Introduction

Hearing loss may result from genetic causes, complications at birth, certain infectious diseases, chronic ear infections, the use of particular drugs, exposure to excessive noise and aging, which is represent a major global health problem. According to the World Health Organization (WHO), about 360 million people worldwide have disabling hearing loss, and 32 million of tríese are children ^{[1]}. There are many infections and contagious diseases related to hearing loss such as mumps. Mumps is an enveloped, single-stranded RNA virus belonging to the family paramyxoviridae and causes an acute infectious disease mainly in children and young adults ^{[2]}. Mumps is transmitted through infected respiratory secretions and is highly contagious ^{[3]}. The most common clinical manifestations of infection include a u-like illness and bilateral swelling of the parotid glands. Mumps infection occasionally induces the potential for complications such as pancreatitis, orchitis, ophoritis, aseptic meningitis, encephalitis and sensorineural hearing loss. Hearing loss due to mumps is thought to be unilateral and profound with rapid onset ^{[4]}.

On the other hand, one of the most social factor of hearing loss is noise exposure (eg, through personal music players). This is a major cause of hearing loss worldwide ^{[5]}. In addition, the characteristic pathological feature of noise-induced hearing loss is the loss of auditory sensory cells in the cochlea. Because these hair cells cannot regenerate in mammals, no remission can occur, prevention of noise-induced hearing loss is the only option to preserve hearing ^{[6]}.

Abnormalities or damage in the structure inside the inner ear can result by many diseases that can load to the hearing loss. An example of such a disease is the Alport syndrome ^{[7]} and Meniere's disease ^{[8]}. For these raisons, many mathematical models have been proposed to model the function and the dysfunction of the inner ear by using partial differential equation (PDEs) ^{[7}^{,}^{8}^{,}^{9}^{,}^{10}^{,}^{11]}. The epidemiological models in the beginning has started by Graunt ^{[12]} then described by Kermack and Mckendric ^{[13]} by considering the total population into three classes namely susceptible (S) individuals, infected (I) individuals and recovered (R) individuals which is known to us as SIR epidemic model ^{[14}^{,}^{15}^{,}^{16]}. Today, this SIR epidemic model is very important in analysis of many diseases. Referring to the simplest version of the most classical epidemiological model for directly-transmitted infectious diseases ^{[17}^{,}^{18}^{,}^{19}^{,}^{20}^{,}^{21}^{,}^{22]}, as well as, the social and epidemiological factors of hearing loss existing in the biological studies and in the literature ^{[23}^{,}^{24}^{,}^{25]}, we proposed a new mathematical model by using (ODEs) that describe the time dependence of the dynamics of hearing loss by considering the contagion factor is the mumps virus and the social factor is the exposure to noise. This model is described by the following system:

The population is divided into three epidemiological classes that are: H(t) is the number of susceptible individuals at time t (normal hearing), L(t) is the number of infected individuals at time t (loss of hearing), and R(t) is the number of removed individuals at time t (recovered hearing). Further *Λ* is the recruitment rate of the population, *μ* is the natural death rate of the population, *β* is the transmission rate due to social contagion of hearing loss (mumps), *ε* is the non-contagion risk of hearing loss due to noise exposure and *γ* is the recovery rate of the infective individuals.

The rest of paper is organized as follows. In the next section, positivity and boundedness of solutions are studied. In Section 3, the basic reproduction number is derived, also the local and the global asymptotic stability of the equilibria are analyzed. The numerical results are given in Section 4. Lastly, we give a conclusion of our results in Section 5.

Positivity and boundedness of solutions

In this section, we will establish the positivity and boundedness of solutions of model (1), which proves that our model is well posed.

**PROPOSITION 2.1.** All solutions starting from non-negative initial conditions
exist for all t > 0 and remain bounded and non-negative.

**PROOF.** For the positivity, we show that any solution starting in non-negative
orthant,

This proves the positivity of solutions. Now, we prove that the solutions are bounded. We defined T(t)= H(t)+L(t) By non-negativity of the solution, it follows that;

Then

From the last equation (1) and since L is bounded, we deduce that R is bounded. This completes the proof.

Stability analysis of hearing loss model

Local stability of endemic equilibrium point for ε > 0

The following theorem presents the existence and uniqueness of endemic equilibrium if R_{0} > 1.

**THEOREM 3.1.** System (1) has a unique equilibrium point *,* which is locally asymptotically stable.

**PROOF.** By equalizing to zero the rights members of the system (1), we find one
endemic point that exists for above model:

Where

With

The Jacobian matrix of the system (1) is given by:

The characteristic equation of the endemic equilibrium point is given by;

Note that coefficient

are both positive if

From the Routh-Hurwitz theorem given in ^{[26]}, all roots of equation (5) have negative real parts. Consequently E * is locally asymptotically stable whenever R_{0} > 1.

Global stability of endemic equilibrium point for ε > 0

The following theorem discusses the global stability of the endemic equilibrium.

