MODELLING AND STABILITY OF EPIDEMIC MODEL WITH FREE-LIVING PATHOGENS GROWING IN THE ENVIRONMENT

To understand the impact of free-living pathogens (FLP) on the epidemics, an epidemic model with FLP is constructed. The global dynamics of our model are determined by the basic reproduction number R0. If R0 < 1, the disease free equilibrium is globally asymptotically stable, and if R0 > 1, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.


Introduction
Epidemics are diseases caused by various pathogens which can spread between humans. Each infectious disease is caused by its specific pathogen, which can be a microbe or a parasite. Including viruses, bacteria, fungi or parasites, etc. Although the development of medical science has been able to prevent and control many infectious diseases, there are still some outbreaks or epidemics that danger people's health and lives. Some infectious diseases are highly contagious, leading to high disability rates and great harm, for example, there are 1.5 million deaths from respiratory infectious worldwide annually [6]. Therefore, it is of great practical significance to study the process of epidemics.
Following the pioneering of Kermack and Mckendrick [15], many researchers' attention have been attracted by epidemic models. In order to better understand the dynamics of infectious diseases, many different epidemic models have emerged and have been studied in different ways. For example, we can find susceptibleinfectious-recovered (SIR) model (see for example [1,16]), apart from this, there is also susceptible-infectious-susceptible (SIS) model [4,8,9,21]. Some researchers added the exposed compartment E or the treatment T to the model, just like susceptible-exposed-infectious-recovered (SEIR) model in [31]; A. Ricardo. [22] proposed a susceptible-exposed-infectious-recovered-treatment (SEIRT ) model and studied the effects of treatment and exposure on epidemics; in addition, there is a mixture of the above models, like [2,13,20,[28][29][30]. Furthermore, some authors have studied the virus model (see [26]) to study the virus infection dynamics. From the work of S.A. Boone, et al. [6] and A. Gabbuti, et al. [10], we know that vaccination against hepatitis B is the most effective measure to control the spread of hepatitis B due to prevention and management of viral disease heavily relies upon vaccines and antiviral medications. So, many researchers incorporated the new compartment, that is, the vaccination V to their model, such as [14,19].
However, vaccines and drugs are not absolutely effective, and there are still no vaccines or drugs for many common viruses [6]. So, avoiding exposure to the virus is more important to prevent infectious disease. Viral transmission is not only depend on interaction with the host, but also interaction with the environment. A contaminated environment such as air, water, food may transmit infection to susceptible hosts [3,7,23]. If we are contacting with infections who is sneezing or coughing, we will be very susceptible to the pathogens. The pathogens in a freeliving can survive even growth in the environment [5,18]. Additionally, a free-living pathogen (FLP) in the environment will naturally die and be replenished by the infected people [3].
By taking into account the above mentioned factors, in this paper, we extend a simple SIRT model to an SIRT P model that include the FLP capable of growth and survival in the environment. The organization of this paper is as follows: In section 2, the model is derived under some assumptions. In section 3, we get the basic reproduction number and obtain the stability of all the equilibria. Some numerical simulations are given in section 4 to illustrate our analytical result. Some discussions are given in the last section. In this section, a simple model with treatment and FLP is introduced. The total population N (t) is divided into four compartments, namely, S(t) represents the number of susceptible individuals. I(t) represents the number of infectious individuals. Note that some people who are infected with pathogens are able to heal themselves. R(t) represents the number of recovered individuals and the recovered individuals may be infected again. T (t) represent the number of individuals being treated. In our model, we assume that not all people can be cured completely for some of practical reasons. If the treatment is successful, individuals in compartment T enter into the compartment of R. If the treatment is terminated or failure, people who failed the treatment still carry and transmit the virus. And the compartment P (t) indicates the FLP load in the environment, people who are susceptible to infection can become infected not only by adequate contacts with infectious people but also contacting pathogens in the environment. Hence, the total population is given by

System description
The transfer diagram of the model is shown in Figure 1 (in this figure, the dotted line and the solid line represent the dynamics of host and FLP, respectively). According to Figure 1, we have following model: (2.1) We assume that α is non-negative and all other parameters in the model are positive constants. The parameters are described in Tabel 1. In this paper, we assume that the FLP population cannot maintain itself through growth in the environment (i.e. the FLP growth rate g is always less than the FLP decay rate r) [3]. For system (2.1), In order to ensure the solutions of the system with positive initial conditions remain positive for all t > 0, it is necessary for us to prove that all the variables are non-negative. Therefore, we give the following lemma. Proof. Under the giving initial conditions, it is easy to prove that the solution of the system (2.1) are positive; if not, we assume a condition: that there exists a first time t 1 such that there exists a t 2 there exists a t 3 there exists a t 5 which is contradictory to the assumption In the second case, we have which is contradictory to the assumption I (t 2 ) < 0, it means that Hence, the solutions S(t), I(t), T (t), P (t), R(t) of system (2.1) remain positive for all t > 0.

