https://orcid.org/0000-0002-8443-8848
Figures
Abstract
This study presents an innovative approach to analyzing finite-time stability (FTS) and synchronization (FTSYN) in integer-order reaction-diffusion systems (RDs), particularly in the context of epidemiological modeling. By integrating Gronwall’s inequality, Lyapunov functionals (LFs), and linear control strategies, a comprehensive framework is developed to address transient dynamics within finite time frames. The proposed methodology advances the theoretical understanding of FTS and FTSYN by addressing the relatively unexplored dynamics of spatially extended systems. MATLAB simulations validate the theoretical findings, demonstrating the effectiveness of the control schemes and their practical applicability in modeling real-world disease transmission. Integrating spatial diffusion and disease dynamics provides critical insights into the influence of parameters such as diffusion rates and mortality on system behavior. This work contributes a robust framework for enhancing the analysis and management of nonlinear systems, with significant implications for epidemiology and other fields requiring rapid convergence and synchronization.
Citation: Batiha I, Anakira N, Bendib I, Ouannas A, Hioual A, Irianto I, et al. (2025) Finite-time analysis of epidemic reaction-diffusion models: Stability, synchronization, and numerical insights. PLoS One 20(5): e0321132. https://doi.org/10.1371/journal.pone.0321132
Editor: Jun Ma, Lanzhou University of Technology, CHINA
Received: September 2, 2024; Accepted: March 2, 2025; Published: May 27, 2025
Copyright: © 2025 Batiha et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Introduction
The study of infectious diseases has garnered considerable attention, with significant efforts directed toward advancing their understanding and management. This interest arises from the need to predict and mitigate disease transmission patterns to reduce mortality rates. Recent progress in mathematical modeling, particularly using the susceptible-infected-susceptible (SIS) epidemic RDs, demonstrates the potential for substantial reductions in disease-related fatalities [1–4]. De Jong et al. [5] introduced the standard incidence term , diverging from the mass action principle and inspiring further investigations by Allen et al. into its broader applicability. Another study [6] proposed a frequency-dependent SIS RDs for continuous spatial domains, offering a refined approach to disease transmission by integrating spatial dynamics and interaction frequency. Peng and Liu [7] rigorously analyzed the endemic equilibrium in Allen et al.’s framework, elucidating its stability conditions and advancing the understanding of disease persistence and transmission. These findings form a robust foundation for designing effective intervention strategies, with implications for improving public health outcomes.
This study analyzes a RDs (1) described by:
The function is a continuously differentiable function that is positive on the interval
, satisfying:
The terms and variables in the studied models significantly influence the system’s dynamics, as discussed in [8,9]. Key parameters include the following:
- The diffusion coefficients d1 and d2 represent the diffusion rates of susceptible and infected individuals across a spatial domain. These coefficients are influenced by real-world factors such as population mobility, migration, and travel patterns. Specifically, d1 denotes the rate at which susceptible individuals (e.g., uninfected people) move or are exposed to different areas, while d2 captures the rate at which infected individuals spread the disease spatially. For instance, d2 may be affected by the mobility of sick individuals traveling to hospitals or areas of high foot traffic, where disease transmission can occur more rapidly.
- The rate of new exposures (
) represents the frequency at which susceptible individuals come into contact with sources of infection, becoming exposed. This rate is influenced by social behavior, population density, and the effectiveness of public health interventions. For example, during a flu outbreak,
might be particularly high in crowded areas such as public transportation or schools, where encounters with infected individuals are more frequent.
- The disease spread frequency (
) governs the speed at which the infection spreads between individuals, making it a critical factor in understanding the growth of an epidemic. A higher
indicates faster disease transmission. This parameter is closely related to the transmission rate, which depends on factors such as the contagion level of the virus, its mode of transmission (e.g., airborne or surface contact), and protective measures like masks. Public health interventions, such as quarantine measures, vaccination campaigns, or isolation, can reduce
by limiting contact between susceptible and infected individuals.
- The mortality rate (
) quantifies the rate at which infected individuals succumb to the disease. This rate reflects the disease’s lethality, depending on regional, population-specific, or healthcare-related contexts. For example, a more developed healthcare system may exhibit lower mortality rates due to better access to treatment. Additionally, factors such as the availability of medical resources, the severity of the disease, and population demographics (e.g., the higher vulnerability of older individuals) can significantly impact
.
