀  ;  �̃� = 𝛼 (𝑆𝑢 + 𝐼) 

 

By a similar way we get: 



  

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𝑀(𝑡) ≤ 
𝛼𝐻

𝛾𝑞
,                        𝑎𝑠 𝑡 →  ∞ 

We obtain that, the solution of system (1) is confined in the following region 

             Ω = {(𝑆𝑢, 𝐼,𝑆𝑎,𝐵,𝑀) ∈ 𝑅+
5:𝑁 ≤

𝐻

𝑞
,0 ≤ 𝑀 ≤

𝛼𝐻

𝛾𝑞
} 

Thus, these solutions are uniformly bounded and the proof is complete.                     ∎ 

3. The number of equilibrium points: 

It is easy that aware susceptible 𝑆𝑎 is related with variable 𝑆𝑢(𝑡) and 𝑀(𝑡) only. Hence for 

fixed values of, 𝑆𝑢(𝑡) and 𝑀(𝑡),  the calculate value of 𝑆𝑎 can be found simply by solving the 

system (1). In fact, we can determine the value of  𝑆𝑎 by the following equation 

 𝑆𝑎 =
𝑎2�̃�𝑢�̃�

𝜇
                                                                                                          (2) 

Consequently, we can reduce system (1) and rewrite it to the following system  

 

𝑆�̇� =  𝜃 −𝑎1𝑆𝑢𝐵 −𝑎2𝑆𝑢𝑀 −𝑎3𝑆𝑢𝐼 −𝑎4𝑆𝑢 −𝜇𝑆𝑢  

𝐼̇ = 𝑎1𝑆𝑢𝐵 +𝑎3𝑆𝑢𝐼 +𝑎4𝑆𝑢 −(𝜇 +𝑑 +𝜂)𝐼       

�̇� = 𝑟𝐵 ( 1− 
𝐵

𝐾
) −𝜇𝐵 +𝜖𝐼                                       

  �̇� =  𝛼 ( 𝑆𝑢 + 𝐼 )–𝛾𝑀                                                    

     
                                   (3) 

Clearly, there are only two equilibrium points of system (3) under the following conditions: 

 The first equilibrium point exists when  𝐼 = 0 (when 𝑎4 = 0) and 𝐵 =  0, and is called 

disease free equilibrium point which is denoted by 𝐸0 = (𝑆𝑢
^ ,0 ,0 ,𝑀^), where 

 𝑀^ =
𝛼

𝛾
𝑆𝑢
^                                                                                                           (4) 

While 𝑆𝑢 is a positive real root of the following quadratic equation 

 𝐴1𝑆𝑢
2 +𝐴2𝑆𝑢 +𝐴3 = 0                                                                                    (5a) 

Here 

 𝑆𝑢 =
−(𝐴2+√𝐴2

2−4𝐴1𝐴3)

2𝐴1
                                                                                      (5b) 

 

𝐴1 =
−𝑎2𝛼

𝛾
             

𝐴2 = −𝜇                
𝐴3 = 𝜃                  

 

 The endemic equilibrium point, denoted by 𝐸1 = (𝑆𝑢
∗ , 𝐼∗,𝐵∗,𝑀∗) where 

𝐼∗ =
𝐵∗( 𝜇𝑘+𝑟𝐵∗−𝑟𝑘 )

𝑘𝜖
                                                                                         (6a) 

 𝑀∗ =
𝛼(𝑘𝜖𝑆𝑢

∗+(𝜇−𝑟)𝑘𝐵∗+𝑟𝐵∗
2
)

𝑘𝜖𝛾
                                                                              (6b) 



  

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While  (𝑆𝑢
∗,𝐵∗) represents a positive intersection point of the following two isoclines: 

 𝑓(𝑆𝑢,𝐵) = 𝑚1𝑆𝑢𝐵
2  + 𝑚2𝐵

2  + 𝑚3𝑆𝑢𝐵 + 𝑚4𝑆𝑢  + 𝑚5𝐵 =  0              (7) 

 𝑔(𝑆𝑢,𝐵) = 𝑛1𝑆𝑢𝐵
2  + 𝑛2𝑆𝑢

2 +𝑛3𝑆𝑢𝐵 + 𝑛4𝑆𝑢 + 𝑛5  =  0                         (8) 

