A study of stability of SEIHR model of infectious disease transmission

: We develop in this paper a Susceptible Exposed Infectious Hospitalized and Recovered (SEIHR), spread model. In the model studied, we introduce a recruitment constant, to take into account the fact that newborns can transmit disease. The disease-free and endemic equilibrium points are computed and ana-lyzed. The basic reproduction number R 0 is acquired, when R 0 ≤ 1, the disease dies out and persists in the community whenever R 0 > 1. From numerical simulation, we illustrate our theoretical analysis.


Introduction
Infectious diseases are a public health problem for populations around the world. For a better understanding of the dynamics of these infectious diseases, several tools are used among which we have the mathematical modeling [3]. Mathematical modeling is a decision support tool used in several elds such as economics, biology, physics and medicine. We have two types of transmission in terms of infectious diseases. The diseases with horizontal transmission whose infection requires the presence of a intermediate host generally a vector and a de nitive host which is the human in general and the diseases with vertical transmission of which the infection is inter-human. In this work, we are interested in the study of a continuous model of a vertically transmitted disease, a Susceptible Exposed Infectious Hospitalized and Recovered (SEIHR), spread model. In [12,21,22], the authors formulated and studied mathematical models giving the dynamics of the transmission of infectious diseases. They study the stability of steady states when the basic reproduction rate R is less than one and greater than one. Also, they study the impact of quarantine on the dynamics of infectious disease transmission (this method is extensively applied to the outbreak of Corona Virus Diseases 2019 ).
Corona Virus Diseases 2019 (COVID- 19) is an infectious disease with vertical transmission, the rst case of which appeared towards the end of 2019 in Wuhan, China [13,20,25]. There exist a wide variety of models that can be used to describe the evolution of an epidemic. Main standard is held by the so called compartmental models, i.e., the family of SIR based models (SIR, SEIR, SIRS, etc) [9,10]. These models consist of a set of di erential equations that take into account transitions between di erent compartments. The rst classical SIR model was proposed by Kermack and McKendrick [11]. Due to the recent coronavirus pandemic , most of the scienti c community is dedicated to study its behaviour, both by using SIR based models and introducing new ones. In [1], a stochastic term is introduced in the system of di erential equa-tions to simulate noise in the detection process. D. K. Mamo [18], proposes an ordinary di erential equation system of the SHEIQRD (Susceptible-Stay-at-Home-Exposed-Infected-Quarantine-Recovered-Death) type to describe the transmission of Corona Virus Disease 2019 . The author calculates the basic reproduction rate R associated with the model and shows that when R < , the disease-free equilibrium is globally asymptotically stable and when R > the endemic equilibrium is globally asymptotically stable. In this paper, we formulate and study a mathematical model of the SEIHR type of the transmission of the infectious disease. We propose a study of the overall stability of equilibria states when the basic reproduction number associated to the mathematical model is less than or greater than .
The paper is organized as it follows: In the section 2, we are interested in the formulation of the mathematical model. We give a description of the model, the equilibria states of the model as well as the basic reproduction number associated to the model. The section 3 is devoted to the study of the stability of the model. We rst of all look at the positivity as well as the boundedness of the solutions and we study the global stability according to the value of the basic reproduction number. In the section 4, we propose a numerical simulation to illustrate our theoretical results. We end with a conclusion and some research perspectives.

Mathematical model
In this section, we give in subsection 2.1, the description and the formulation of mathematical model which will be investigate. The subsection 2.2 is devoted to the determination of the equilibria states and the basic reproduction number.

. Description of model
According to the known characteristics of infectious diseases, we assume that each person is in one of the following compartments: -Susceptible (denoted by S): The person is not infected by the disease pathogen. -Exposed (denoted by E): The person is in the incubation period after being infected by the disease pathogen and has no visible clinical signs. The individual could infect other people but with a lower probability than people in the infectious compartments. -Infectious (denoted by I), this compartment will be divided in two compartments which was: Infectious that will be detected (denoted by I d ), and the Infectious that will not be detected (denoted by Iu). The infectious that will be detected starts developing clinical signs and will be detected and reported by the authorities. After this period, people in this compartment are taken in charge by sanitary authorities and we classify them as Hospitalized. The infectious that will not be detected can infect other people and may start developing clinical signs but will not be detected and reported by authorities. After this period, people in this compartment pass to the Recovered state (the person who survive are in the compartment denoted by R Iu ). -Hospitalized or in quarantine at home (but detected and reported by the authorities) that will recover (denoted by H R ). At the end of this state, a person passes to the Recovered state (denoted by R I d ). -Hospitalized that will die (denoted by H d ): The person is hospitalized and still infect other people. After the general description given above, we get the transfer diagram at follow.
The function f (S, X) is given by: By using the Figure 1, we obtain the dynamical system: The parameters used in the System 1 are given in the Table 1.

. Equilibrium and basic reproduction number
In this subsection, we determine the equilibria points and the basic reproduction number associated with System (1).  (1) is equivalent the following system: Let E = (S, E, I, I d , Iu , H R , H d ) be the equilibrium point of System (2). Then, System (2) can be rewritten as follows: ( From the third equation of System (3), we have: By adding the rst two equations of System (3) and the relation (4), we get: By using the others equations of System (3) we obtain: Let E and E * be respectively the disease-free and endemic equilibrium points of System (2). The diseasefree equilibrium correspond to the case were there is not infected individual. In this case, we have I = and the disease-free equilibrium is given by: We design by I * the infectious at endemic equilibrium point, so with We also make this assumption:

