Journal of Nonlinear Modeling and Analysis
The Journal of Nonlinear Modeling and Analysis (JNMA), started in 2019, aims to publish original research papers, communications as well as survey papers on theory and application of nonlinear analysis related to differential equations, dynamical systems and other areas in mathematics and mathematical sciences, including mathematical biology. The journal was published quarterly in March, June, September and December between 2019 and 2024. From 2025 on, it will be published bimonthly with its issues released in February, April, June, August, October and December.
Global Attractors for the BBM Equation with Fading Memory
This paper considers the existence of global attractors for BBM equation with fading memory. Using some new estimate technique to prove the existence of global attractors in topological space $H_1 × L^2_{\mu}(\mathbb{R}^+;H_1).$
Copyright (c) 2026 Journal of Nonlinear Modeling and Analysis
Steady State Solution and Its Stability Analysis of a Reaction-Diffusion Mathematical Model for LDL, Lipoprotein(a), and CRP in Atherosclerosis
This study presents a reaction-diffusion mathematical model to investigate the spatiotemporal dynamics of low-density lipoprotein (LDL), lipoprotein(a), and C-reactive protein (CRP) within the arterial wall, key mediators in the pathogenesis of atherosclerosis and related cardiovascular diseases. The model employs a system of partial differential equations to capture both the diffusion of these biomolecules and their biochemical interactions, including LDL oxidation, CRP-mediated inflammatory responses, and plaque initiation. Numerical simulations, implemented using the finite difference method, demonstrate that CRP substantially accelerates LDL oxidation and promotes plaque deposition, particularly in regions of locally elevated lipoprotein concentrations. Furthermore, the results indicate that elevated CRP levels synergistically enhance the pro-atherogenic effects of LDL and lipoprotein(a), highlighting the critical role of inflammation in plaque progression. These findings underscore the potential of anti-inflammatory therapeutic strategies, alongside lipid-lowering interventions, in mitigating atherosclerotic risk. The proposed model provides a robust computational framework for elucidating the interactive effects of lipid metabolism and vascular inflammation in the progression of cardiovascular disease.
Forced Epidemic Waves in a Nonlocal Dispersal SIR Model with Shifting Transmission and Time Delay
This paper is concerned with the forced epidemic waves of a nonlocal dispersal SIR model with shifting transmission and time delay. We first demonstrate that the existence of forced waves can be reduced to a fixed point problem. Then by constructing different pairs of upper and lower solutions and using Schauder's fixed point theorem, we establish two types of forced epidemic waves that reveal different state conversions of the disease. Moreover, we prove the nonexistence of forced epidemic waves when the basic reproduction number is less than unity. Finally, some biological explanations for the theoretical results are given in the discussion.
Solving the Kuramoto-Sivashinsky Equation via a Modified Adomian Decomposition Method
The Kuramoto-Sivashinsky equation (KSE) is a well-known nonlinear partial differential equation (PDE) that plays a significant role in various scientific fields, particularly in fluid dynamics and reaction-diffusion systems. In this study, we employ the Modified Adomian Decomposition Method (MADM) to derive an analytical solution to the KSE over the domain $t \in \mathbb{R}$ and $0 \le x < 1$. The research provides a comprehensive analysis of the applicability of MADM in generating explicit series solutions for the KSE. The equation is formulated under proper initial conditions, and the systematic implementation of MADM is demonstrated involving the decomposition of nonlinear terms and successive approximation. The convergence and stability of the resulting series solutions are investigated, highlighting the method's efficiency in capturing the dynamics described by the KSE. Numerical simulations are presented to validate the analytical results, illustrating the effectiveness of the MADM in solving the KSE within the given parameters. This study not only enhances the theoretical understanding of the KSE but also serves as a practical guide for applying analytical techniques to complex nonlinear PDEs. Moreover, the proposed method represents a significant advancement in solving highly non-linear models by providing a straightforward recursive scheme that avoids linearization or small-parameter assumptions. The physical relevance of the obtained solutions is reflected in their ability to capture key features of instability and chaotic behavior characteristic of the KSE. These findings underscore the potential of MADM as a powerful and versatile tool in the study of spatiotemporal phenomena governed by nonlinear PDEs.