Here, 

𝑚1 = 𝑎3𝑟 ,𝑚2 = −(𝜇 +𝑑 + η),m3  = 𝑎1𝑘𝜖 + 𝑎3(𝜇 −𝑟)𝑘                   

𝑚4 = 𝑎4𝑘𝜖 ,    𝑚5 = −(𝜇 +𝑑 + 𝜂)(𝜇 −𝑟)                                                    

 

n1 = − ( 𝑎2r+ 𝑎3𝛾𝑟),            n2 = −𝑎2𝛼𝑘𝜖    

   n3 = − [ 𝑎1𝑘ϵ𝛾 + 𝑎2𝑘(𝜇 −𝑟)+𝑎3𝛾𝑘(𝜇 −𝑟)] 

n4 = −𝑘ϵ𝛾(𝑎4 +𝜇),n5 =  𝑘ϵ𝛾𝜃                         

                                              

 

Clearly as 𝐵 → 0 the first isoclines (7) intersects the 𝑆𝑢 − axis at zero 

However when 𝐵 → 0 the second isoclines (8) will intersect the 𝑆𝑢 − axis at a unique positive 

point, say  𝑆𝑢1 

Consequently, these two isoclines (7) and (8) have an intersection point in the interior of the 

positive quadrant of 𝑆𝑢𝐵 – plane, namely (𝑆𝑢 
∗ ,𝐵∗), 

Provided that the following conditions are satisfied 

 

𝜕𝑓

𝑑𝑆𝑢
> 0        𝑎𝑛𝑑                 

𝜕𝑓

𝑑𝐵
 < 0

𝑜𝑟            
𝜕𝑓

𝑑𝑆𝑢
< 0       𝑎𝑛𝑑                   

𝜕𝑓

𝑑𝐵
 > 0

   }                                                              (9a) 

 

𝜕𝑔

𝑑𝑆𝑢
> 0         𝑎𝑛𝑑                  

𝜕𝑔

𝑑𝐵
 > 0

𝑜𝑟            
𝜕𝑔

𝑑𝑆𝑢
< 0          𝑎𝑛𝑑                

𝜕𝑔

𝑑𝐵
 < 0

   }                                                             (9b) 

Therefore, we have the endemic equilibrium point 𝐸1 = (𝑆𝑢
∗ , 𝐼∗ ,𝐵∗ ,𝑀∗) if the above 

conditions (9a)-(9b) hold and the following condition is satisfied too 

 𝑟𝑘 < 𝜇𝑘 +𝑟𝐵∗                                                                                                (10) 

4. Local stability analysis: 

In this section, the local stability conditions of system (3) near 𝐸𝑖 , 𝑖 = 0,1 are established in 

the following theorems. 

Theorem (2): If the 𝐸0 point exists,  it is locally asymptotically stable if the following 

conditions are held 

𝑟 < 𝜇                                                                                                            (11a) 

𝛼 < 𝑎2𝑀
^ +𝜇                                                                                             (11b) 

𝑆𝑢
^ < 𝑚𝑖𝑛{

μ+d+η−(𝜖+𝛼)

2𝑎3
,
𝜇−𝑟

2𝑎1
,
𝛾

𝑎2
}                                                                 (11c) 



  

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Proof: Clearly the Jacobian matrix of system (3) at 𝐸0 can be written as 

𝐽(𝐸0) = (𝑤𝑖𝑗)4𝑥4 

where 

=

(

 

−𝑎2𝑀
^ −𝜇 −𝑎3𝑆𝑢

^ −𝑎1𝑆𝑢
^ −𝑎2𝑆𝑢

^

0 𝑎3𝑆𝑢
^ −𝜇 −𝑑 −η 𝑎1Su

^ 0
0 𝜖 𝑟 −𝜇 0
𝛼 𝛼 0 −𝛾 )

  

Now, by applying the condition in the Gersgorin theorem [19]. 

|𝑤𝑖𝑖| > ∑ |𝑤𝑖𝑗|
4
𝑖=1
𝑖≠𝑗

                                                                                            (12) 

