THE ROLE OF HOUSEFLIES IN CHOLERA TRANSMISSION

In this paper, we propose and analyse a mathematical model that describes the dynamics of Cholera. The main aim of this model is to investigate the role of houseflies in the transmission of Cholera. Our analysis showed that the disease free equilibrium is globally asymptotically stable whenever the basic reproduction number is less than unity; and unstable otherwise; and our model posses only one endemic equilibrium which is locally asymptotically stable whenever basic reproduction number is greater than unity. Our sensitivity analysis showed that the basic reproduction number is very sensitive to ingesting vibrios rate from aquatic environment by vectors, the rate of contribution to V. cholera in the aquatic environment and the death rate of vector and death rate of vibrios in aquatic environment which indicates that the vector (i.e. the houseflies) play a very important role in the transmission procedure.Numerically, we shown that the rate of water contamination by infectious people shedding V. cholera into the environment has no impact in the infection because it depends on both bacteria shedding of the infected individuals and the level of sanitation in the environment and since the environment is safe, then it is obviously has no effects. In addition, if the contact rate of vectors with contaminated water is high in the presence of increased contribution of each infected vector to the aquatic environment then cholera will persist in the population. Therefore, to obtain a significant and effective control, the contribution of each infected vector to the aquatic environment and the rate of exposure to contaminated water must be reduced.


INTRODUCTION
Enteric diseases are considered to be one of the greatest threat to human race, since it causes mortality of millions of people as well as huge impact on social and economic aspects of populations [15]. Enteric diseases are defined as infections caused by viruses or bacteria that enter the body through the mouth or intestinal system, primarily as a result of eating, drinking and/or digesting contaminated foods or liquids. Cholera is a waterborne enteric disease which has a main symptom; acute, watery diarrhea that caused by a bacterium (gram-negative rod), Vibrio cholerae. It can be developed to severe watery diarrhoea with vomiting. If people are not treated promptly, they can lose large amounts of fluid and salts which lead to severe dehydration and death within hours [22]. The species V. cholerae is subdivided into serogroups, which are toxigenic (O1 & O139) and non-toxigenic (non-O1) [10]. Strains that have the potential to cause epidemic cholera and thus are of public health significance belong to serogroups O1 or O139 and produce cholera toxin (CT) [24,22]. In its most severe form, cholera is one of the rapid lethal infectious diseases which can lead to death within hours, especially in places where drinking water is unprotected from faecal contamination. These characteristics of cholera have yielded a reputation that cause fear. However, with appropriate treatment, mortality can be kept low.
In 2014, 190549 cases were notified from 43 countries with 55% in Africa and 2231 deaths were reported in that year [27]. In 2015, a total of 172454 cholera cases were reported in 42 countries, 41% of which in Africa, 37% in Asia and 21% in Hispaniola (Central American island) [26]. In the same period, 1304 deaths were reported [26]. During 2016, 132 121 cases were reported from 38 countries, including 2420 deaths [26]. The year 2017 was remarkable for cholera because it marked 200 years since the onset of the first recognized cholera pandemic in 1817, while the current seventh pandemic continues as the longest ever recorded [25]. Globally, in 2017, 71 countries provided data on cholera with 34 countries reported a total of 1 227 391 cases and 5654 deaths and the remaining countries reported no cases. 84% of all suspected cases reported were in Yemen with a total of 1 032 481 cases and 2261 death [25].
The prevalence of cholera depends on numerous environmental and biological variables, including seasonal environmental drivers, host immunity and infectivity of the bacteria [9,24].
Cholera is usually transmitted through faecally contaminated water, hands or food, and remains an ever-present risk in many countries. New outbreaks can occur where water supply, sanitation, food safety, and hygiene are insufficient. The dynamics of cholera is much more complex as it involves multiple interactions between the human host, other organisms, and the environment [24]. The transmission of cholera include both indirect (i.e., environment-to-human) and direct (i.e., human-to-human) routes [24]. The indirect exposure occurs when people ingest water or food from the environment that is contaminated by the vibrios. The direct transmission may occur when the vibrios are transmitted from an infected person directly to a healthy person by close contacts (such as shaking hands or hugging) or by eating food prepared or consumed by individuals with dirty hands [9,23,21]. Some studies show that the infected person or vector typically shed vibrios in their stool for only 1 day, at approximately 10 3 vibrios per gram of stool [24]. Therefore, vectors (e.g. house flies ) can play an important role in the transmission of the cholera. The mechanism of transmitting cholera infection from house flies (Musca domestica fly [14]) among humans is such that the flies feed, crawl and lay eggs on human food [14,28,11].
To understand the complex dynamics of cholera, several mathematical models have been developed [6,9,17,23]. Capasso and Paveri-Fontana [6]  contribution is the division of the environment into two sub-environments according to the concentration of cholera vibrios. The V. cholerae is associated with contaminated water and food such as rivers, dams, wells, and ponds [9,23]. In contrast, there are some places in the environment such that it can be considered as uncontaminated and safe places (households and market places). Fundamental in our assumption is that people are well informed of the development and severity of the disease outbreak, thus will take action to reduce contact with other individuals and/or the contaminated environment. However, they can get infection from safe places as the vectors can transmit the infection to them..