A Robust Fixed Point Method to Solve System of Nonlinear Equations
Solving nonlinear partial differential equations requires addressing systems of nonlinear algebraic equations. In this article, we propose a new fixed-point method for solving these systems. We assume that the method is both fast and globally convergent. Additionally, this method can be accelerated using Aitken or Anderson acceleration techniques. Several numerical test cases are presented to illustrate the efficiency of the proposed method.
Homotopy Analysis Method on Listeria Infection Model Caused by Pre-Cooked Package Food
In this study, the Listeria infection model involving the fractional derivatives in the Caputo sense is considered and studied through two different techniques called the Homotopy analysis method (HAM) and Runge-Kutta 4th order (RK4) method. HAM is a semi-analytical approach, and the RK4 method is a numerical method proposed for solving fractional order systems of ordinary differential equations (ODEs). We have developed a semi-analytical solution in terms of a series of polynomials by HAM and a numerical solution by the RK4 method for the model. First, we solved the model through HAM by choosing the preferred control parameter and also applied the RK4 method. The model solved by projected methods is compared with the NDSolver solution. The nature of the model is analyzed using different parameters, and the calculations are performed using Mathematica software. The obtained results are expressed in graphs and tables.
Numerical Solutions of Fuzzy Fractional Variable-Order Differential Equations
This paper addresses a class of fuzzy fractional differential equations (FFDEs) with variable-order (VO) derivatives, where the variable-order derivative is defined in the Caputo sense for fuzzy-valued functions. Using the $\Upsilon$-cut representation of fuzzy-valued functions, the original problem is reformulated into a new problem. To solve it, we apply operational matrices (OMs) derived from shifted Chebyshev polynomials of the third kind (SCP3). By approximating the unknown function and its derivative with SCP3, the problem is reduced to a system of nonlinear algebraic equations. A theoretical error analysis of the numerical solution is presented, along with an example to validate the method's accuracy.
shifted Chebyshev polynomials of the third kind
Stability Analysis of a Three-Dimensional Discrete Topp Model
Mathematical models of glucose, insulin, and pancreatic beta cell mass dynamics are essential for understanding the physiological basis of type 2 diabetes. This paper investigates the discrete-time dynamics of the Topp model to represent these interactions. We perform a comprehensive analysis of the system's trajectory, examining both local and global behavior. First, we establish the invariance of the positive trajectory and analyze the existence of fixed points. Then, we conduct a complete stability analysis, determining the local and global asymptotic stability of these fixed points. Finally, numerical examples validate the effectiveness and applicability of our theoretical findings. Additionally, we provide biological interpretations of our results.
On Constacyclic Codes of Length $4p^s$ over $F_{p^m}[u,v]/\langle u^2, v^2, uv-vu \rangle$
In this paper, we determine the algebraic structure of all $\lambda$-constacyclic codes of length $4p^s$ over the ring $R_{u^2,v^2,p^m} = F_{p^m}[u,v]/\langle u^2, v^2, uv-vu \rangle$ and $u^2 = 0, v^2 = 0$ where $\lambda = (\alpha+\beta u+\gamma v+\delta uv)$ with $\beta,\gamma,\delta \in F_{p^m}$, $\alpha \in F_{p^m}^*$ and $\beta, \gamma$ are not both zero. If $\lambda$ is a square, each $\lambda$-constacyclic codes of length $4p^s$ is expressed as a direct sum of an -$\alpha$-constacyclic code and $\alpha$-constacyclic code of length $2p^s$. In the primary case where the unit $\lambda$ is not square, it is shown that any non-zero polynomial of degree $\leq 4$ over $F_{p^m}$ is invertible in the ring $R_{\alpha,\beta,\gamma,\delta} = R_{u^2,v^2,p^m}[x]/\langle x^{4p^s}-\lambda \rangle$ when $\lambda = (\alpha+\beta u+\gamma v +\delta uv)$ for non-zero elements $\alpha, \beta,\gamma \in F_{p^m}^*$, $\delta \in F_{p^m}$, it follows that the ring $R_{\alpha,\beta,\gamma,\delta}$ is a chain ring with maximal ideal $(x^4-\alpha_0)$ and $(\alpha+\beta u+\gamma v+\delta uv)$-constacyclic codes are $\langle (x^4-\alpha_0)^i \rangle$, for $0 \leq i \leq 2p^s$. In the second case where $\lambda$ is not square and $\lambda =\gamma$ for $\gamma \in F_{p^m}$, it is obtain that $R_{u^2,v^2,p^m}[x]/\langle x^{4p^s}-\gamma \rangle$ is a local ring with maximal ideal $\langle x^4-\gamma_0,u \rangle$ but not a chain ring, such $\lambda$-constacyclic codes are classified into four distinct types. We provide the number of codewords for each type, and give the details of their dual codes.
