Stability of a Discrete-Time Pathogen Infection Model with Adaptive Immune Response

Alshaikh, M. A.; Elaiw, A. M.

doi:https://doi.org/10.1155/2020/1264175

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2020 | Article ID 1264175 | https://doi.org/10.1155/2020/1264175

Stability of a Discrete-Time Pathogen Infection Model with Adaptive Immune Response

M. A. Alshaikh^1,2and A. M. Elaiw^1,3

Academic Editor: Nikos I. Karachalios

Received01 Jan 2020

Accepted23 Jun 2020

Published25 Aug 2020

Abstract

This paper studies the global stability of a discrete-time pathogen dynamic model with both cell-mediated and antibody immune responses. Both latently and actively infected cells are incorporated into the model. We discretize the continuous-time model by using the nonstandard finite difference (NSFD) method. We establish that NSFD preserves the nonnegativity and boundedness of the solutions of the model. We derive four threshold parameters which govern the existence and stability of the steady states. We establish by using the Lyapunov method, the global stability of the five steady states of the model. We illustrate our theoretical results by using numerical simulations.

1. Introduction

Studying and analysing the inward host dynamics of pathogens that infect the human body such as the human immunodeficiency virus (HIV), hepatitis C virus (HCV), hepatitis B virus (HBV), and human T-cell leukemia virus (HTLV) have received great attention from scientists and researchers (see, e.g., [1–12]). The dynamical behavior of pathogens can be understood by using mathematical modeling. The adaptive immune system plays an important role in controlling pathogenic infections. B cells and cytotoxic T lymphocytes (CTLs) are the main types of lymphocytes. B cells produce antibodies (antibody immune response) to neutralize the pathogens. CTLs (cell-mediated immune response) attack and kill cells infected by pathogen. In the literature, the impact of antibody immune response on pathogenic infections has been studied in several mathematical models (see, e.g., [13–20]). Further, the effect of cell-mediated immune response on pathogen dynamics has been modeled in several works (see, e.g., [21–27]). Wodarz [28] has combined both CTL and antibody immune responses in the following pathogen dynamic model:where , , , , and are the concentrations of healthy cells, infected cells, free pathogens, antibodies, and CTLs at time , respectively; the healthy cells are generated, dead, and become infected at rates , , and ; the term represents the death rate of infected cells; the infected cells are killed by CTLs at a rate ; the free pathogens are produced from infected cells at a rate , die at a rate , and neutralized by the antibodies at a rate ; the antibodies and CTLs are proliferated at rates and , respectively, while they die at rates and , respectively.

In models (1)–(5), it is assumed that cells generate mature pathogens directly after they are infected. However, it has been reported in [29] that a number of intracellular processes are needed to produce new mature pathogens. It has been shown in [30–32] that latent HIV reservoirs serve as a major barrier in HIV treatment. Pathogen dynamic models with latently infected cells and adaptive immunity have been studied in [33–35].

In the previous mentioned works, the pathogen dynamics are modeled by systems of nonlinear differential equations which are difficult to be solved analytically. Therefore, we can use an approximate solution which is obtained by discretizing the continuous-time model. Nonstandard finite difference (NSFD) has become one of the famous discretization methods which has been utilized in solving different types of differential equations [36]. NSFD can preserve the basic and global properties of several types of nonlinear systems [37]. NSFD has been used to discretize different pathogen dynamic models (see, e.g., [37–40]). The effect of the cell-mediated immune response has been considered of discrete-time pathogen dynamic models in [41, 42]. In addition, Elaiw and Alshaikh [43–45] have studied discrete-time pathogen infection models with antibodies.

The present paper aims to study the impact of both antibody and cell-mediated immune responses on the pathogen dynamic model. The model takes into account two types of infected cells, latently infected cells and actively infected cells. We use the NSFD method to discretize the continuous-time model. We show that the model is well posed. Further, we construct Lyapunov functions to prove the global stability of the steady states.

2. The Model

We consider the following continuous pathogen infection model with cell-mediated immune response, antibody immune response, and latency:where is the concentration of latently infected cells at time ; a fraction of healthy cells becomes actively infected cells, and the remaining fraction will become latently infected cells; the terms and represent the activation and death rates of the latently infected cells, respectively.

We apply the NSFD method [36] on systems (6)–(11), and we obtainwhere is the time step size and are the approximations of the solution of systems (6)–(11) at the discrete-time points , . The denominator function is chosen such that [46, 47]. Let us consider the following form of [47]:

The initial conditions of systems (12)–(17) are

2.1. Preliminaries

We define the setwhere , , , , and .