- The average disease duration (
) describes the typical length of time an individual remains infectious and symptomatic. This duration is influenced by the progression of the disease and the availability of medical treatments. For example, individuals infected with influenza may be infectious for 5–7 days, whereas diseases like Ebola often involve longer contagious periods. Advances in medical treatments can shorten
, while chronic illnesses or limited access to healthcare can prolong it, see Fig 1.
The study of stability theory remains a cornerstone in system analysis, with FTS gaining prominence due to its rapid convergence and enhanced robustness [10,11]. In [12], FTS for linear time-invariant fractional-order systems was introduced using the Lyapunov–Razumikhin technique [13]. In contrast, a novel criterion for FTS in integer-order nonlinear systems was proposed based on the generalized Gronwall inequality [14]. However, the absence of FTS equilibrium points (EPs) in most nonlinear systems underscores the inherent challenges in their stability analysis, emphasizing the need for innovative analytical techniques [15–23]. Synchronization of nonlinear systems has garnered significant attention over recent decades [24–31], with FTSYN emerging as a compelling research area [32]. Despite these advancements, the FTSYN of spatially extended systems, particularly RDs, remains under explored. This gap highlights an important avenue for future investigation into the dynamics and synchronization of such systems, with the potential to advance nonlinear dynamics and synchronization theory [33,34]. Recent developments underscore the role of finite-time control in managing nonlinear systems influenced by complex dynamics, such as event-triggered outputs [35]. This approach ensures that system states achieve desired behaviors or synchronization within a prescribed time, making it critical for applications requiring precision and rapid responses. Integer and fractional-order systems have also gained attention for their ability to model memory and hereditary effects prevalent in RD processes [36,37]. Event-triggered control reduces communication and computation overheads by activating control actions based on specific conditions, proving vital for large-scale networks like those in epidemiology and biology. Synchronization across integer and fractional-order RD nodes is crucial for maintaining coherence and stability in such systems [38,39]. Although advancements in synchronization techniques for nonlinear systems are notable, the FTSYN of integer-order RD networks remains relatively underexplored. This study addresses this gap by integrating finite-time control with integer-order dynamics to achieve synchronization in complex RD systems. The proposed approach provides valuable insights into the interplay between dynamics, synchronization, and FTS, with applications spanning epidemiology, physics, and engineering.
The paper is structured as follows: Mathematical background discusses the foundational mathematical framework and key theoretical tools required for the study. Finite time stability result focuses on analyzing the stability of EPs within a finite period, utilizing LF and related techniques. Finite-time synchronization scheme explores the synchronization dynamics of master-slave systems, emphasizing FTSYN with control strategies. Numerical simulations presents practical examples to validate theoretical results, showcasing the applicability of the proposed methods in real-world scenarios using MATLAB.
Mathematical background
This section establishes the foundational mathematical framework and theoretical tools for analyzing FTS and FTSYN in RDss. Key concepts, such as LF, eigenvalue properties, and Gronwall’s inequality, are introduced to derivate stability conditions and synchronization criteria. Additionally, lemmas and definitions pertinent to the boundedness and convergence of solutions are presented to build a comprehensive theoretical basis for the subsequent sections.
Lemma 1. [40,41] Assume that function satisfies:
where and
. If
is non-decreasing, then
Lemma 2. [42] Let satisfying the boundary condition
. Then, for the eigenvalue
associated with the following system:
the following inequality holds:
Lemma 3. If the subsequent conditions hold:
- There exist constant C1>0 such that the function
satisfies the Lipschitz condition:
(7)
- There exist constant C2>0 such that the function
is uniformly bounded:
(8)
Then, the inequality (9) holds:
where
Proof 1. We estimate the term as follows:
Definition 1. [40] The system given by (1) is FTS with respect to if
implies
where , and
.
Lemma 4. [43] The conditions stated in (2) imply
Finite time stability result
In this section, we analyze the FTS of the EPs for the proposed RD epidemic model. FTS ensures that system trajectories converge to the equilibrium states within a bounded time, which is critical for applications requiring rapid stabilization of dynamic processes.
Two key EPs are considered:
- The disease-free EP
representing a scenario where the infection is eradicated from the population.