Solvability for a Class of Logarithmic Type Initial Conditions of Hadamard Fractional Differential System on an Infinite Interval
This paper investigates a class of Hadamard fractional differential system involving logarithmic type initial conditions on an infinite interval. Based on some obtained properties of the Green's functions, Schauder fixed point theorem and Banach contraction mapping principle, we establish the existence and uniqueness results for the system. Finally, we illustrate examples which show our main results.
Dynamical Analysis of Virulence Evolution in Multistrain Infection Model Within Hosts
Understanding the evolutionary patterns of viral virulence characteristics from a microscopic perspective is crucial for effectively combating viral mutations. This paper investigates the dynamics of virulence trait development within healthy cells following pathogen invasion using adaptive dynamics, building on a viral dynamics model that accounts for multiple infectious strains within hosts. Ignoring viral evolution, stability analysis of equilibria reveals the competitive exclusion principle. When viral virulence evolves, we assume that the infection rate of healthy cells and the mortality rate of infected cells are functions of virulence. We establish global stability conditions for the system and examine the evolutionary trajectory of viral virulence using adaptive dynamics. Our results indicate that mutant viruses can cause trait substitution. The evolutionary singular strategy is identified as a continuously stable strategy without producing evolutionary branching. Furthermore, we consider the influence of certain parameters in the system on the evolution of singular strategies.
A Predator-Prey Biological Model of Multiple Species with Linear Growth Rates
The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to an elliptic system of the Dirichlet problem on a bounded domain $\Omega$ in $R^n$. Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.
Nonlinear Dynamics and Chaos in Fractional-Order Cardiac Action Potential Duration Mapping Model
This study introduces a novel one-dimensional fractional-order model for cardiac action potential duration (APD) dynamics, incorporating memory effects through discrete fractional calculus. By generalizing the classical APD map using the Caputo fractional difference operator, we uncover complex nonlinear behaviors not observed in traditional integer-order models. Through comprehensive numerical simulations, including bifurcation analysis and Lyapunov exponent calculations validated by the 0-1 test, we demonstrate that the fractional-order system exhibits: 1) Early onset of chaos (at $t_s = 307ms$) without preceding period-doubling bifurcations. 2) Novel rhythm alternations between 5 : 5 and 3 : 3 patterns. 3) Unique bistability phenomena, including 2 : 2 $\leftrightarrow$ chaos and 5 : 5 $\leftrightarrow$ 3 : 3 states. 4) Memory-dependent dynamics where current APD depends on all previous states. Our results reveal that fractional calculus provides a more physiologically realistic framework for modeling cardiac dynamics by naturally incorporating memory effects. The identified dynamical regimes offer new insights into the transition mechanisms from normal rhythms to potentially arrhythmic states, with particular clinical relevance to understanding alternans as precursors to ventricular fibrillation. The fractional-order approach demonstrates superior capability for capturing the complex, history-dependent nature of cardiac excitation compared with conventional models.
A Semi-Analytical Study on SEIR Model with Vaccinated Effects
This research describes the dynamics of COVID-19 propagation using a mathematical model that considers vaccination and self-defence. Furthermore, the impact of vaccination on the development of disease transmission has been investigated. Our findings specifically show that putting into practice the best control measures, including time-dependent interventions, lowers the total infection load and disease transmission. Five different compartments (Susceptible, Exposed, Vaccinated, Infected, and Recovered) are used in this study to examine an epidemiological model of COVID-19 dynamics. Approximate analytical solutions to the model’s system of equations were obtained using the homotopy analysis method (HAM). The numerical simulation using MATLAB was employed to validate the accuracy and effectiveness of the solutions obtained through the Homotopy Analysis Method (HAM) by comparing the results. Excellent agreement is found when comparing the approximate analytical solution and the numerical simulation. The five-compartment model includes many more aspect parameters that are explored and graphically represented. These parameters include recovery rate and vaccination rate, among others. Moreover, it underscores the potential of the HAM as a powerful tool for exploring epidemic models and formulating control strategies.