Lemma 1. Any solution of models (12)–(17) with initial condition (19) is nonnegative and ultimately bounded.

Proof. From equations (12)–(17), we obtainWe show that exists uniquely and is positive. If , then from equations (21) and (22) and the initial condition (19), we get and . From equations (23) and (26), we getThen,whereSince and , then , and hence equation (27) admits a unique positive root, . It follows from equation (23) thatNext, we consider . It follows from equations (24) and (25) thatThen, we getwhereSince and , then , and hence there exists a unique positive root of equation (32), .
From equation (24), we getTherefore, the solution exists uniquely and is nonnegative.
For , repeat the above process; we can show that exists uniquely and is positive. Utilizing the mathematical induction, for all , we know that exists uniquely and is positive.
To investigate the boundedness of solutions, we defineUsing equations (12)–(17), we getThen,Hence,Using lemma 2.2 in [48], we haveThen, , , , , , , and . Therefore, the solution converges to as .

2.2. Steady States

Systems (12)–(17) have five steady states:(i)Pathogen-free steady state , where .(ii)Chronic pathogen steady state without immune response , where in which is the basic reproduction number given as Thus, exists if .(iii)Chronic pathogen steady state with only antibody immune response , where in which Clearly, . If , then exists.(iv)Chronic pathogen steady state with only cell-mediated immune response , where in which Therefore, exists if .(v)Chronic pathogen steady state with both antibody and cell-mediated immune responses , where in which Thus, exists when and .

2.3. Global Stability Analysis

In this section, we study the global stability of the steady states by using the Lyapunov method. We define a function aswhich yields

Theorem 1. Let , then of models (12)–(17) is globally asymptotically stable (GAS).

Proof. Define a function aswhereThe solution of equation (51) is given byClearly, for all and at . We calculate asUsing inequality (49), we getFrom equations (12)–(17), we haveSince, , then for all , we have . Hence, the sequence is monotonically decreasing. Since , then , and hence . This yields , , and . For the case , we have , , , and . From equations (14) and (15), we obtain and . If , we have , , and . From equations (12), (14), and (15), we obtain , , and . Hence, the pathogen-free steady state is GAS.

Theorem 2. If and , then is GAS.

Proof. Define asWe have for all , and moreover . Computing asUsing inequality (49), we getFrom equations (12)–(17), we haveFrom the steady state conditions of , we haveand we getSince and , then is monotonically decreasing. We have , and then and hence , which implies , , , , and . We have four cases:(i) and , and then from equation (14), Applying the steady state condition (60), we obtain . In addition, from equation (15), we get From equation (60), we get .(ii), , and . From equation (63), we get .(iii), , and . From equation (62), we get .(iv), , , and . It follows that if and , then , , , , and . Then, is GAS.

Theorem 3. Let and , then of models (12)–(17) is GAS.

Proof. Define asClearly, for all and . Computing asUsing inequality (49), we getFrom equations (12)–(17), we haveWe haveand we getSince , then . Then, the sequence is monotonically decreasing. Since , then and hence . Thus, , , , , and . We have two cases:(i), and then from equation (14), and we get . From equation (15), we get This gives .(ii) and . From equation (71), we get . Hence, is GAS.

Theorem 4. Let and , then of models (12)–(17) is GAS.

Proof. Define asClearly, for all and . We have asUsing inequality (49), we getFrom equations (12)–(17), we haveWe haveand we getSince and if , then . Hence, the sequence is monotonically decreasing. Since , then and hence . Thus, , , , , and . We have two cases:(i), and from equation (14), and this gives ; moreover, from equation (15), we have then we get .(ii) and . From equation (78), we get and hence is GAS.

Theorem 5. Let , then of models (12)–(17) is GAS.

Proof. Define asClearly, for all and . Computing asUsing inequality (49), we getFrom equations (12)–(17), we haveWe haveand we getWe note that . It follows that is a monotone decreasing sequence. Since , then and hence . Thus, , , , and . From equations (14) and (15), we havethen and . Hence, is GAS.

3. Numerical Simulations

To conduct numerical simulations for models (12)–(17), we use the following data: , , , , , , , , , , , , and . The remaining parameters will be chosen as below.