Remark 2.2.
In biological view point, hypothesis H tells us that at endemic equilibrium the number of severely hospitalized individuals is more than this number at any other time. As well as less severe hospitalized, infected detected and infected undetected. Technically, hypothesis H allows us to conclude that the derivative of the candidate Lyapunov function is negative and therefore the overall stability of the endemic equilibrium.
In this work, the basic reproduction number associated to System (2) is denoted by R .
Proposition 2.1. The basic reproduction number R associated to the System (2) is de ne by: Proof. For this proof we use the technique of Van den Driessche and Watmough [23]. In our model the infected classes correspond to states E, I, I d , Iu, H R and H d . Thus, we can rewrite system (2) aṡ F is the rate of appearance of new infections in each class, and V is the rate of transfer of individuals out of (for positive values) or into (for negative values) compartment by all other means.
Hence, in our case, we have that The jacobian matrix of F and V at the disease-free equilibrium E are given by The matrix V is invertible (non-zero determinant) and its inverse V − is de ned by: were So the next-generation matrix −FV − is given by: The basic reproduction ratio is given by R = ρ(−FV − ), the spectral radius of next-generation matrix −FV − . In our case, we have that .

Positivity and boundness of solution
In this subsection we prove the positivity and boundedness of the solutions of system (1) with the initial conditions (14).  (14). Then, (S(t) > , E(t) > , Proof. The system (1) can be rewrite as follows: where f (X(t)) = (f (X), . . . , f (X)) T . We note that Then it follows from the Lemma 3.1 that R + is an invariant set.

. Global stability of the equilibria
This subsection is devoted to the studied of global stability of disease-free equilibrium E , when R < and the global stability of endemic equilibrium E * , if R > .
The matrix M associate to the linearised system (19) is given by: and the linearization system (19) can be rewrite at followṡ where Y = (E, I, The matrix V is an invertible matrix and it invert V − is given by the relation (13). We can also see that F ≥ and V − ≥ .
Thus, R = ρ(−F V − ) < and from the theorem of Varga (see [24]) the matrix M is asymptotically stable. The eigenvalue of matrix M has negative real part, by a standard comparison theorem [14], when t → +∞, E → , I → , I d → , Iu → , H R → and H d → for system (19) and substituting E = , I = , I d = ,  (2), when R < . Therefore diseasefree equilibrium E is globally asymptotically stable in the positively setΓ when R < .
Theorem 3.2. The endemic equilibrium E * of system (2) is globally asymptotically stable, when R > .
Proof. Let E * = (S * , E * , I * , I * d , I * u , H R , H d ) be the endemic equilibrium of system (2). From system (2), we have: Let de ne the function ψ on R * + by The function ψ is non-negative for all x ∈ R * + . Let us consider the Lyapunov candidate function de ne by: Now, we have to di erentiate the function V with respect to the time. Let us computeV S : By using the rst equation of (22), we have: Let us computeV E : From the second equation of system (22), we get: By adding the equations (24) and (25), we obtain: Let us computeV I :V By using the third equation of (22), we get: Then, by the asymptotic stability theorem [15], the endemic equilibrium E * of System (2) is globally asymptotically stable.

Numerical result
In this section, our aim is to discuss numerically, the dynamic of the di erent compartment of system (1). We have to distinguish two, at rst we illustrate numerically the dynamic of the di erent class when the basic reproduction number R is less then and we also present the dynamic of the di erents class when the basic reproduction number R is more then . . estimated β I .
[ ] . estimated Figure 2 gives the dynamics of susceptible and exposed humans, when the basic reproduction R < . The black and green curves, respectively give the dynamics of susceptible humans and exposed humans. We see the decrease in exposures and an increase in susceptible, which means the disease tend to disappear from the population. Figure 3 shows us the dynamics of infectious humans, when R < . The black, green and blue curves respectively give the dynamics of infectious humans, detected infectious and undetected infectious. We see the decrease of the infectious class, which means that the disease is tending to disappear from the population. Figures 4 and 5 respectively give the dynamics of hospitalized and recovered humans when R < . In the Figure 4, the black and green curves give respectively dynamic hospitalized H R and the hospitalized H d , in the gure 5 the black and green curves give respectively the dynamic of recovered R d and recovered Ru. The decrease in the curves observed at the level of the di erent gures means that the disease tends to disappear from the population.   Figure 6 gives the dynamics of susceptible and exposed humans, when the basic reproduction R > . The black and green curves, respectively give the dynamics of susceptible humans and exposed human. The state of the curves tells us that the disease remains persistent in the population. Figure 7 shows us the dynamics of infectious humans, when R > . The black, green and blue curves respectively give the dynamics of infectious humans, detected infectious and undetected infectious. We see the increase in infectious class, which means that the disease remains persistent in the population. Figures 8 and 9 respectively give the dynamics of hospitalized and recovered humans when R > . In the Figure 8, the black and green curves give respectively dynamic hospitalized H R and the hospitalized H d , in the gure 9 the black and green curves give respectively the dynamic of recovered R d and recovered Ru. The increase in the curves observed at the level of the di erent gures means that the disease remains persistent in the population.

Conclusion
In this paper, we have studied a continuous mathematical model which modeling an infectious disease. The mathematical model studied in this work is the SEIHR (Susceptible-Exposed-Infectious-Hospitalized-Recover) type. We have calculated the basic reproduction number (R ) associated with the model, the basic properties of the model (positivity of the solutions and their bounditude). We also prove the stability of the model according to the values of R . We used the Varga theorem [24] to proved that the disease-free equilibrium E is globally asymptotically stable, when R < . The technique of Lyapunov function is used to obtain the global stability of endemic equilibrium E * , when R > . In addition we illustrate our theoretical results by the numerical simulation. In the following we would like to propose a control system to which we will associate a problem of minimising the cost related to the use of the di erent candidate vaccines and the other preventive measures such as the wearing of masks, hand washing etc... To solve the control problem we use the Pontryagin maximum principle.