Geometric Inequality for CR-Slant Warped Product Submanifolds in Nearly Lorentzian Para-Sasakian Manifold
This paper examines CR-slant warped product submanifolds of the form $B \times_f N_\theta$ within a nearly Lorentzian para-Sasakian manifold. Here, $B$ represents a CR-product submanifold, $N_\theta$ is a slant submanifold and $f$ denotes the warping function. We establish an inequality relating the squared norm of the second fundamental form to the warping function, considering two cases based on the behavior of the structure vector field. Additionally, the conditions under which equality holds are investigated.
nearly Lorentzian para-Sasakian manifold
Dynamical Analysis in a Discrete Fractional Order Harvested Predator-Prey Model Incorporating Fear Effect and Refuge
This paper examines the dynamics of a discrete fractional-order predator-prey model, incorporating the effects of fear, prey refuge, and population harvesting. The fractional-order framework, which accounts for memory-dependent processes, provides a more accurate depiction of biological interactions than traditional integer-order models. Fear is modeled as a factor that reduces predator-prey encounters, while prey refuge offers a sanctuary, altering population dynamics. The inclusion of harvesting further complicates prey population regulation. We identify the system’s equilibrium points and conduct a local stability analysis, focusing on the conditions that lead to stability, instability, and bifurcation. Specifically, we examine the Neimark-Sacker bifurcation, where a stable fixed point evolves into a closed periodic orbit. Numerical simulations corroborate the analytical findings, illustrating the pivotal roles that fear, refuge, and harvesting play in shaping system stability and long-term behavior.
Modelling Zika Virus, Dengue Fever and Chikungunya Virus: Gaining Insights into the Co-Dynamics of the Three Diseases
The recent spike in the concurrent circulation of Zika, Dengue fever, and the Chikungunya virus poses a severe threat to public health, both at home and in the diaspora. This study dives into the interplay of these three diseases, applying mathematical analysis to explain their co-dynamics within a population where they coexist. A comprehensive investigation of the model indicates that the sub-models experience backward bifurcation when the relevant reproduction numbers for each disease fall below unity. To gain insights, real-life data from Espírito Santo State in Brazil, where both diseases are co-circulating and endemic, was gathered and incorporated into our model. This allowed us to estimate important parameter values that were embedded in the model. Through uncertainty and sensitivity analysis, we identified the top-ranked parameters that drive the spread of these three diseases, which are, effective contact rate of infected mosquitoes with susceptible humans, infected humans interacting with susceptible mosquitoes, and sexual transmission (specifically for the Zika virus). Simulations of the comprehensive Zika-Dengue-Chikungunya model revealed that minimizing the biting and contact rates of mosquitoes with humans decreases the disease load. Conversely, the absence of pesticide spraying and failure to utilize treated nets, together with unprotected sexual intercourse with sick individuals, result in the co-circulation of the three diseases, considerably aggravating the overall disease burden.
ESS in a River Network with Two Branches
In a recent work by Jiang et al. [Bull. Math. Biol., 2020, Paper No. 131, 42pp], to discuss the impact of differing topology of river networks on the evolution of dispersal, the authors proposed three different models and found the winning of slower/faster diffuser or evolutionarily singular strategies. However, it is unknown whether there is an evolutionarily stable strategy (ESS, a central concept in evolution game theory) for these three models. In this paper, focusing on the Model (III) proposed by them (the most complicated one as they stated), we give a confirmed answer to this issue, that is, there does exist ESS.
Cross-Invariant Sets of the Cubic Nonlinear Schrödinger System with Partial Confinement
This paper studies the cubic nonlinear Schrödinger system with partial confinement: \begin{cases}-i\varphi_t + (x_1^2 + x_2^2)\varphi = \Delta\varphi +\mu_1|\varphi|^2\varphi + \beta|\psi|^2\varphi, & (t,x) \in \mathbb{R}^+\times \mathbb{R}^3, \\ -i\psi_t + (x_1^2 + x_2^2)\psi =\Delta\psi + \mu_2|\psi|^2\psi + \beta|\varphi|^2\psi, & (t,x) \in\mathbb{R}^+ \times \mathbb{R}^3,\end{cases} which models the Bose-Einstein condensates with multiple states or the propagation of mutually incoherent wave packets in nonlinear optics. The cross-invariant sets of the evolution flow are obtained by constructing the cross-constrained variational problem. Furthermore, the sharp condition for global existence and blowup of the solutions is derived.