3.1. Stability of Steady States

We consider the following initial values: IV1: , , , , , and IV2: , , , , , and IV3: , , , , , and

We choose varied , , and : Case (1) , , and . This yields . Figure 1 shows that the concentration of healthy cells increases and tends to the value , while the concentrations of the other compartments decrease and tend to zero for all initial IV1–IV3. Therefore, is GAS. This illustrates the result of Theorem 1. Case (2) , , and . These values give and . From Figure 2, it can be seen that the solutions of the system tend to the steady state , for all the three initial values IV1–IV3. As a result, exists, and based on Theorem 2, is GAS. Case (3) , , and , and then and . Figure 3 shows that the trajectories of the system reach the steady state for all the initial values IV1–IV3. This shows the existence of . According to Theorem 3, is GAS. Case (4) , , and , and then and . Figure 4 shows that the solutions starting with initial values IV1–IV3 converge to the steady state . This shows that exists, and it is GAS according to Theorem 4. Case (5) , , and , and then , and . Figure 5 displays that the solutions with initial values IV1–IV3 approach the steady state . Hence, exists; moreover, on the basis of Theorem 5, is GAS.

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3.2. Effect of Antipathogenic Drugs

In this section, we study the effect of antipathogenic drugs on the pathogen dynamics. We develop models (12)–(17) by incorporating two categories of antipathogenic drugs with efficacies and as

The basic reproduction number for this system is given by

Without the loss of generality, we let , and then

It follows that

Since the target of antipathogenic drugs is to clear the pathogens from the body, we have to determine the minimum drug efficacy such that for all . We can find the value of by solving the following algebraic equation:

The positive root of equation (91) is given by

Therefore,

We take , , and . We get . We consider the initial IV3 and choose different values of . Figure 6 shows the effect of antipathogenic drugs on the pathogen dynamics. It is observed that as is increased, the concentration of the healthy cells is increased, while the concentrations of the other compartments are decreased. In addition, we have(i)If , then and is GAS. In this case, the antipathogenic drugs can remove the pathogens from the body.(ii)If , then and had lost its stability and the solution trajectories of the system converge to one of the other steady states of the system. This means that using the drug treatment with less efficacy will not able to clear the pathogens from the body.

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3.3. Comparison Results

In this section, we perform a comparison between the NSFD method and Runge–Kutta method with a variable step size presented in MATLAB (ODE45). We compare the solutions of NSFD and ODE45 for Case (5). The numerical results show that there is a very good agreement between the result of NSFD ( and ) and that of ODE45 (see Figure 7). Of course, if one considers the Runge–Kutta method with a fixed numerical step size , then the numerical error will increase by increasing . Moreover, the global stability of the discrete-time model obtained by the Runge–Kutta method may not guaranteed. In the case of NSFD, we can see that the increasing of does not affect the stability of the steady states. This is in agreement with the results of several papers [37–40] that NSFD preserves the essential qualitative features of these models independently of the chosen step size .

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4. Conclusion

We studied a discrete-time pathogen dynamic model with adaptive immunity. The model incorporated two classes of infected cells, latently infected cells and actively infected cells. We proved that the solutions of the proposed model are positive and bounded. We showed that the model has five steady states, and their existence are governed by four threshold parameters. We constructed Lyapunov functions to prove the global stability of the steady states. In addition, we conducted numerical simulations to illustrate the global stability of the steady states. We studied the effect of antipathogenic treatment on the pathogen dynamics. We observed that as is increased, the concentration of the healthy cells is increased while the concentration of pathogens is decreased. Since the pathogen-free steady state is the desired steady state to be stabilized, we determined the critical drug efficacy by solving the equation and showed that is GAS when . We presented a comparison between the NSFD method and the Runge–Kutta method. The numerical results showed that there is a very good agreement between the result of NSFD and that of the Runge–Kutta method.