- The endemic EP
which characterizes the persistence of the infection in the population under certain conditions.
We employ LF, Gronwall’s inequality, and eigenvalue analysis to derive sufficient conditions for FTS at these EPs. The analysis focuses on the role of model parameters, such as diffusion rates, mortality, and recovery rates, in determining the EPs between the absence of the disease and its persistent presence in the population. Furthermore, explicit formulas for the settling time are presented, offering insights into the temporal dynamics of the system under different parameter settings.
Theorem 1. The EP of the system (1) is FTS, if the following conditions are satisfied:
where Additionally,
is defined as:
where and
are positive eigenvalues.
Proof 2. We utilize a positive LF defined as:
Then,
Next, we express the functions as:
where
By applying Green’s formula, we simplify each term as follows:
Combining the simplified expressions and applying Lemmas 1–4, we estimate:
Therefore, we have
where is a non-negative constant defined as:
By Lemma 1 and Definition 1,we can conclude that:
which implies
Hence, the settling time can be expressed as:
This completes the proof.
Theorem 2. The EP of system (1) is FTS if :
and
The settling time for FTS is defined as:
C is as defined in Lemma 3.
Proof 3. Consider the following positive definite function:
Let be a LF defined by:
Using Lemmas 2 and 3, we obtain:
This, consequently, yields
Hence, we conclude:
Utilizing Lemma 1, we obtain:
Thus, the setting time is given by:
Therefore, by Definition 1, it can be concluded that system (1) achieves stability within a finite duration, provided .
Finite-time synchronization scheme
This section presents a synchronization framework for master-slave RDs, focusing on achieving FTSYN. The proposed scheme ensures rapid convergence of the slave system’s states to those of the master system within a finite time. By employing LFs and designing state-dependent control laws, the approach addresses synchronization discrepancies robustly, suitable for complex nonlinear systems.
We delve into the FTSYN dynamics of the master-slave systems (1) and (31), where the slave system (31) is described by:
The control systems and
play a vital role in achieving FTSYN in the master-slave RDs. Specifically, they are designed to ensure that the states of the slave system synchronize with those of the master system within a finite time. These control functions are incorporated into the equations governing the slave system to address synchronization discrepancies.
Their key contributions include:
- By employing feedback mechanisms, the control terms
and
ensure that the error terms converge to zero, leading to synchronization.
- The controllers are designed based on LF and stability criteria to guarantee that synchronization is achieved within a finite period. This rapid convergence is essential for systems requiring precise and timely synchronization.
- The control strategies are adaptable to the nonlinear and spatially extended nature of RDs, addressing the complexities of these models.
The choice and formulation of and
are critical, as highlighted in the provided equations and proofs, where specific conditions and feedback laws are derived to achieve FTSYN efficiently.
We address the synchronization discrepancies present in Eqs (1) and (31):
We aim to demonstrate that the discrepancy tends to zero as time approaches . This is accomplished by substituting the expression derived from Eq (1) into the error system delineated in Eq (33):
Theorem 3. [44] is a FTS EP of the nonlinear system (31) if there exists a positive definite LF
three class
functions
and
such that:
Definition 2. [45,46] The systems (1) and (31) are said to be FTSYN if there exists a settling time such that:
and for all ,
Theorem 4. The systems described by Eqs (1) and (31) achieve FTSYN by implementing the following linear feedback controller:
where C is as defined in Lemma 3. The settling time of FTSYN is given by:
Proof 4. We have chosen a LF represented by:
Using Lemmas 2–3 and Green’s formula, we can calculate , leading us to conclude:
By defining , we obtain :
Using Theorem 1, we establish that the zero solution of the error system (33) signifies the FTS of the EP . Thus, we have:
This implies
Applying Lemma 1, we deduce the following inequality:
Finally, the synchronization time is estimated as :
Consequently, according to the criteria specified in Definition 2, the systems described by (Eqs 1) and (31) achieve synchronization within a finite time .
Numerical simulations
To validate the theoretical findings, this section presents numerical simulations of the proposed stability and synchronization methods. Examples illustrating the dynamic behavior of RDs are provided, with parameters tailored to demonstrate finite-time convergence. The results are visualized through spatiotemporal plots and LF trajectories, showcasing the practical applicability and accuracy of the developed methodologies.
