Abstract
A novel mathematical fractional model of multistrain tuberculosis with time delay memory is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the sense of the Grünwald–Letinkov definition. Modified parameters are introduced to account for the fractional order. The stability of the equilibrium points is investigated for any time delay. Nonstandard finite deference method is proposed to solve the resulting system of fractional-order delay differential equations. Numerical simulations show that nonstandard finite difference method can be applied to solve such fractional delay differential equations simply and effectively.
1. Introduction
It is known that tuberculosis (TB) is one of the most important infectious diseases and is considered as the second largest cause of mortality by infectious diseases and a challenging disease to control [1]. Time delays required to treatment of active TB present a major obstacle to the control of a TB epidemic [2]; it worsens the disease, increases the risk of death, and enhances tuberculosis transmission to the community [3, 4]. Both patient and the health system may be responsible for the treatment delay [3]. On the other hand, mathematical models are quite important and efficient tool to describe and investigate TB diseases; see [5–9]. In [10], Silva et al. presented TB model with time-delay memory. Herein, we consider a general model of multistrain TB diseases with time-delay memory. A discrete time delay is incorporated, in the variables of active TB infection of two and three strains, to represent the required time to commencement of treatment and diagnosis [11].
The multistrain TB model incorporates three strains: extensively drug-resistant (XDR), emerging multidrug resistant (MDR), and drug sensitive, and has been developed by Arino and Soliman [12] in 2015. Several factors of spreading TB such as the exogenous reinfection, the fast infection, and secondary infection are included in this model. Sweilam et al. introduced some numerical studies for this model in [13–16].
Fractional differential equations have been the focus of many studies due to their frequent appearance in various sciences [13–20]. The general theory of differential equations with delays (DDEs) is widely developed and discussed in the literature [21–25]. Delayed fractional differential equations (DFDEs) are also used to describe dynamical systems [26–28]. Recently, DFDEs begin to raise the attention of many researchers [29–33]. Relatable and efficient numerical techniques for DFDEs are very necessary and important [34]. Nonstandard finite difference method (NSFDM) was firstly proposed by Mickens [35] in 1980s to solve numerically the ordinary differential equations (ODEs) and partial differential equations (PDEs) with more accuracy than standard finite difference method (SFDM). It is considered as a powerful numerical scheme that preserves properties of exact solutions of the differential equation [36].
The main aim of work is to study numerically the solutions of fractional-order model of multistrain TB with time delay memory. The presence of fractional-order and time delays in the model can lead to a notable increase in the complexity of the observed behavior, and the solution continuously depends on all the previous states. An efficient numerical method, NSFDM, is used to numerically solve the fractional-order delay model. The rest of the paper is organized as follows: In Section 2, we present a fractional order model with time delay for multistrain TB. Stability of equilibrium points is presented in Section 3. NSFD for fractional-order delay differential equations is introduced in Section 4. Some numerical simulations are given in Section 5, and conclusion in Section 6. Some definitions on fractional calculus and some properties of nonstandard discretization are given in Appendix.
2. Fractional Multistrain TB Model with Time Delay
In this section, a multistrain TB model of fractional-order and time delay memory is presented. The population of interest is divided into eight compartments depending on their epidemiological stages as follows: susceptible ; latently infected with drug sensitive TB ; latently infected with MDR TB ; latently infected with XDR TB ; sensitive drug TB infectious ; MDR TB infectious ; XDR TB infectious ; recovered . One biological meaning of the given parameters is given in Table 1. One of the main assumptions of this model is that the total population , with , is variable of the time. We introduce a discrete time delay in the state variables and , denoted by , that represents the time required for diagnosis and commencement of treatment of active TB infection of two and three strains. The parameters in the modified the model are described in Table 2; see [37]. The modified system of multistrain TB model of fractional-order and time delay isThe initial conditions for system (1) are , , , , , , where , where is the Banach space . From biological meaning, we further assume that for . Throughout this paper, we focus on the dynamics of the solutions of (1) in the restricted region, We refer here to [24, 31], for local existence, uniqueness, and continuation results.
The unique solution , of (1) with initial condition exists for all time . Consider the solutions of (1), for , where is the interior of , for all . Then the solutions stay in the interior of the region for all time ; that is, the region is positively invariant with respect to system (1) (see, e.g., [31]). Model (1) has a disease-free equilibrium given by ; see [32].
2.1. Basic Reproduction Number
The basic reproduction number, , is defined as the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [32]. Herein, we apply the method in [32] to drive . The order of the infected variables is The vector representing new infections into the infected classes is given by The vector representing other flows within and out of the infected classes is given by The matrix of new infections and the matrix of transfers between compartments are the Jacobian matrices obtained by differentiating and with respect to the infected variables and evaluating at the disease-free equilibrium. They take the form whereThen the basic reproduction number for system (1) is the spectral radius of the next generation matrix and is given bywhere
3. Equilibrium Points and Their Asymptotic Stability
To discuss the local asymptotic stability for evaluating the equilibrium points, let us consider the following [38]:Then, from (A.1)where denotes any equilibrium point.
3.1. Stability of the Disease-Free Equilibrium
If then , , and . Then the disease-free equilibrium (DFE) is .