We mention that the proposed pathogen dynamic model is constructed at a supermacroscopic scale, namely, by means of population dynamics with deterministic interactions. This pathogen model can be extended to incorporate stochastic interactions (see [49]). Recently, many authors argued that the pathogens move freely in the body and follow the Fickian diffusion (see, e.g., [50–53]). Therefore, it is more reasonable to study reaction-diffusion versions of our discrete-time model. In certain circumstances, some pathogens such as HIV can suppress immune response or even destroy it especially when the concentration of the pathogens is too high. In the literature, several pathogen dynamic models were proposed by considering either CTL immune impairment (see, e.g., [54–56]) or antibody immune impairment (see, e.g., [57, 58]). However, modeling the impairment of both CTL and antibody immune responses has not been studied before. We leave these extensions as a possible future work.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74–79, 1996.
View at: Publisher Site | Google Scholar
A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999.
View at: Publisher Site | Google Scholar
D. Callaway and A. S. Perelson, “HIV-1 infection and low steady state viral loads,” Bulletin of Mathematical Biology, vol. 64, no. 1, pp. 29–64, 2002.
View at: Publisher Site | Google Scholar
N. Bellomo, K. J. Painter, Y. Tao, and M. Winkler, “Occurrence vs. Absence of taxis-driven instabilities in a may–nowak model for virus infection,” SIAM Journal on Applied Mathematics, vol. 79, no. 5, pp. 1990–2010, 2019.
View at: Publisher Site | Google Scholar
A. M. Elaiw, A. Almatrafi, A. D. Hobiny, and K. Hattaf, “Global properties of a general latent pathogen dynamics model with delayed pathogenic and cellular infections,” Discrete Dynamics in Nature and Society, vol. 2019, Article ID 9585497, 18 pages, 2019.
View at: Publisher Site | Google Scholar
A. M. Elaiw and S. A. Azoz, “Global properties of a class of HIV infection models with Beddington-DeAngelis functional response,” Mathematical Methods in the Applied Sciences, vol. 36, no. 4, pp. 383–394, 2013.
View at: Publisher Site | Google Scholar
A. M. Elaiw and N. A. Almuallem, “Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells,” Mathematical Methods in the Applied Sciences, vol. 39, no. 1, pp. 4–31, 2016.
View at: Publisher Site | Google Scholar
A. U. Neumann, N. P. Lam, H. Dahari et al., “Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon- therapy,” Science, vol. 282, no. 5386, pp. 103–107, 1998.
View at: Publisher Site | Google Scholar
S. Zhang and X. Xu, “Dynamic analysis and optimal control for a model of hepatitis C with treatment,” Communications in Nonlinear Science and Numerical Simulation, vol. 46, pp. 14–25, 2017.
View at: Publisher Site | Google Scholar
N. Yousfi, K. Hattaf, and A. Tridane, “Modeling the adaptive immune response in HBV infection,” Journal of Mathematical Biology, vol. 63, no. 5, pp. 933–957, 2011.
View at: Publisher Site | Google Scholar
A. M. Elaiw and A. A. Raezah, “Stability of general virus dynamics models with both cellular and viral infections and delays,” Mathematical Methods in the Applied Sciences, vol. 40, no. 16, pp. 5863–5880, 2017.
View at: Publisher Site | Google Scholar
Y. Wang, J. Liu, and J. M. Heffernan, “Viral dynamics of an HTLV-I infection model with intracellular delay and CTL immune response delay,” Journal of Mathematical Analysis and Applications, vol. 459, no. 1, pp. 506–527, 2018.
View at: Publisher Site | Google Scholar
A. M. Elaiw and N. H. AlShamrani, “Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal,” Nonlinear Analysis: Real World Applications, vol. 26, pp. 161–190, 2015.
View at: Publisher Site | Google Scholar
A. M. Elaiw and N. H. AlShamrani, “Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response,” Mathematical Methods in the Applied Sciences, vol. 40, no. 3, pp. 699–719, 2017.
View at: Publisher Site | Google Scholar
H. Miao, Z. Teng, C. Kang, and A. Muhammadhaji, “Stability analysis of a virus infection model with humoral immunity response and two time delays,” Mathematical Methods in the Applied Sciences, vol. 39, no. 12, pp. 3434–3449, 2016.
View at: Publisher Site | Google Scholar
A. M. Elaiw, E. Kh. Elnahary, and A. A. Raezah, “Effect of cellular reservoirs and delays on the global dynamics of HIV,” Advances in Difference Equations, vol. 2018, no. 1, p. 85, 2018.
View at: Publisher Site | Google Scholar
A. Elaiw and E. Elnahary, “Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays,” Mathematics, vol. 7, no. 2, p. 157, 2019.