Example 1. In the specified domain and
, the parameter values are set as follows:
The initial conditions are defined as :
The function is given by:
with satisfies the Lipschitz condition:
and remains uniformly bounded :
From the setup, the parameters and
are determined as:
The stability condition of Theorem 1 is satisfied:
where
The settling time is calculated as:
Figs 2 and 3 present, respectively, the solutions and
over space and time, demonstrating the dynamics under homogeneous Neumann boundary conditions. The EP
is determined based on Theorem 1, confirming the system’s FTS.
Figs 4, 5, 6, and 7 depict numerical validation, showing that errors and LF converge to zero as t approaches .
Example 2. Consider the intervals and
. The parameters for this example are chosen as
The initial conditions are given by:
The function is redefined for this scenario as:
This function satisfies the Lipschitz condition:
and is uniformly bounded as:
For this configuration, the computed values of and
are:
The stability condition outlined in Theorem 2 is verified with
and
The settling time for FTS is computed as:
Figs 8, 9, 10, and 11 illustrate the spatiotemporal dynamics of the solutions and
, highlighting the system’s behavior under the homogeneous Neumann boundary conditions. An EP
is identified, demonstrating the system’s FTS as per Theorem 2.
Numerical simulations verify the theoretical findings, showing that the LF converges to zero as time approaches
in Figs 12 and 13. The convergence of the error terms further corroborates the system’s FTS.
Example 3. Consider the spatial domain , and temporal domain
. The parameters are selected as follows:
The initial conditions are defined as:
and
The infection rate function is modified to account for spatial variability:
This function satisfies the Lipschitz condition and boundedness. Then, The computed stability parameters are:
Using Theorem 4, the stability condition is verified:
The settling time for FTSYN is calculated as:
Numerical simulations were conducted to validate the FTS and FTSYN of the proposed RDs, with results presented in the subsequent figures. These visualizations provide insights into the system dynamics and synchronization behavior under specified initial conditions and parameters.
- Figs 14 and 15 illustrate the spatiotemporal evolution of the master system’s state variables,
and
, over
and
, demonstrating stabilization towards the equilibrium point as predicted by Theorem 4, confirming FTS.
- Figs 16 and 17 show the slave system’s state variables,
and
, synchronizing with the master system under control laws, supporting the FTSYN claim from Theorem 4.
- Figs 18 and 19 display synchronization errors
and
converging to zero as
, verifying the control scheme’s effectiveness in achieving FTSYN.
- Figs 20 and 21 offer a one-dimensional perspective, plotting state trajectories and synchronization errors at x = 100, where rapid error decay validates FTSYN at a fixed spatial location.
- Fig 22 depicts the LF
, showing monotonic decrease and convergence to zero within
s, confirming synchronization error stability and satisfying Theorem 4.
These results collectively confirm the theoretical findings of Example 3, demonstrating the robustness and practical applicability of the proposed methodology in SIS RDs.
Conclusion and future work
This study has comprehensively analyzed FTS and FTSYN in integer-order epidemic RDs, addressing significant gaps in the existing literature. By integrating LF, Gronwall’s inequality, and linear control strategies, we derived sufficient conditions for the FTS of EPs and FTSYN of master-slave systems. Numerical simulations demonstrated the effectiveness of the proposed methodologies, providing valuable insights into the dynamic behavior and control of epidemic models. The results underline the critical role of diffusion rates, interaction frequencies, and control parameters in achieving rapid stabilization and synchronization within finite time. Practical examples and MATLAB simulations validate the theoretical findings and highlight the real-world applicability of the proposed framework in modeling and managing infectious disease transmission.
Despite the advancements made, several areas remain open for further exploration:
- Extending the current framework to fractional-order RDs to account for memory and hereditary effects in dynamic processes.
- Investigating more complex control strategies, including nonlinear and time-varying approaches, to enhance system robustness against uncertainties and external perturbations.
- Applying the proposed methods to larger, more complex networks, including multi-patch epidemic models and agent-based systems.
- Incorporating stochastic elements into the models to capture the randomness and uncertainty inherent in real-world epidemic scenarios.
- Developing hardware and software implementations of the control strategies to facilitate their deployment in real-time monitoring and control systems.
This work lays a robust foundation for future research into finite-time dynamics and synchronization, with potential applications in epidemiology, ecology, and engineering.