**THEOREM 3.2.** The endemic equilibrium

**PROOF.** Consider the following Lyapunov functional;

Where

To simplify the presentation, we shall use the following notation: *H=H(t)* and *L=L(t).*

Note that

And

Hence,

Then, we obtain the following equation:

From equation (6), we have:

Since,

We have

Thus

if and only if H =

From LaSalle invariance principle ^{[27]}, we conclude that

Stability analysis for special case ε = 0

Notice that the system (8) has a basic reproductive number

R_{0} represents the average number of secondary infections caused by an infective individual introduced into a group of susceptible. By equalizing to zero the rights members of the system (8), we find two equilibrium points that exists for above model:

The endemic equilibrium point exist only when

i.e the infection rate must be greater than the death rate of the infected individuals or R_{0} > 1.

Local stability of equilibria

The following theorems discuss the local stability of the equilibrium point.

**THEOREM 3.3.**

1. If R

_{0}< 1, then the disease-free equilibrium,*E*_{f}is locally asymptotically stable.2. If R

_{0}> 1, E_{f}is unstable.

**PROOF.** The Jacobian matrix evaluated in the disease-free equilibrium

is given by

Whose eigenvalues are

then R_{0} < 1 and therefore E_{f} is locally asymptotically stable.

The disease-free equilibrium point is unstable if

which translate into *R*
_{
0
} > 1.

Now, we focus on local stability of the endemic infection equilibrium *E*
^{
*
}
*.*

**THEOREM 3.4.**

1. If R

_{0}> 1, E* is locally asymptotically stable.2. If R

_{0}< 1, then the endemic equilibrium*E*^{*}does not exist.

**PROOF.** By substituting the endemic equilibrium
*E*^{*}*=
(H*^{*}*,
L*^{*}*,
R*^{*}*)* in the
Jacobien matrix of the system (8)

The characteristic equation of the endemic equilibrium point is given by

Clearly when R_{0} > 1, both

And

then all roots of the characteristic equation have negative real parts. Consequently
*E*^{*} is locally
asymptotically stable.

Global stability of equilibria

In this section, we establish the global stability of the equilibria. Firstly, we have the following

**THEOREM 3.5.** The disease-free equilibrium
*E*_{f} is globally
asymptotically stable when *R*_{0}
< 1.

**PROOF.** Consider the following Lyapunov functional

Where

and calculating the time derivative of V_{1}(t) along the positive solution of system (8), we get;

Where

Since R_{0} ≤ 1, we have

Thus, the disease-free equilibrium *E*_{f} is
stable, and

if and only if *H = H*_{f} and
*L*(*R*_{0}-1)=0. We discuss two
cases:

> If R

_{0}< 1, then L = 0.> If R

_{0}= 1. From*H = H*_{f}and the first equation of the system (8), we have

Then *βH*_{f}*L* = 0, Hence
*L* = 0.

From LaSalle invariance principle ^{[27]}, we
conclude that *E*_{f} is globally
asymptotically stable.

Note that the disease-free equilibrium *E*_{f} is
unstable when *R*_{0} > 1.

Now, we establish a set of conditions which are sufficient for the global stability of the endemic equilibrium E^{*}.

**THEOREM 3.6.** The endemic equilibrium E* of the system (8) is globally
asymptotically stable.

**PROOF.** See the proof of theorem 3.2, and we replace *ε* by zero,
we conclude that *E*^{*} is globally
asymptotically stable.

Numerical simulations

In this section, we show the numerical simulations and the graphs of system (1) to illustrate
the different result obtained for each of the two cases *ε* > 0
and *ε* = 0 previously analyzed.

We choose the following data set of system (1) as follows: *Λ*= 0.5, *μ*= 0.1 and *γ*= 1/17. By using the values of *Λ*, *μ*, *
γ
* and

*ε*= 0.0025 > 0

^{[28]}, we find

*β*= 0.1433 for R

_{0}> 1 already defined for mumps disease

^{[29]}. Therefore, according to Theorem 3.2, the system (1) has a unique endemic equilibrium which is globally asymptotically stable. So, the solution of system (1) is persistent in population and converge to

*ε*= 0 which means the hearing loss result only from mumps diseases. So, the system (1) has a disease free equilibrium E

_{f}= (5,0) which is globally asymptotically stable when R

_{0}< 1, then the disease dies out. Numerical simulations illustrated our results (see Figure 2). For R

_{0}= 4.4 > 1, the endemic E* is globally asymptotically stable which satisfy Theorem 3.6, (see Figure 3). In Figure 4, we use different values of

*e*> 0 and we find the solution of system (1), in this case, we observe that when we increase the value of

*e*the number of normal hearing individuals decrease and the numbers of hearing loss increase.

Conclusions

In this paper, we have presented a mathematical model of hearing loss based on a nonlinear system of differential equations. We analysis the hearing loss resulting from two factors, the first factor is contagious due to Mumps disease and the second is social caused by exposure to noise. By analysis the model, we have proved the existence, positivity and the boundedness of solutions of the problem, which implies that the model is well posed. We have shown in the case of *ε* = 0 that the disease free equilibrium is globally asymptotically stable if the basic reproductive number R_{0} < 1 and the endemic point is globally asymptotically stable when R_{0} > 1. In the case of hearing loss with the both factors mumps and noise *ε* > 0 the system has a unique endemic point exists and is globally asymptotically stable, which means that the disease persists in the population. In addition, the simulation of this model provides that the number of individuals with hearing loss increase when we introduce the risk factor noise.