View at: Publisher Site | Google Scholar
A. Murase, T. Sasaki, and T. Kajiwara, “Stability analysis of pathogen-immune interaction dynamics,” Journal of Mathematical Biology, vol. 51, no. 3, pp. 247–267, 2005.
View at: Publisher Site | Google Scholar
A. D. Hobiny, A. M. Elaiw, and A. Almatrafi, “Stability of delayed pathogen dynamics models with latency and two routes of infection,” Advances in Difference Equations, vol. 2018, p. 276, 2018.
View at: Google Scholar
J. Xu, Y. Zhou, Y. Li, and Y. Yang, “Global dynamics of a intracellular infection model with delays and humoral immunity,” Mathematical Methods in the Applied Sciences, vol. 39, no. 18, pp. 5427–5435, 2016.
View at: Publisher Site | Google Scholar
M. Y. Li and H. Shu, “Global dynamics of a mathematical model for HTLV-I infection of CD4⁺ T cells with delayed CTL response,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1080–1092, 2012.
View at: Publisher Site | Google Scholar
X. Shi, X. Zhou, and X. Song, “Dynamical behavior of a delay virus dynamics model with CTL immune response,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1795–1809, 2010.
View at: Publisher Site | Google Scholar
D. Huang, X. Zhang, Y. Guo, and H. Wang, “Analysis of an HIV infection model with treatments and delayed immune response,” Applied Mathematical Modelling, vol. 40, no. 4, pp. 3081–3089, 2016.
View at: Publisher Site | Google Scholar
C. Lv, L. Huang, and Z. Yuan, “Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 1, pp. 121–127, 2014.
View at: Publisher Site | Google Scholar
J. Pang, J.-A. Cui, and J. Hui, “The importance of immune responses in a model of hepatitis B virus,” Nonlinear Dynamics, vol. 67, no. 1, pp. 723–734, 2012.
View at: Publisher Site | Google Scholar
H. Shu, L. Wang, and J. Watmough, “Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses,” SIAM Journal on Applied Mathematics, vol. 73, no. 3, pp. 1280–1302, 2013.
View at: Publisher Site | Google Scholar
Y. Zhao and Z. Xu, “Global dynamics for a delayed hepatitis C virus, infection model,” Electronic Journal of Differential Equations, vol. 2014, no. 132, pp. 1–18, 2014.
View at: Google Scholar
D. Wodarz, “Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses,” Journal of General Virology, vol. 84, no. 7, pp. 1743–1750, 2003.
View at: Publisher Site | Google Scholar
A. S. Perelson, “Modelling viral and immune system dynamics,” Nature Reviews Immunology, vol. 2, no. 1, pp. 28–36, 2002.
View at: Publisher Site | Google Scholar
X. Wang, S. Tang, X. Song, and L. Rong, “Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission,” Journal of Biological Dynamics, vol. 11, no. sup2, pp. 455–483, 2017.
View at: Publisher Site | Google Scholar
T.-W. Chun, L. Stuyver, S. B. Mizell et al., “Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy,” Proceedings of the National Academy of Sciences, vol. 94, no. 24, pp. 13193–13197, 1997.
View at: Publisher Site | Google Scholar
J. K. Wong, M. Hezareh, H. F. Gunthard et al., “Recovery of replication-competent HIV despite prolonged suppression of plasma viremia,” Science, vol. 278, no. 5341, pp. 1291–1295, 1997.
View at: Publisher Site | Google Scholar
A. Elaiw and N. AlShamrani, “Stability of a general adaptive immunity virus dynamics model with multistages of infected cells and two routes of infection,” Mathematical Methods in the Applied Sciences, vol. 43, no. 3, pp. 1145–1175, 2020.
View at: Publisher Site | Google Scholar
A. M. Elaiw and N. H. AlShamrani, “Global stability of a delayed adaptive immunity viral infection with two routes of infection and multi-stages of infected cells,” Communications in Nonlinear Science and Numerical Simulation, vol. 86, Article ID 105259, 2020.
View at: Publisher Site | Google Scholar
A. M. Elaiw and N. H. AlShamrani, “Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays,” Mathematical Methods in the Applied Science, vol. 36, pp. 125–142, 2018.
View at: Google Scholar
R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.
A. Korpusik, “A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection,” Communications in Nonlinear Science and Numerical Simulation, vol. 43, pp. 369–384, 2017.
View at: Publisher Site | Google Scholar
A. M. Elaiw and M. A. Alshaikh, “Stability of discrete-time HIV dynamics models with three categories of infected CD4⁺ T-cells,” Advances in Difference Equations, vol. 2019, no. 1, p. 407, 2019.
View at: Publisher Site | Google Scholar
Y. Yang, X. Li, and L. Yahui, “Global stability of a discrete virus dynamics model with Holling type-II infection function,” Mathematical Methods in the Applied Sciences, vol. 39, no. 8, pp. 2078–2082, 2016.
