Fractional Calculus Model for the Hepatitis C with Different Types of Virus Genome
Moustafa El-Shahed^{1}, Ahmed. M. Ahmed^{2}, Ibrahim. M. E. Abdelstar^{2, 3}
^{1}Department of Mathematics, Faculty of Art and Sceinces, Qassim University, Qassim, Unizah, Saudi Arabia
^{2}Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt
^{3}Quantitative methods Unit, Faculty of Business & Economice, Qassim University, Almulyda, Saudi Arabia
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Moustafa El-Shahed, Ahmed. M. Ahmed, Ibrahim. M. E. Abdelstar. Fractional Calculus Model for the Hepatitis C with Different Types of Virus Genome. International Journal of Systems Science and Applied Mathematics. Vol. 1, No. 3, 2016, pp. 23-29. doi: 10.11648/j.ijssam.20160103.12
Received: September 6, 2016; Accepted: September 18, 2016; Published: October 11, 2016
Abstract: In this paper, a fractional order model to study the spread of HCV-subtype 4a amongst the Egyptian population is constructed. The stability of the boundary and positive fixed points is studied. The generalized Adams-Bashforth-Moulton method is used to solve and simulate the system of fractional differential equations.
Keywords: Hepatitis C Virus, Fractional Order, Stability, Numerical Method, Sovaldi
1. Introduction
Egypt has possibly the highest HCV prevalence in the world; 10-20% of the general population are infected and HCV is the leading cause of HCC and chronic liver disease in the country [17]. The genomes of HCV display significant sequence heterogeneity and have been classified into types and subtypes. Six types from 1 to 6 have been recognized, each type having a different number of subtypes like a, b, c, etc. Recently, new variants have been identified and assigned to proposed types 7-11. The worldwide presence of the virus and the geographic distribution of genotypes clearly indicate that HCV is an old companion of human kind [20].
Moneim and Mosa [17], constructed a mathematical model to study the spread of HC V-subtype 4a amongst the Egyptian population, we assume that people from the Egyptian population have some factors which lead to substitutions or mutations of the different genotypes of HCV into HCV-subtype 4a. They analyzed and solved the model to derive new results about the behavior of the spread of HCV. In recent decades, the fractional calculus and Fractional differential equations have attracted much attention and increasing interest due to their potential applications in science and engineering [5, 14, 18]. In this paper, we consider the fractional order model for hepatitis C virus and antiviral medication (Sovaldi). We give a detailed analysis for the asymptotic stability of the model. Adams-Bashforth-Moulton algorithm have been used to solve and simulate the system of differential equations.
2. Model for Mulation
The model for the spread of virus HCV-subtype 4a can be written as a set of four coupled nonlinear ordinary differential equations as follows [17]:
(1)
Where
1. R denote the densities (or fractions) of recovered individuals.
2. The birth rate is equal a positive constant rate (c), and death rate is equal a positive constant rate (b),
3. g, d be the rate at which susceptible individuals a removed after taking the medicine from , respectively.
The population is mixing in a homogenous manner, i.e. every person has the same chance of coming in contact with an infected person.
Where g, d³0 and all the other parameters are positive. If , which means there are no cure by treatment, then . System (1) will reduce to the standard SR model.
Fractional order models are more accurate than integer-order models as fractional order models allow more degrees of freedom. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. Fractional differential equations are also regarded as an alternative model to nonlinear differential equations. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent work on fractional differential equations, see [5,14,18]. Now we introduce fractional order in to the ODE model by Liu et al. [16]. The new system is described by the following set of fractional order differential equations:
(2)
where is the Caputo fractional derivative and A=cN. Because model (2) monitors the dynamics of human populations, all the parameters are assumed to be non-negative. Furthermore, it can be shown that all state variables of the model are non-negative for all time t³0 (see, for instance, [4,10].
Lemma 1 The closed set is positively invariant with respect to model (2).
Proof. The fractional derivative of the total population, obtained by adding all the equations of model (2), is given by
(3)
The solution to Eq. (3) is given by
(4)
where is the Mittag-Leffler function. Therefore, all solutions of the model with initial conditions in W remain in W for all t >0. Thus, region W is positively invariant with respect to model (2).
In the following, we will study the dynamics of system (2).
3. Equilibrium Point and Stability
In the following, we discuss the stability of the commensurate fractional ordered dynamical system:
(5)
Let be an equilibrium point of system (3.1) and , where is a small disturbance from a fixed point. Then
(6)
System (6) can be written as:
(7)
where and is the Jacobian matrix evaluated at the equilibrium points. Using Matignon's results [23], it follows that the linear autonomous system is asymptotically stable if is satisfied for all eigenvalues of matrix at the equilibrium point. If Let denote the discriminant of a polynomial, then
Following [1,2,3,15], we have the proposition.
Proposition 3.1 One assumes that exists in.
1. If the discriminant of, is positive and Routh-Hurwitz are satisfied, that is, , then is locally asymptotically stable.