We get all the eigenvalues of above Jacobian exist in the region 

𝛺 =∪{𝑢∗ ∈ 𝐶:|𝑢∗ −𝑤𝑖𝑖| < ∑|𝑤𝑖𝑗|

4

𝑖=1
𝑖≠𝑗

} 

Then all the eigenvalues of 𝐽(𝐸0) exist in the disc centered at 𝑤𝑖𝑖. Thus if the diagonal 

elements are negative and the conditions (11a) and (11c) are held, all the eigenvalues will 

exist in the left half plane and the  𝐸0 is locally asymptotically stable                                                                                                                             

∎ 

Theorem (3): The 𝐸1 = (𝑆𝑢
∗, 𝐼∗,𝐵∗,𝑀∗) of system (3) is locally asymptotically stable under 

the following conditions held 

𝑆𝑢
∗ < 𝑚𝑖𝑛{

μ+d+η−(𝜖+𝛼)

2𝑎3
,
2𝑟𝐵∗+𝑘(𝜇−𝑟)

2𝑎1𝑘
,
𝛾

𝑎2
}                                                    (13a) 

𝛼 < 𝑎2𝑀
∗ +𝜇                                                                                             (13b) 

Proof: It is easy from Jacobian matrix of system (3) at 𝐸1 = (𝑆𝑢
∗, 𝐼∗,𝐵∗,𝑀∗)  that can be 

written as 𝐽(𝐸1) = (𝑑𝑖𝑗)4𝑋4 

Here, 

𝑑11 = −𝑎1𝐵
∗ −𝑎2𝑀

∗ −𝑎3𝐼
∗ −𝑎4 −𝜇,𝑑12 = −𝑎3𝑆𝑢

∗,𝑑13 = −𝑎1𝑆𝑢
∗ 

𝑑14 = −𝑎2𝑆𝑢
∗,𝑑21 = 𝑎1𝐵

∗ +𝑎3𝐼
∗ +𝑎4 ,𝑑22 = 𝑎3𝑆𝑢

∗ −(𝜇 +𝑑)−η  

d23 = 𝑎1Su 
∗ ,𝑑32 = 𝜖,𝑑33 =

𝑘𝑟 −2𝑟𝐵∗ −𝑘𝜇

𝑘
,𝑑41 = 𝛼,𝑑42 = 𝛼             

 

𝑑44 = −𝛾,𝑑31 = d24 = 𝑑34 = 𝑑43 = 0 

Now, by applying the condition in the Gersgorin theorem [19]. 

|𝑑𝑖𝑖| > ∑ |𝑑𝑖𝑗|
4
𝑖=1
𝑖≠𝑗

                                                                                             (14) 

We get all the eigenvalues of above Jacobian exist in the region 



  

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𝛺 =∪{𝑢∗ ∈ 𝐶:|𝑢∗ −𝑑𝑖𝑖| < ∑|𝑑𝑖𝑗|

4

𝑖=1
𝑖≠𝑗

} 

Then all the eigenvalues of 𝐽(𝐸1) exist in the disc centered at  𝑑𝑖𝑖 . 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 𝐸1 is locally asymptotically stable.                                                                                                                           

∎ 

5. 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. 

Theorem (4): The 𝐸0 of system (3) is globally asymptotically in the sub region of 𝑅+
4 under 

the following sufficient conditions 

(𝑎2 −
𝛼

𝑀
)
2

< 4(
𝑎2𝑀

^+𝜇

𝑆𝑢
)(

𝛾

𝑀
)                                                                     (15a) 

𝑆𝑢
^ < 𝑚𝑖𝑛{

𝜇−𝑟

𝑎1
,
(𝜇+𝑑+𝜂−𝛼−𝜖)𝑀+𝛼𝑀^

𝑎3𝑀
}                                                          (15b) 

Proof: consider the following positive definite function 

𝑉0(𝑆𝑢, 𝐼,𝐵,𝑀) = (𝑆𝑢 −𝑆𝑢
^ −𝑆𝑢

^ ln
𝑆𝑢
𝑆𝑢
^
)+𝐼 +𝐵 +(𝑀 −𝑀^ −𝑀^ ln

𝑀

𝑀^
) 

It is easy to see that 𝑉0(𝑆𝑢, 𝐼,𝐵,𝑀) ∈ 𝐶
1(𝑅+

4,𝑅)  and  𝑉0(𝑆𝑢
^,0,0,𝑀^) = 0, while 

𝑉0(𝑆𝑢, 𝐼,𝐵,𝑀) > 0 ∀(𝑆𝑢, 𝐼,𝐵,𝑀) ∈ 𝑅+
4 and (𝑆𝑢, 𝐼,𝐵,𝑀) ≠ (𝑆𝑢

^,0,0,𝑀^) . 