Analysis of a Fractional Order Eco-Epidemiological Model with Prey Infection and Predator Cannibalism
This study proposes a fractional-order prey-predator model that integrates disease dynamics within prey populations, fear induced behavioral responses, and predator cannibalism. The prey population is divided into susceptible and infected classes to represent disease dynamics effectively, while predator cannibalism is modeled as a survival strategy during prey scarcity. Essential properties such as non-negativity, boundedness, and the existence and uniqueness of solutions are thoroughly examined. Stability of equilibrium points is established, and global stability is proved using Lyapunov functions. Numerical simulations validate the theoretical results, highlighting how variations in cannibalism rates and fractional-order parameters influence population dynamics and system behavior.
Optimal Control Approach for Bilateral Elastic Contact Problem with Power-Law Friction
We use the control variational technique to examine an elastic contact model, subject to a non-penetration condition in the normal direction and to power-law friction, proving the unique existence of the solution. This method uses optimal control theory to minimize the energy functional of the nonlinear equation. A multivalued equation $f\in \mathcal{F}y+∂\Phi(y)$ for the displacement field describes the problem in a weak formulation, where a linear mapping is represented by $\mathcal{F},$ and the Clarke’s subdifferential of the mapping $\Phi$ is indicated by $∂\Phi.$ We employ abstract existence theorems to verify the unique weak solution to the contact model.
Coulomb dry friction (power-law friction)
Nonlinear Dynamics and Pull-in Phenomena in a Magneto-Electro MEMS Actuator with Hardening Spring
This paper investigates the dynamic behavior and pull-in instability of a magneto-electro Micro-Electro-Mechanical System (MEMS) actuator, focusing on the nonlinear effects arising from magnetic forces and spring stiffness. The system consists of a movable wire attracted toward a stationary wire due to magnetic forces generated by applied currents. A critical equilibrium, known as the pull-in point, is reached when the currents exceed a threshold, leading to instability. We consider the governing equation based on Newton’s Second Law, incorporating a nonlinear restoring force for the spring, which exhibits hardening behavior. The resulting second-order differential equation is analyzed using qualitative and bifurcation theories, revealing the critical bifurcation values determined by the currents and spring stiffness. Through a dynamical systems approach, we characterize the phase portraits and solutions, identifying distinct dynamical behaviors and the conditions for pull-in instability. Numerical simulations are performed to validate the analytical predictions, demonstrating excellent agreement with the theoretically derived threshold.
On the Numerical Solution of Solidification of Water
The objective of this paper is to find an economical computer algorithm using a third-degree polynomial for the free boundary problem corresponding to phase change problem. We solve the problem to obtain the approximate analytical expression for the temperature distribution and the movement of the interface using the constrained integral method. Results are validated against existing numerical methods, demonstrating strong agreement with other techniques.
Generalized Variational Inclusion Problem Involving Averaged Operator and Cayley Operator
In this paper, we consider and study a variational inclusion problem involving averaged and Cayley operators in the context of a Hilbert space. It is shown that our problem is equivalent to a fixed-point equation. We define an iterative algorithm based on a fixed-point formulation to obtain the solution. Our result is supported by a numerical example, computation tables, and a convergence graph are also provided.
Stability Analysis of a Diffusive Ratio-Dependent Predator-Prey Model with Nonlocal Perception
In this paper, we propose a diffusive ratio-dependent predator-prey model with a nonlocal perceptual term. We first discuss the well-posedness of solutions. Then for the model without the nonlocal perception term, we obtain stability conditions of the spatially homogeneous steady state. Under these conditions, when considering the effects of nonlocal perception, our result indicates that if the system without nonlocal perception term is stable, and even after introducing nonlocal perception term, the system remains stable. However, if the system without nonlocal perception term is unstable, and the introduction of nonlocal perception term will make the system become stable.