References
- 1. Zafar ZUA, Hussain MT, Inc M, Baleanu D, Almohsen B, Oke AS, et al. Fractional-order dynamics of human papillomavirus. Results Phys. 2022;34:105281.
- 2. Zafar ZUA, Zaib S, Hussain MT, Tunç C, Javeed S. Analysis and numerical simulation of tuberculosis model using different fractional derivatives. Chaos Solitons Fractals. 2022;160:112202.
- 3. Lei C, Zhou X. Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection. DCDS-B. 2022;27(6):3077.
- 4. Castillo-Chavez C, Cooke K, Huang W, Levin SA. On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). Part 1: Single population models. J Math Biol. 1989;27(4):373–98. pmid:2769085
- 5.
De Jong M, Diekmann O, Heesterbeek H. How does transmission of infection depend on population size. Cambridge: Cambridge University Press; 1995.
- 6. Allen LJS, Bolker BM, Lou Y, Nevai AL. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. DCDS-A. 2008;21(1):1–20.
- 7. Peng R, Liu S. Global stability of the steady states of an SIS epidemic reaction-diffusion model. Nonlinear Anal. 2009;71:239–47.
- 8. Piqueira J, Castano M, Monteiro L. Modeling the spreading of HIV in homosexual populations with heterogeneous preventive attitude. J Biol Syst. 2004;12:439–56.
- 9. Kim KI, Lin Z, Zhang Q. An SIR epidemic model with free boundary. Nonlinear Anal Real World Appl. 2013;14(5):1992–2001.
- 10. Almatroud OA, Ouannas A. Finite-time stability analysis of a discrete-time generalized reaction–diffusion system. Mathematics. 2024;12(23):3679.
- 11. Anakira N, Almalki A, Katatbeh D, Hani G, Jameel A, Al Kalbani K, et al. An algorithm for solving linear and non-linear Volterra integro-differential equations. IJASCA. 2023;15(3):77–83.
- 12. Ma Y, Wu B, Wang Y-E. Finite-time stability and finite-time boundedness of fractional order linear systems. Neurocomputing. 2016;173:2076–82.
- 13. Gokul P, Rakkiyappan R. New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach. J Franklin Inst. 2022;359(14):7620–31.
- 14. Phat VN, Thanh NT. New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach. Appl Math Lett. 2018;83:169–75.
- 15. Batiha I, Benguesmia A, Oussaeif T, Jebril I, Ouannas A, Momani S. Study of a superlinear problem for a time fractional parabolic equation under integral over-determination condition. Int J Appl Math. 2024;54(2):187–93.
- 16. Fatma A, Batiha I, Imad R, Takieddine O, Ouannas A. Solvability and weak controllability of fractional degenerate singular problem. JRC. 2024;5(2):542–50.
- 17. Almatroud O, Hioual A, Ouannas A, Batiha I. Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models. Comput Math Appl. 2024;170:25–32.
- 18. Hammad M, Bendib I, Alshanti W, Alshanty A, Ouannas A, Hioual A, et al. Fractional-order degn-harrison reaction-diffusion model: finite-time dynamics of stability and synchronization. Computation. 2024;12(1):144.
- 19. Almatroud OA, Bendib I, Hioual A, Ouannas A. On stability of a reaction diffusion system described by difference equations. J Differ Equ Appl. 2024;30(1):706–20.
- 20. Bendib I, Batiha I, Hioual A, Anakira N, Dalah M, Ouannas A. On a new version of Gierer-Meinhardt model using fractional discrete calculus. RNA. 2024;7:1–15.
- 21. Almatroud OA, Hioual A, Ouannas A, Grassi G. On fractional-order discrete-time reaction diffusion systems. Mathematics. 2023;11(11):2447.
- 22. Farraj G, Maayah B, Khalil R, Beghami W. An algorithm for solving fractional differential equations using conformable optimized decomposition method. IJASCA. 2023;15(1):187–96.
- 23. Alsayyed O, Hioual A, Gharib GM, Abualhomos M, Al-Tarawneh H, Alsauodi MS, et al. On stability of a fractional discrete reaction–diffusion epidemic model. Fractal Fract. 2023;7(10):729.