𝑑𝑉0
𝑑𝑡

= (
𝑆𝑢 −𝑆𝑢

^

𝑆𝑢
)
𝑑𝑆𝑢
𝑑𝑡

+
𝑑𝐼

𝑑𝑡
+
𝑑𝐵

𝑑𝑡
+(

𝑀 −𝑀^

𝑀
)
𝑑𝑀

𝑑𝑡
 

𝑑𝑉0
𝑑𝑡

= (
𝑆𝑢 −𝑆𝑢

^

𝑆𝑢
)[𝜃 −𝑎1𝑆𝑢𝐵 −𝑎2𝑆𝑢𝑀 −𝑎3𝑆𝑢𝐼 −𝑎4𝑆𝑢 −𝜇𝑆𝑢]        

           +[𝑎1𝑆𝑢𝐵 +𝑎3𝑆𝑢𝐼 −(𝜇 +𝑑 +𝜂)𝐼] +[𝑟𝐵 ( 1− 
𝐵

𝑘
)−𝜇𝐵 +𝜖𝐼]

          +(
𝑀 −𝑀^

𝑀
)[𝛼 (𝑆𝑢 + 𝐼)–𝛾𝑀]                                                          

 

Furthermore by taking the derivative and simplifying the resulting terms, we obtain  

That 

𝑑𝑉0
𝑑𝑡

= −[(
𝑎2𝑀

^ +𝜇

𝑆𝑢
)(𝑆𝑢 −𝑆𝑢

^)2 +(𝑎2 −
𝛼

𝑀
)(𝑀 −𝑀^)(𝑆𝑢 −𝑆𝑢

^)+
𝛾

𝑀
(𝑀−𝑀^)2]

−(𝜇 −(𝑟 +𝑎1𝑆𝑢
^))𝐵 −[(𝜇 +𝑑 +𝜂)+

𝛼𝑀^

𝑀
−(𝑎3𝑆𝑢

^+∈ +𝛼)]𝐼     

 



  

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By using the above condition, we obtain that 

𝑑𝑉0
𝑑𝑡

≤ −[√
𝑎2𝑀

^ +𝜇

𝑆𝑢
(𝑆𝑢 −𝑆𝑢

^)+√
𝛾

𝑀
(𝑀 −𝑀^)]

2

−(𝜇 −(𝑟 +𝑎1𝑆𝑢
^))𝐵

−[(𝜇 +𝑑 +𝜂)+
𝛼𝑀^

𝑀
−(𝑎3𝑆𝑢

^+∈ +𝛼)]𝐼                                  

 

Clearly, 𝑉0̇ = 0 at 𝐸0 = (𝑆𝑢
^,0,0,𝑀^), moreover 𝑉0̇ < 0 otherwise. Hence 𝑉0̇  is negative 

definite and then the solution starting from any initial point satisfies the conditions (15a) and 

(15b), will converge to 𝐸0  point.                                                ∎ 

Theorem (5): The 𝐸1 of system (3) is globally asymptotically stable under the following 

conditions held. 

 𝑏12
2 <

4

9
𝑏11𝑏22                                                                                              (16a) 

𝑏13
2 <

4

6
𝑏11𝑏33                                                                                              (16b) 

𝑏14
2 <

4

6
𝑏11𝑏44                                                                                              (16c) 

𝑏23
2 <

4

6
𝑏22𝑏33                                                                                             (16d) 

𝑏24
2 <

4

6
𝑏22𝑏44                                                                                              (16e) 

𝑎3𝑆𝑢 <  𝜇 +𝑑 +𝜂                                                                                       (16f) 

𝑟 < 
𝑟

𝑘
(𝐵 +𝐵∗)+𝜇                                                                                     (16g) 

Proof: Consider the following function 

𝑉1(𝑆𝑢, 𝐼,𝐵,𝑀) =
1

2
(𝑆𝑢−𝑆𝑢

∗)2 +
1

2
(𝐼−𝐼∗)2 +

1

2
(𝐵−𝐵∗)2 +

1

2
(𝑀−𝑀∗)2 

It is easy to see that 𝑉1 = (𝑆𝑢, 𝐼,𝐵,𝑀) ∈ 𝐶
1(𝑅+

4,𝑅)  in addition                  

𝑉1(𝑆𝑢
∗, 𝐼∗,𝐵∗,𝑀∗) = 0 while 𝑉1(𝑆𝑢, 𝐼,𝐵,𝑀) > 0 ∀ (𝑆𝑢, 𝐼,𝐵,𝑀) ∈ 𝑅+

4  and (𝑆𝑢, 𝐼,𝐵,𝑀) ≠

(𝑆𝑢
∗, 𝐼∗,𝐵∗,𝑀∗). 