MODEL FORMULATION
To formulate the model we consider two host populations, human population (N h ) and vector population (N v ). Since the model incorporates the indirect environmental transmission, we add the dynamics of the concentration of free living Cholera vibrios in safe and unsafe environments. Let the human host population be divided into the following, susceptible individuals S h (t) ,those who are infected with cholera I h (t) , and those who are recovered and have permanent immunity R h (t). This implies: Similarly, let the vector population have two categories, susceptible vector S v (t), and infected vector I h (t), such that The total population for humans and vectors are assumed to be a constant, which is a reasonable assumption for a relatively short period of time and for low-mortality diseases such as cholera.
Note that we have considered the infection through contact with environmental free Cholera vibrios. As it is the case for most models involving free-living pathogens in the environment [9,23,17,30], the environmental-related forces of infection,e.g. β 1 S h B κ 1 +B , is modelled using Michealis-Menten or Holling type II functional responses. The constants κ 1 , κ 2 represent the minimum amount of vibrios in the environment capable of ensuring 50% chance of contracting the disease.
The parameters used for system (1) and their biological interpretations are giving in Table ( (1) with non-negative initial data will remain non-negative for all time t > 0. This yields to the following theorem [5].
> 0 is the integration factor. Hence integrating the last relation with respect to t we get: so that the division of both sides by ρ(t) yields The same arguments can be used to prove The dynamics of model (1) is dynamical system in the biological feasible compact set The disease-free equilibrium (DFE) for the cholera model (1) is given by To compute the basic reproduction number of the model, we use the standard method of the next generation matrix developed in [29,12] by separating the infected states form the uninfected states. Here, the associated next generation matrices are given by: The expression of the basic reproductive number is given by: Using Theorem 2 in [29], the following result is established: Lemma 3.2. The DFE of system (1) is locally asymptotically stable (LAS) whenever R 0 < 1, and unstable whenever R 0 > 1. However, to guarantee that the disease will be eliminated independently of the initial sizes of host populations, the DFE must be global asymptotically stable (GAS) of the DFE when R 0 < 1 as showing in the following theorem.  [2] as and solving these three ordinary differential equations gives which is clearly an M-matrix. Meanwhile, we find and since 0 ≤ S h ≤ N h and 0 ≤ S v ≤ N v , then it follows thatĜ(X, Z) ≥ 0 Thus, P 0 is GAS whenever R 0 < 1. This lemma is considered as an important result because it eliminates the possibility for the model (1) to have non-trivial equilibriums on the boundary . Therefore, the cholera model (1) could have a unique interior (endemic) equilibrium with the disease being present in all the populations under consideration [20]. This lemma with the existence and uniqueness of interior equilibrium for system (1) related sub-models suggested in [4,20,19,5], yields to the following conjecture [20] Conjecture 3.5. Assume that R 0 > 1 for system (1). Then there exists a unique interior (endemic) equilibrium.

The endemic equilibrium of the model is denoted by
and it satisfies the following: The endemic equilibrium is LAS whenever R 0 > 1 [4,5] and this will be shown numerically at a later stage.

Bifurcation analysis of the model.
To study the possibility of backward bifurcation, we use the theorem in Castillo-Chavez and Song [8]. Introducing (1) becomes: Consider the case when R 0 = 1 and suppose that φ = λ 2 is chosen as a bifurcation parameter.
Then, R 0 can be seen, in terms of the parameter . The Jacobian of the system (10) at the disease-free equilibrium is given by the following: Calculation of the eigenvectors of J φ . It can be shown that the Jacobian of the system (10) at Φ = λ 2 (denoted by J φ ) has a right eigenvector given by and a left eigenvector given by (10), it can be shown that:

Computation of a and b: From system
and and all the other second-order partial derivatives are equal to zero. Thus, we can compute the coefficient a and b defined in (thereom 4.1 in [8]), that is, and since w 1 &w 4 are negative,then a < 0. and Therefore, we have the following result: Theorem 3.6. The direction of the bifurcation of system (10) (or system (1)) at R 0 = 1 is forward. Since the bifurcation parameter changes from negative to positive, then the DFE changes its stability from stable to unstable. Therefore, a negative unstable equilibrium becomes positive and locally asymptotically stable Theorem 3.6 proves that the unique endemic point is locally asymptotically stable.