The Impact of Educational Campaigns on the Global Dynamics of Cholera Epidemics
In this study, we examine a cholera epidemic model that incorporates educational transmission and delve into the long-term dynamics of the epidemic equilibrium point, particularly in relation to the spread of information about disease transmission. We find that if the basic reproduction number $R_0$ is less than or equal to 1, the disease is destined to be eradicated, and the unique disease-free equilibrium point becomes globally stable. It is noteworthy that the stability of this disease-free equilibrium is not influenced by the extent of disease education, the variety of infections, or the rate of education. On the other hand, if the basic reproduction number is greater than 1, we study the existence of the endemic equilibrium, and demonstrate that the disease will persist if $β_0$ > $D_0$. To assess the impact of educational campaigns on disease control, we further establish the asymptotic behavior of both infected and susceptible populations in response to educational interventions. Our findings indicate that well-executed educational campaigns can substantially contribute to the management and mitigation of the disease’s effects.
Existence Results for a Generalized $p$($x$)-Biharmonic Problem Type with No-Flux Boundary Condition
This paper aims to study the existence and multiplicity of weak solutions for a problem involving a generalized $p$($x$)-biharmonic operator with no flux boundary condition. By using the variational techniques and the theory of the variable exponent Lebesgue spaces, we obtain the existence of at least one nontrivial solution and at least $n$ distinct pairs of nontrivial weak solutions to this problem, respectively.
MEK Inhibition May Prevent the Recruitment of NK Cells in the Tumor Microenvironment: A Mathematical Model of Glioma Treatment
In this paper, we analyze the dynamics of oncolytic virotherapy combined with MEK inhibitors for the treatment of glioma. Although the therapeutic effect of oncolytic viruses (OV) depends on the recruitment of NK cells, these viruses are frequently subject to NK cell clearance, which may reduce the efficacy of oncolytic virotherapy. Furthermore, it is more challenging for OV to enter glioma cells because these cells lack Coxsackievirus and Adenovirus Receptors (CAR). MEK inhibitors, however, are able to compensate for this deficiency in CAR molecules. Herein, we propose and analyze a mathematical model of the dynamics of cancer cells, oncolytic viruses, activated NK cells, and CAR molecules while combining oncolytic virotherapy with MEK inhibition. Our goal is to identify the circumstances in which the treatment may be effective. In this research, we investigated the existence of equilibrium points. The two endemic equilibriums and the virus infection-free equilibrium stability conditions are given. The findings demonstrate that, in a dynamical system, NK cell activation can either establish or destroy equilibrium points and that substantial recruitment of activated NK cells might have detrimental effects on oncolytic virotherapy. However, MEK inhibitors boost OV effectiveness and may prevent NK cell recruitment.
Local Ground-State and Mountain Pass Solutions for a $p$-Kirchhoff Equation with Critical Exponent
We study a Kirchhoff-type equation where the diffusion coefficient is non-locally affected, the nonlinear diffusion phenomenon is governed by the $p$-Laplace operator and the population supply presents critical growth. The energy functional associated with the equation is not bounded from below so that there is no global ground-state; however, we prove the existence of a positive local ground-state. We also prove that the equation has a positive solution of mountain pass type. The concentration-compactness principle is a main tool in our approach.
A Mathematical Modeling of Two-Strain Tuberculosis Transmission: Deterministic and Stochastic Approaches to Screening, Treatment, and Quarantine Strategies
This research observes the transmission dynamics of drug-sensitive tuberculosis (DS-TB), drug-resistant tuberculosis (DR-TB) and undiagnosed tuberculosis infections using deterministic and stochastic models. The study focuses on the impacts of contact rates, screening of latent individuals or high-risk groups, treatment, and quarantine measures on the basic reproduction rate ($R_0$) and the estimated number of infected individuals ($m$). The results show that undiagnosed infected individuals are the greatest factor on the spread of tuberculosis within the system, which emphasizes the importance of screening on latent individuals. Analysis on the effects of screening on latent individuals also emphasizes the importance of treatment and quarantine for both DS-TB and DR-TB infections, respectively. However, analysis on the effects of both treatment and quarantine states that relying on either treatment or quarantine efforts is not enough to stop the spread of the two-strain TB within the system; thus a combined strategies is required to help eradicate the disease.