𝑑𝑉1
𝑑𝑡

= (𝑆𝑢 −𝑆𝑢
∗)
𝑑𝑆𝑢
𝑑𝑡

+(𝐼 −𝐼∗)
𝑑𝐼

𝑑𝑡
+(𝐵 −𝐵∗)

𝑑𝐵

𝑑𝑡
+(𝑀 −𝑀∗)

𝑑𝑀

𝑑𝑡
𝑑𝑉1
𝑑𝑡

= (𝑆𝑢 −𝑆𝑢
∗)[𝜃 −𝑎1𝑆𝑢𝐵 −𝑎2𝑆𝑢𝑀 −𝑎3𝑆𝑢𝐼 −𝜇𝑆𝑢]                        

+(I− I∗)[𝑎1𝑆u𝐵 +𝑎3𝑆u𝐼 −(μ+d+η)𝐼]

  +(𝐵 −𝐵∗)[r𝐵 ( 1−
𝐵

k
)−μ𝐵 +ϵ𝐼]           

  +(𝑀 −𝑀∗)[α ( 𝑆u + 𝐼 )–γ𝑀]                      

                      

 

Furthermore, by the derivative and simplifying the 



  

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𝑑𝑉1
𝑑𝑡

= −[
𝑏11
3
(𝑆𝑢−𝑆𝑢

∗)2 +𝑏12(𝐼 −𝐼
∗)(𝑆𝑢 −𝑆𝑢

∗)+
𝑏22
3
(𝐼−𝐼∗)2]    

                −[
𝑏11
3
(𝑆𝑢−𝑆𝑢

∗)2 +𝑏13(𝐵 −𝐵
∗)(𝑆𝑢 −𝑆𝑢

∗)+
𝑏33
2
(𝐵−𝐵∗)2]  

                   −[
𝑏11
3
(𝑆𝑢−𝑆𝑢

∗)2 +𝑏14(𝑀 −𝑀
∗)(𝑆𝑢 −𝑆𝑢

∗)+
𝑏44
2
(𝑀 −𝑀∗)2]

           −[
𝑏22
3
(𝐼 −𝐼∗)2 −𝑏23(𝐵 −𝐵

∗)(𝐼 −𝐼∗)+
𝑏33
2
(𝐵 −𝐵∗)2] 

               −[
𝑏22
3
(𝐼 −𝐼∗)2 −𝑏24(𝑀 −𝑀

∗)(𝐼 −𝐼∗)+
𝑏44
2
(𝑀 −𝑀∗)2] 

 

Therefore according to the condition (16a) and (16g) we obtain that: 

𝑑𝑉1
𝑑𝑡

< −[√
𝑏11
3
(𝑆𝑢 −𝑆𝑢

∗)+√
𝑏22
3
(𝐼 −𝐼∗)]

2

−[√
𝑏11
3
(𝑆𝑢 −𝑆𝑢

∗)+√
𝑏33
2
(𝐵 −𝐵∗)]

2

 

              

      

       −[√
𝑏11
3
(𝑆𝑢 −𝑆𝑢

∗)+√
𝑏44
2
(𝑀 −𝑀∗)]

2

−[√
𝑏22
3
(𝐼 −𝐼∗)−√

𝑏33
2
(𝐵 −𝐵∗)]

2

     

 −[√
𝑏22
3
(𝐼 −𝐼∗)−√

𝑏44
2
(𝑀 −𝑀∗)]

2

                                                                  

 

Where 

𝑏11 = 𝑎1𝐵
∗ +𝑎2𝑀

∗ +𝑎3𝐼
∗ +𝑎4 +𝜇 ,    𝑏12 = 𝑎3𝑆𝑢 −𝑎1𝐵

∗ −𝑎3𝐼
∗ −𝑎4

𝑏22 = 𝜇 +𝑑 +𝜂 −𝑎3𝑆𝑢,    𝑏23 = 𝑎1𝑆𝑢 +𝜖,   𝑏13 = 𝑎1𝑆𝑢                                          

𝑏33 =
𝑟

𝑘
(𝐵 +𝐵∗)−(𝑟 −𝜇),   𝑏14 = 𝑎2𝑆𝑢 −𝛼,   𝑏24 = 𝛼,   𝑏44 = 𝛾           

 