View at: Publisher Site | Google Scholar
J. Xu, J. Hou, Y. Geng, and S. Zhang, “Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence,” Advances in Difference Equations, vol. 2018, p. 108, 2018.
View at: Google Scholar
J. Wang, Z. Teng, and H. Miao, “Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response,” Advances in Difference Equations, vol. 2016, no. 1, p. 143, 2016.
View at: Publisher Site | Google Scholar
J. Xu and Y. Geng, “Stability preserving NSFD scheme for a delayed viral infection model with cell-to-cell transmission and general nonlinear incidence,” Journal of Difference Equations and Applications, vol. 23, no. 5, pp. 893–916, 2017.
View at: Publisher Site | Google Scholar
A. M. Elaiw and M. A. Alshaikh, “Stability analysis of a general discrete-time pathogen infection model with humoral immunity,” Journal of Difference Equations and Applications, vol. 25, no. 8, pp. 1149–1172, 2019.
View at: Publisher Site | Google Scholar
A. M. Elaiw and M. A. Alshaikh, “Global stability of discrete virus dynamics models with humoural immunity and latency,” Journal of Biological Dynamics, vol. 13, no. 1, pp. 639–674, 2019.
View at: Publisher Site | Google Scholar
A. M. Elaiw and M. A. Alshaikh, “Global stability of discrete pathogen infection model with humoral immunity and cell-to-cell transmission,” Chaos, Solitons & Fractals, vol. 130, Article ID 109458, 2020.
View at: Publisher Site | Google Scholar
R. E. Mickens, Application of Nonstandard Finite Difference Scheme, World Scientific, Singapore, 2000.
R. E. Mickens, “Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition,” Numerical Methods for Partial Differential Equations, vol. 23, no. 3, pp. 672–691, 2007.
View at: Publisher Site | Google Scholar
P. Shi and L. Dong, “Dynamical behaviors of a discrete HIV-1 virus model with bilinear infective rate,” Mathematical Methods in the Applied Sciences, vol. 37, no. 15, pp. 2271–2280, 2014.
View at: Publisher Site | Google Scholar
L. Gibelli, A. Ełaiw, M. A. Alghamdi, and A. M. Althiabi, “Heterogeneous population dynamics of active particles: progression, mutations, and selection dynamics,” Mathematical Models and Methods in Applied Sciences, vol. 27, no. 4, pp. 617–640, 2017.
View at: Publisher Site | Google Scholar
N. Bellomo, Y. Tao, and Y. Tao, “Stabilization in a chemotaxis model for virus infection,” Discrete & Continuous Dynamical Systems-S, vol. 13, no. 2, pp. 105–117, 2020.
View at: Publisher Site | Google Scholar
A. M. Elaiw and A. D. Al Agha, “Analysis of a delayed and diffusive oncolytic M1 virotherapy model with immune response,” Nonlinear Analysis: Real World Applications, vol. 55, Article ID 103116, 2020.
View at: Publisher Site | Google Scholar
A. M. Elaiw, A. D. Hobiny, and A. D. Al Agha, “Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response,” Applied Mathematics and Computation, vol. 367, Article ID 124758, 2020.
View at: Publisher Site | Google Scholar
A. Elaiw and A. A. Agha, “Global analysis of a reaction-diffusion within-host malaria infection model with adaptive immune response,” Mathematics, vol. 8, p. 563, 2020.
View at: Publisher Site | Google Scholar
P. Krishnapriya and M. Pitchaimani, “Analysis of time delay in viral infection model with immune impairment,” Journal of Applied Mathematics and Computing, vol. 55, no. 1-2, pp. 421–453, 2017.
View at: Publisher Site | Google Scholar
A. M. Elaiw, A. A. Raezah, and S. A. Azoz, “Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment,” Advances in Difference Equations, vol. 2018, p. 414, 2018.
View at: Google Scholar
P. Krishnapriya and M. Pitchaimani, “Modeling and bifurcation analysis of a viral infection with time delay and immune impairment,” Japan Journal of Industrial and Applied Mathematics, vol. 34, no. 1, pp. 99–139, 2017.
View at: Publisher Site | Google Scholar
A. M. Elaiw, S. F. Alshehaiween, and A. D. Hobiny, “Global properties of a delay-distributed HIV dynamics model including impairment of B-cell functions,” Mathematics, vol. 7, no. 9, p. 837, 2019.
View at: Publisher Site | Google Scholar
A. M. Elaiw and S. F. Alshehaiween, “Global stability of delay-distributed viral infection model with two modes of viral transmission and B-cell impairment,” Mathematical Methods in the Applied Sciences, vol. 43, no. 11, pp. 6677–6701, 2020.
View at: Publisher Site | Google Scholar

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Copyright © 2020 M. A. Alshaikh and A. M. Elaiw. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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