Invariant regions
Lemma 2.2. We assume that there is a constant H, then, all feasible solution of the system (2.1) are bounded and enter the region Ω = {(S(t), We assume that (S, I, T, R) is any solution with non-negative initial conditions, adding the first four equations of the system (2.1), we have where It follows that where N (0) represents initial value of the total population.
For the last equation of the system (2.1), we have and thus there exists H > 0 so that lim sup t→∞ P (t) ≤ H. The constant H can be chosen as the unique positive zero of the quadratic polynomial γ b m −(r −g)x−gcx 2 . Therefore all feasible solutions of system (2.1) enter the region So we consider dynamic of system (2.1) on the set Ω in this paper.

Disease free equilibrium and the reproduction number
It's easy for us to get the disease free equilibrium of the system (2.1) In the following, the basic reproduction number of system (2.1) will be derived by using the next generation matrix method formulated in [3,12,25,27]. At first, we rearrange system (2.1) as following: Then let x = (I, T, P, S, R) T , thus the system (3.1) can be written as: The Jacobian matrices of F (x) and V (x) at the disease free equilibrium E 0 are respectively, The model reproduction number, denoted by R 0 is thus given by Following Theorem 2 of [25], we have the following result on the local stability of E 0 .
Theorem 3.1. The disease free equilibrium E 0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.

Global stability of
Proof. The comparison theorem can be used to prove the global stability of the disease free equilibrium. The rate of change of the variables (I, T, P ) of system (3.1) can be written as where F and V are defined in (3.2).
Since γI + (g − r)P − gcP 2 ≤ γI + (g − r)P and S ≤ b m for all t ≥ 0 in Ω, then Note that both F and V −1 are non-negative. Following the Perron-Frobenius Theorem, let µ T be a non-negative left eigenvector of [11], give a Lyapunov function L = µ T V −1 (I, T, P ).
The derivative of L is given by T , If R 0 < 1, then L = 0 implies that µ T (I, P, T ) = 0, and thus I = 0 or T = 0 or P = 0. It follows from the system (2.1) that the largest invariant set where L = 0 satisfies δSP + k 2 T = 0 or kI = 0 or γI = 0; Therefore, I = T = P = 0 because of δ ≥ 0, k ≥ 0, γ ≥ 0. The first and the fourth equations of the system (2.1) lead to S = b m and R = 0 , respectively, in the above invariant set, which is the singleton {( b m , 0, 0, 0, 0)}. By LaSalle's invariance principle [17], the disease free equilibrium E 0 of system (2.1) is globally asymptotically stable in Ω if R 0 < 1. Proof. The equilibrium equation of the system (2.1) are given by

Epidemic equilibrium
It follows that the endemic equilibrium satisfies I = ψ(P ) := gc γ P 2 + r − g γ P, To see if there is a endemic equilibrium solution in the above formulas (3.3), we only need to judge whether the two lines determined by the functions I = ψ(P ) and P = φ(I) have intersections. Because of the P and I are all positive numbers, so we just consider the graphs in the first quadrant, by analyzing these two functions, we can get that the graph of the function I = ψ(P ) is a parabola, the function P = φ(I) has two vertical asymptotes, take I 0 = b(k2+k1+m+µ2)(α+m) (m+µ1+ν+k)(k2+k1+m+µ2)(α+m)+kk2(α+m)+αν(k2+k1+m+µ2)+αkk1 (Because of the range that P > 0, another vertical asymptote is not considered here). Then, lim I→I0 φ(I) = +∞, so the intersection exists if and only if φ (0) < 1 ψ (0) , see Figure 2 (This is a figure that reflects the positional relationship between the two functions, in this figure, the real curve and the dashed curve represent the function I = ψ(P ) and the function P = φ(I), respectively, while the curve perpendicular to the I axis represents the asymptote of the function P = φ(I). Through calculations, we get and ψ (0) = r−g γ . Then, Organize the above inequality to get Thus, the sufficient and necessary condition of a unique endemic equilibrium exists is R 0 > 1.