Clearly, 
𝑑𝑉1

𝑑𝑡
< 0, and then 𝑉1 is a Lyapunov function provided that the given condition (16a) 

and (16g) held . Therefore, 𝐸1 is globally asymptotically stable.       ∎ 

6. 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  

𝜃 = 1.2 , 𝑎1 = 1.5 ,  𝑎2 = 1.19 ,  𝑎3 = 0.15 , 𝑎4 = 1.15 , 𝜇 = 0.5   
𝑑 = 0.3 , η = 0.5 , r = 0.3 , k = 0.1 , ϵ = 0.02 , α = 0.1 , γ =  0.3

              (17) 

The dynamical behaviors of system (1) converge to the 𝐸1 = (0.54,0.53,0.46,0.03,0.35) and 

the investigation of the global stability, are shown in Figure 1. starting from different sets of 

initial points. 



  

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Figure 1. The trajectory of system (1) approaches asymptotically to a globally stable to endemic equilibrium 

point of system (1) for the parameter set in eq. (17), started from different sets of initial point. (a) for Su(t), (b) 

for I(t), (c) for Sa(t), (d) for B(t), (e) for M(t). 

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 𝑎4 = 0, 𝑘 = 0.01 the solution of system (1) 

converge to the disease free equilibrium point is shown in the following Figure 2. 



  

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Figure 2. The trajectory of system (1) approaches asymptotically to a globally stable to E0 =

(1.21,0,1.18,0,0.40), started from different sets of initial point. (a) for Su(t), (b) for I(t), (c) for Sa(t), (d) for 

B(t), (e) for M(t).  

It is easy by keeping fixed the parameters values given in Eq. (17) with putting  𝜇 𝑎𝑛𝑑 𝑘 in 

the range 0.5 ≤ 𝜇 ≤ 16.8,0.001 ≤ 𝑘 ≤ 0.01 and 𝑎4 = 0, the solution of the system (1) 

converge to 𝐸0 = (1.21,0,1.18,0,0.40) is shown in the typical figure given by Figure 3. 

below. 



  

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Figure 3. The solution of system (1) for the data (17) (a) for μ = 0.5 and k = 0.1  (b) for μ = 0.7 and k =  0.01. 

According to 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 𝑎3 is investigated.  System (1) is solved numerically by choosing the parameters 

values given by equation (17) with  𝑎3 = 3.15,6.15,9.15  respectively and then the 

trajectories of the system (1) are shown in  Figure 4. 

 

 

Figure 4. The solutions of the system (1) (a) for a3 = 3.15  (b) for a3 = 6.15  (c) a3 = 9.15 

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. 



  

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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 𝑎1 = 5.5,20.5,35.5 respectively and then the 

trajectories of the system (1) are shown in the  Figure 5. 

 

 

Figure 5. Time series of solutions of the system (1) (a) for a1 = 5.5  (b) for a1 = 20.5  (c) a1 = 35.5 

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. 

7. 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 𝑆𝑢, infected individuals 𝐼, aware susceptible  

individuals 𝑆𝑎 and people carrying bacteria 𝐵. 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 (𝐸0) exists when 𝑎4 = 0 and 𝐼 = 𝐵 = 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 (𝐸1) exists if the conditions 



  

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(9a-9b) are held and locally stable if the conditions (13a-13b) are held more than it is globally 

stable if the conditions (16a-16g) are held. Finally, to understand the effect of varying each 

parameter on the global system (1) and confirm our above analytical results, system (1) has 

been solved numerically for different sets of initial points and different sets of parameter 

given by equation (17), and the following observation are made: 

1. For the set of hypothetical parameters values given by Eq. (17), the system (1) 

approached asymptotically to the global stable endemic equilibrium point, 𝐸1 . 

2. Equation (17), when we set the following values 𝑎4 = 0 and 𝑘 = 0.01  the system (1) 

converge to disease free equilibrium point 𝐸0 and satisfied the global stability. 

3. If the contact rates 𝑎3 and 𝑎1 increase, the solution of system (1) still converge to the 

endemic equilibrium point.  

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