Let us consider the coordinate transformation: , , , , , , , . The corresponding characteristic equation for DFE is given as follows:where , , , , , , , , , , , , , , , , , , , , , .
The characteristic equation associated with above matrix is [38]
Lemma 1. If , then the disease-free equilibrium is locally asymptotically stable for .
Proof. When , the associated transcendental characteristic equation of system (1) at becomes , and then the eigenvalues of the Jacobian matrix are and by using Routh Hurwitz Theorem [28], these roots are negative or have negative real parts and all eigenvalues satisfy Matignon’s conditions [39], given by so the disease-free equilibrium is locally asymptotically stable.
Lemma 2. Let , and then the disease-free equilibrium is locally asymptotically stable for .
Proof. Let us consider , and we noted that second and third factor of the characteristic equation (12), which are and , have no pure imaginary roots for any value of the delay , if . Hence all the roots of the characteristic equation have negative real parts and we get that DFE is locally asymptotically stable regardless of the value of the delay and all eigenvalues satisfy Matignon’s conditions [39], given by so the disease-free equilibrium is locally asymptotically stable.
3.2. Stability of the Endemic Equilibrium
System (1) has an endemic equilibrium if at least one of the infected variables is not zero. The expression “analytic” is complexity and not useful for our purposes. Consider the values of parameters from Table 3. Then the basic reproduction number is . The endemic equilibrium , , , , , , , . The matrices and associated with the linearized system at the endemic equilibrium are computed as The transcendental characteristic equation is given bywhen , and we have the following characteristic equation: The roots of (16) are , , , , , , , and , and these roots are negative or have negative real parts and all eigenvalues satisfy Matignon’s conditions [39], given by so the endemic equilibrium is locally asymptotically stable.
Consider now the case ; we noted that the roots of the characteristic equation (15) have no pure imaginary roots for any value of the delay , if . Hence all the roots of the characteristic equation have negative real parts and all eigenvalues satisfy Matignon’s conditions [39]. Therefore, the endemic equilibrium is locally asymptotically.
4. NSFD for Fractional Delay Differential Equations
In this section, we apply NSFD method with GLFDs to obtain the discretization of the delay fractional multistrain TB model (1), which will yield the following equations: where and , , , where the nonlocal approximations are used for the nonlinear terms and the following functions of denominator:Then we obtain
5. Numerical Results and Simulations
In this section, we show the effectiveness of the numerical technique for delay fractional differential equations. Throughout this section, all simulations are performed with initial conditions , with the parameters in Table 3. The approximate solutions of the proposed system are given in Figures 1–12 at different values of and . Figure 1 shows the behavior of the approximate solutions of in two cases with and without delay using dde23 at and . In Figure 2, we use the same data in Figure 1 and use NSFDM; we noted that the number of individuals increases in the case of nondelay, that is, the delays in diagnosis and commencement of treatment to the individuals of and causing a shortage in the number of individuals of . Figure 3 shows the relationship between and and chaotic attractors at and . Figure 4, shows the relationship between and at in case of integer order. Figures 5 and 6 show the relationship between and and with , respectively, in case of fraction order where . Figures 7 and 8 show the approximate solutions for and at , by using NSFDM. Figures 9 and 10 show the approximate solutions of different in both fraction and integer cases; we noted that increasing the value of causes decreasing the values of . Figures 11 and 12 show the behavior of the approximate solutions with different value of , which are given to demonstrate how the fractional model is a generalization of the integer order model.
6. Conclusion
Fractional models have the potential to describe more complex dynamics than the integer models and can include easily the memory effect present in many real world phenomena. In this paper, multistrain TB model of fractional order derivatives with time delay memory is presented. A nonstandard numerical scheme is introduced to numerically study the approximate solution of proposed model problem. The obtained results show that the delays in diagnosis and commencement of treatment to the individuals of and cause a shortage in the number of individuals of . The approximate solution of proposed model changes when and take different values. Some figures are given to demonstrate how the fractional delay model is a generalization of the integer order model. It is concluded that NSFDM can be applied to solve such fractional delay differential equations simply and effectively. All results are obtained by using MATLAB programming.
Appendix
A. Preliminaries and Notations
In this section, some basic definitions and properties in the theory of the fractional calculus are presented. Moreover, we introduce the main aspects concerning nonstandard discretization methods.
A.1. Grünwald–Letinkov Fractional Derivatives (GLFDs)
We will begin with the signal fractional differential equation (see [17, 40, 41])where . is the final time and denotes the fractional derivative, where , defined by and is the th-order Riemann–Liouville integral operator: and . The Grünwald–Letinkov approximation of the fractional derivative is defined as follows [42]:where denotes the integer part of and is the step-size. Equation (A.4) can be discretized as follows:where and are the Grünwald–Letinkov coefficients defined as
A.2. NSFD Discretization
It is known that the numerical scheme is called nonstandard method if at least one of the following conditions is satisfied [36]:(1)the discretization of derivatives is not traditional and uses a nonnegative function [35, 43],(2)nonlocal approximations are used. To construct the numerical scheme for system (1) using NSFDM, the approximations of temporal derivatives are made based on generalized forward scheme of first order. Hence, if , we define its derivative as follows: where is a real-valued function on and . In the following, the denominator functions are little complex functions of the step-size of time than the classical one [44].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.