Global stability of the endemic equilibrium
Theorem 3.4. Assume that α = 0, if R 0 > 1 the unique endemic equilibrium E * of system (2.1) is globally asymptotically stable.
Proof. If R 0 > 1, by Theorem 3.3, we know that there exist a unique endemic equilibrium E * (S * , I * , T * , R * , P * ), where S * , I * , T * , R * , P * satisfy the equilibrium equations (3.3). Following [19,24], we introduce a Lyapunov function V as follows V = S − S * ln S + B(I − I * ln I) + D(T − T * ln T ) + E(P − P * ln P ). Where B, D, E are positive constants to be determined later, the derivative of V is given by By denoting S S * = x, I I * = y, T T * = z, P P * = u, we have The variable terms that appear in (3.4) with positive coefficients are xy, y, xu, and u. If all of the coefficients are positive, then there is a possibility that .
V could be positive. Making the coefficients of xy, y, xu, and u equal to zero, we get that From (3.5), we obtain that Hence, we have It's easy for us to see that −mS . V = 0} is the singleton (1, 1, 1, 1). It means that, the largest invariant set where . V = 0 is the singleton {(S * , I * , T * , P * , R * )}. Thus, by Laselle's invariance principle [17], for system (2.1), the endemic equilibrium E * is globally asymptotically stable if α = 0 and R 0 > 1.
Remark 3.1. Theorem 3.4 tells us that when α = 0 (1/α is immune period) and R 0 > 1, the endemic equilibrium E * is globally asymptotically stable. Note that the global stability of the endemic equilibrium E * is an open problem when α > 0. But, when α = 0, the same result is validated by numerical simulations (Figure 6).

Numerical simulation
In this section, to illustrate the analytic results obtained above, we have presented some simulations of system (2.1) using the parameter values in Table 1. The parameter values are mainly taken from [3].
We choose b = 12.05, β = 0.003, δ = 0.0002, k 1 = 0.6, k 2 = 0.3, α = 0, numerical simulation gives R 0 = 12.2 > 1, then the endemic equilibrium E * is global asymptotically stable (Figure 4). From Figure 4, we can clearly see as time going on, the number of people finally tends to a constant, however, we can't see some details clearly from Figure 4. Change the axis range, we can obtain the partial enlarged imagines as Figure 5, from Figure 5 we can see that four curves reach their equilibrium positions are not zero. It reveals that our proof of the existence of E * is correct. Figure 6 shows when α = 0, the E * is also globally asymptotically stable.  Figure 3. The disease free equilibrium E0 is globally asymptotically stable.

Discussion
In this paper, we have formulated a novel epidemic model, which is incorporated a new compartment, that is, FLP capable of growth and survival in the environment. With the help of the next generation matrix method, we obtained the basic reproduction number R 0 which play a important role, and we derived the globally   Figure 6. When α = 0, the endemic equilibrium E * is globally asymptotically stable.    Figure 9. The relationship among R0, δ and γ dynamics of the model by constructing Lyapunov function. When the basic reproduction number R 0 is less than unity, the disease free equilibrium is globally asymptotically stable, that is, the disease will be extinct; When the basic reproduction number R 0 is greater than unity, the endemic equilibrium is globally asymptotically stable, which means that the disease will be permanent.
We all know that the basic reproduction number can be used to distinguish  Figure 11. The relationship among R0, β and δ whether the disease disappears or not. For system (2.1), r reflects pathogens decay rate, γ reflects pathogens shedding rate while δ and β denote the environment-tohost transmission and the host-to-host transmission, respectively. In the event of an epidemic outbreaks, we can control the disease by reducing contact with pathogens, the measures may including: (1) more frequent purification of the environment (i.e. increasing the value of r by washing our hands often or disinfecting and so on); (2) enhance people's physical and antiviral ability by antibiotic treatments (i.e. the shedding rate γ will be decreased); (3) reducing contacts with infectious host (i.e. the host-to-host transmission β is reduced); (4) reducing contacts with contaminated environment (i.e. the environment-tohost transmission δ is reduced). Figure 7 shows the change in the basic reproduction number R 0 due to changes in r, γ, δ, β. All other parameter values are as given in Table 1. Note that when r increase, infectious individuals will decrease, as stated in (1). Moreover, as shown in Figure 7(b)-7(d), R 0 grows with γ, δ, β, it means that if we want to control the disease, we should reduce the γ, δ and β, just like described in (2) − (4). Comparing Figure 7(c) with Figure 7(d), we can see that for some infectious diseases, the environment-to-host transmission is even more serious than the infection between people. Figure 8 shows the relation among the basic reproduction number R 0 , r, and γ. Figure 9 shows the relation among the basic reproduction number R 0 , δ, and γ. Figure 10 shows the relation among the basic reproduction number R 0 , δ, and r. Figure 11 shows the relation among the basic reproduction number R 0 , β, and δ. From Figures 8-11, we can also see that if γ, δ and β are increase, then R 0 will increase, and if r increase, then R 0 will decrease.
Our results show that the role of environment-to-host disease transmission is becoming evident, it is necessary and meaningful to take the FLP into account. In addition, in this paper we discuss the global dynamic behaviors of system (2.1), we obtain that when the precondition are α = 0 and the basic reproduction number R 0 > 1, the endemic equilibrium is globally asymptotically stable. Although when α = 0, the same result is validated by numerical simulations (Figure 6), we didn't prove by mathematics that it is right. So we may need to conduct further research on this issue in the future.