SENSITIVITY ANALYSIS (SA) OF THE BASIC REPRODUCTION NUMBER WITH RE-SPECT TO THE MODEL PARAMETERS
One of the most important concerns about the infectious disease is its ability to invade a population. The useful and valuable quantity which helps determine whether or not an infectious disease can spread through a population is basic reproduction number (R 0 ) [29,7]. R 0 measures whether a disease can persist in a population. When R 0 is less than 1, on average each infected individual infects less than one individual, and the disease will die out. In contrast, when R 0 exceeds unity there is an exponential rise in the number of cases over time, and an epidemic results [29,7]. Therefore, we studied the sensitivity analysis of the basic reproduction number, with respect to the model parameters in order to discover parameters that have a high impact on R 0 and should be targeted by intervention strategies. There are many ways of conducting sensitivity analysis, all resulting in a slightly different sensitivity ranking [18]. We used the normalized forward sensitivity index which is also called elasticity. The normalized forward sensitivity index of a variable with respect to a parameter is defined as the ratio of the relative change in the R 0 to the relative change in the parameter [3,16,18]. It is given by: Given the explicit formula (3) for R 0 , one can easily derive an analytical expression for the sensitivity of R 0 with respect to each parameter that comprise it. The obtained values are described in Table 2, which presents the sensitivity indices for the baseline parameter values for R 0 < 1 and R 0 > 1 given in

NUMERICAL SIMULATION
In this section, we perform a numerical simulation of model system (1) to confirm our analytical results and to illustrate the asymptotic behaviour of the model. At this stage, we solve the model numerically with three sets of initial conditions with total human population is 10000 and vector population is 30000. We choose a set of parameter values in model system (1) according to Table 4 where some of the parameter's values were obtained from literature, and some of them were assumed or made varying in order to study their role.
The GAS of the disease-free equilibrium P 0 demonstrated in Theorem 3.2 and the existence and stability of a unique endemic equilibrium as stated in Conjecture 1 for the model is shown on  Varying the values of β 1 , λ 1 and λ 2 (the ingesting vibrios rate from the safe environment to human, vectors, and ingesting vibrios rate from the unsafe environment to vectors respectively) Numerical simulation shows that the increase in the value of β 1 leads to an increase in the number of both infected human and infected vector with the effect is more in human population as seen in Figure 4.
It can be seen from Figure

CONCLUSION
We develop a general model for the dynamics of cholera that incorporates an indirect transmission of V. cholera to yhe environmental reservoir. The proposed model divided the environment into two sub-environments according to the concentration of V. cholera. Our analysis of the model showed that the disease free equilibrium is globally asymptotically stable when R 0 is less than unity; and unstable when R 0 is greater than unity, and our system posses only one endemic equilibrium and we showed that it is locally asymptotically stable when R 0 is greater than unity since the direction of the bifurcation is forward.
Our sensitivity analysis showed that R 0 is sensitive to all of model parameters either positively or negatively, and the most influential has been the ingesting vibrios rate from aquatic environment by vectors, the rate of contribution to V. cholera in the aquatic environment and the death rate of vector and death rate of vibrios in aquatic environment which indicates that the best control strategy is by eliminating vector populations and by sanitizing the aquatic environment.
Numerical simulations were used to examine the effect of all of the parameters of the model.
The results showed that β 1 , λ 1 and α 1 have a positive effects in disease transmission as the increase in their values contribute significantly to the spread of the cholera infections in the system. Also, the simulations showed that the rate of water contamination by infectious people shedding V. cholera into the environment (α 2 ) has no impact in the infection because it depends on both bacteria shedding of the infected individuals and the level of sanitation in the environment and since the environment is safe, then it is obviously has no effects. In addition, if the contact rate of vectors with contaminated water (λ 2 ) is high in the presence of increased contribution of each infected vector to the aquatic environment (α 1 ) then cholera will persist in the population. Therefore, to obtain a significant and effective control, the contribution of each infected vector to the aquatic environment and the rate of exposure to contaminated water must be reduced. Moreover, to reduce the epidemic's peak other interventions are needed.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.