- 24. Berir M. Analysis of the effect of white noise on the Halvorsen system of variable-order fractional derivatives using a novel numerical method. IJASCA. 2024;16(3):294–306.
- 25. Kermack WO, McKendrick AG. Contributions to the mathematical theory of epidemics, part 1. Proc R Soc Lond Ser A Math Phys Sci. 1927;115:700–21.
- 26. Keeling MJ, Eames KTD. Networks and epidemic models. J R Soc Interface. 2005;2(4):295–307. pmid:16849187
- 27. Almeida R. Analysis of a fractional SEIR model with treatment. Appl Math Lett. 2018;84:56–62.
- 28. Rao R, Lin Z, Ai X, Wu J. Synchronization of epidemic systems with Neumann boundary value under delayed impulse. Mathematics. 2022;10(10):2064.
- 29. Cui R. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discret Contin Dyn Syst B. 2021;26:2997–3022.
- 30. Zhang L, Wang Z, Zhao X. Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality. J Evol Equ. 2020;20:1029–59.
- 31. Hamadneh T, Hioual A, Saadeh R, Abdoon MA, Almutairi DK, Khalid TA, et al. General methods to synchronize fractional discrete reaction–diffusion systems applied to the glycolysis model. Fractal Fract. 2023;7(11):828.
- 32. Alattas KA, Mostafaee J, Sambas A, Alanazi AK, Mobayen S, Vu MT, et al. Nonsingular integral-type dynamic finite-time synchronization for hyper-chaotic systems. Mathematics. 2021;10(1):115.
- 33. Ouannas A, Mesdoui F, Momani S, Batiha I, Grassi G. Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law. Archiv Control Sci. 2021;31(2):1–13.
- 34. Bouaziz K, Douaifia R, Abdelmalek S. Asymptotic stability of solutions for a diffusive epidemic model. Demonstr Mathematica. 2022;55(1):553–73.
- 35. Ouannas A, Abdelli M, Odibat Z, Wang X, Pham V-T, Grassi G, et al. Synchronization control in reaction-diffusion systems: application to Lengyel-Epstein system. Complexity. 2019;2019(1).
- 36. Ouannas A, Wang X, Pham V-T, Grassi G, Huynh VV. Synchronization results for a class of fractional-order spatiotemporal partial differential systems based on fractional Lyapunov approach. Bound Value Probl. 2019;2019(1).
- 37. Ouannas A, Batiha IM, Bekiros S, Liu J, Jahanshahi H, Aly AA, et al. Synchronization of the glycolysis reaction-diffusion model via linear control law. Entropy (Basel). 2021;23(11):1516. pmid:34828214
- 38. Batiha I, Ogilat O, Bendib I, Ouannas A, Jebril I, Anakira N. Finite-time dynamics of the fractional-order epidemic model: Stability, synchronization, and simulations. Chaos Solitons Fractals X. 2024;13:100118.
- 39. Batiha IM, Bendib I, Ouannas A, Jebril IH, Alkhazaleh S, Momani S. On new results of stability and synchronization in finite-time for Fitiz-Nagamo model using grownal inequality and Lyapunov function. JRC. 2024;5(6):1897–909.
- 40. Yang X, Song Q, Liu Y, Zhao Z. Finite-time stability analysis of fractional-order neural networks with delay. Neurocomputing. 2015;152:19–26.
- 41.
Corduneanu C. Principles of differential and integral equations. Boston: Allyn and Bacon; 1971.
- 42. Rao R, Lin Z, Ai X, Wu J. Synchronization of epidemic systems with neumann boundary value under delayed impulse. Mathematics. 2022;10(12):2064.
- 43. Bouaziz K, Douaifia R, Abdelmalek S. Asymptotic stability of solutions for a diffusive epidemic model. Demonstr Mathematica. 2022;55(1):553–73.
- 44. Feng Z, Xiang Z. Finite-time stability of fractional-order nonlinear systems. Chaos. 2024;34(2):023105. pmid:38305053
- 45. Wang L, Yang X, Liu H, Chen X. Synchronization in finite time of fractional-order complex-valued delayed gene regulatory networks. Fractal Fract. 2023;7(5):347.
- 46. Li Y, Yang X, Shi L. Finite-time synchronization for competitive neural networks with mixed delays and non-identical perturbations. Neurocomputing. 2016;185:242–53.