2. If then is locally asymptotically stable.
3. If < 0,, then is unstable.
4. The necessary condition for the equilibrium point, to be locally asymptotically stable, is .
To evaluate the equilibrium points, let
Then
1. the first disease free equilibrium (DFE) point is, when the disease is absent in the population, in this case (), therefore the population is fully susceptible.
The basic reproductive number is defined as the expected number of secondary cases produced by a single infected individual entering the population at the DFE [5]. It means the average new infections produced by one infected individual during his lifespan when the population is at .
Theorem 3.2 For the system (1) one have the basic reproduction number
Proof We will use the next generation method to find the basic reproduction number, for the system (1), rewrite the equations by which classes infection first and then the rest of the equations secondly, we have
(8)
We make matrices f, v, such that the system (8) in the form
Where
We make matrices F, V, such that
then
at the first disease free equilibrium (DFE) point the matrix F become
we now find the inverse matrix of V
we need to find the multiplying , we have
to get the eigenvalues of we solve the equation
(9)
where l is the eigenvalues and I is the identity matrix, we have
then
where is the next generation matrix for the model (1). It the follows that the spectral radius of matrix is
Then
Let , then
2. By (2), a positive equilibrium point which is the free HCV infection from all types except 4a ().
3. The third point is the endemic from all types of infection. Then . Therefore the third equilibrium point is where
Remark. The free HCV type 4a infection, , does not exist, as there is a positive mutation rate m from infective class, , to infective class, , so if then it should be that, is the (DFE) again.
The Jacobian matrix for system given in (2) evaluated at the disease free equilibrium is as follows:
Theorem 3.3 The disease free equilibrium is locally asymptotically stable if and is unstable if at least one of .
Proof. The disease free equilibrium is locally asymptotically stable if all the eigenvalues, of the Jacobian matrix satisfy the following condition [1, 2, 3, 8, 11, 17]:
(10)
The eigenvalues of the characteristic equation of are and . Hence is locally asymptotically stable if and is unstable if at least one of .
We now discuss the asymptotic stability of a positive equilibrium point of the system given by (2). The Jacobian matrix evaluated at a positive equilibrium is given as:
The eigenvalues of the characteristic equation of are and
.
Theorem 3.4 The equilibrium point is locally asymptotically stable if and only if .
Proof. We start our proof by assuming that the equilibrium point is locally asymptotically stable. Then . Hence,
So, Conversely, assume that and if the equilibrium point is not locally asymptotically stable, then at least one of the following cases holds
· . Thus () which is a contradiction.
· is a complex number with nonnegative real part, i.e. () which is also a contradiction with the fact that ().
· is a real number and nonnegative, i.e.
.
This leads to which contradicts with the assumption. Finally, it is obvious that has a negative real part. Therefore we can deduce that, the equilibrium point is locally asymptotically stable.
We now discuss the asymptotic stability of the endemic (positive) equilibrium point of the system given by (2). The Jacobian matrix evaluated at a endemic (positive) equilibrium point is given as:
The characteristic equation of is
Where
Following from proposition (1), a necessary condition for is . Then one has the following theorem:
Theorem 3.5 The endemic equilibrium point is locally asymptotically stable if , and unstable if
If one take
It follows that . In this case, the endemic equilibrium point
is local asymptotically stable where the eigenvalues are
4. Numerical Methods and Simulations
Since most of the fractional-order differential equations do not have exact analytic solutions, approximation and numerical techniques must be used. Several analytical and numerical methods have been proposed to solve the fractional order differential equations. For numerical solutions of system (2), one can use the generalized Adams-Bashforth-Moulton method. To give the approximate solution by means of this algorithm, consider the following nonlinear fractional differential equation [6,7,15]
This equation is equivalent to the Volterra integral equation
(11)
Diethelm et al. used the predictor-correctors scheme [6, 7], based on the Adams-Bashforth-Moulton algorithm to integrate Eq. (4.1). By applying this scheme to the fractional-order model for Hepatitis C virus, and setting , Eq. (4.1) can be discretized as follows [6, 7, 15]:
where
5. Conclusion
In this paper, we consider the fractional order model for Hepatitis C virus and vaccines. We have obtained a stability condition for equilibrium points. We have also given a numerical example and verified our results. One should note that although the equilibrium points are the same for both integer order and fractional order models, the solution of the fractional order model tends to the fixed point over a longer period of time. One also needs to mention that when dealing with real life problems, the order of the system can be determined by using the collected data. The transformation of a classical model into a fractional one makes it very sensitive to the order of differentiation a: a small change in a may result in a big change in the final result. From the numerical results Figures follows, it is clear that the approximate solutions depend continuously on the fractional derivative a. The simulation results of our model have been performed using Matlab. We use some documented data for some parameters like death rate b = 0.02, birth rate c=0.04 take the number of population N= 1,000,000 and then suggest the other parameters such as mutation rate , the contact rates between S and both of and respectively. Finally, the values of the basic reproductive numbers have been suggested to be first, less than one in value and then larger enough than one to test the stability of the three different equilibrium points of our model.
Figure 1. The approximate solutions and are displayed in Figs. a-d, respectively. In each figure three different values of .
References