Mathematics

Research

29 pages, 1323 KB

Open AccessArticle

Threshold Dynamics of a SIRI Model with Reinfection: Averaged and Periodic Systems and Application to Tuberculosis Data

by Fang Liu, Mingtao Li, Fenfen Zhang and Ruiqiang He

Mathematics 2026, 14(6), 953; https://doi.org/10.3390/math14060953 - 11 Mar 2026

Viewed by 125

Tuberculosis (TB) remains a major public health challenge in high-burden regions, where reinfection and seasonal variation play important roles in disease transmission. In this paper, we study a tuberculosis transmission model with reinfection based on the SIRI framework, with particular emphasis on the [...] Read more.

Tuberculosis (TB) remains a major public health challenge in high-burden regions, where reinfection and seasonal variation play important roles in disease transmission. In this paper, we study a tuberculosis transmission model with reinfection based on the SIRI framework, with particular emphasis on the intrinsic relationship between the averaged system and the periodic system. The averaged system is shown to characterize the long-term epidemiological behavior, whereas the periodic system captures short-term seasonal fluctuations. From a theoretical perspective, we prove that the periodic system and its corresponding averaged system share the same basic reproduction number. We analyze the threshold dynamics of the seasonal model and investigate the dynamical properties of the averaged system, including the existence and stability of equilibria and the occurrence of backward bifurcation. In particular, we show that disease persistence may occur even when the basic reproduction number (

R_{0}

) is less than one, and we examine the stability of equilibrium points at the critical threshold (

R_{0} = 1

). These results reveal how transmission and reinfection jointly determine the disease burden and equilibrium structure. To validate the theoretical findings, numerical simulations are performed using tuberculosis incidence data from Yunnan Province, China, covering the period from 2005 to 2020. The numerical simulations suggest that the seasonal model provides a better fit to the data, while the averaged model may overestimate the transmission potential of the disease. Under the condition that the two models share the same basic reproduction number, a constrained numerical simulation is performed. The results show that, under certain parameter settings, the endemic equilibrium of the averaged system can approximate the mean prevalence of the periodic solution. However, such an approximation cannot be guaranteed in general. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

31 pages, 4562 KB

Open AccessArticle

A Mathematical Model of Within-Host HBV and HTLV-1 Co-Infection Dynamics

by Amani Alsulami and Ebtehal Almohaimeed

Mathematics 2026, 14(5), 912; https://doi.org/10.3390/math14050912 - 7 Mar 2026

Viewed by 198

Abstract

Hepatitis B virus (HBV) and human T-lymphotropic virus type 1 (HTLV-1) are blood-borne pathogens with overlapping transmission routes, resulting in an increased prevalence of HBV among individuals infected with HTLV-1. Notwithstanding the widespread application of mathematical modeling to the study of each virus [...] Read more.

Hepatitis B virus (HBV) and human T-lymphotropic virus type 1 (HTLV-1) are blood-borne pathogens with overlapping transmission routes, resulting in an increased prevalence of HBV among individuals infected with HTLV-1. Notwithstanding the widespread application of mathematical modeling to the study of each virus in isolation, the within-host dynamics of HBV–HTLV-1 co-infection remain insufficiently characterized. This study introduces a novel within-host co-infection model that characterizes the interactions between HBV and HTLV-1, where HTLV-1 infects CD4⁺ T cells and HBV targets hepatocytes. A comprehensive qualitative analysis yields four threshold parameters (

R_{i}, i = 1, 2, 3, 4

) governing the existence and stability of equilibrium points, with global stability established using Lyapunov functions. Numerical simulations validate the analytical results, and sensitivity analysis identifies parameters that most strongly influence the basic reproduction numbers for HBV (

R_{1}

) and HTLV-1 (

R_{2}

) mono-infections. Our results corroborate that, in patients with HBV, the presence of HTLV-1 contributes to an elevated HBV viral load and CD4⁺ T cells play a crucial role in controlling HBV infection. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

25 pages, 3274 KB

Open AccessFeature PaperArticle

Understanding the Impact of Flight Restrictions on Epidemic Dynamics: A Meta-Agent-Based Approach Using the Global Airlines Network

by Alexandru Topîrceanu

Mathematics 2026, 14(2), 219; https://doi.org/10.3390/math14020219 - 6 Jan 2026

Viewed by 317

Abstract

In light of the current advances in computational epidemics and the need for improved epidemic governance strategies, we propose a novel meta-agent-based model (meta-ABM) constructed using the global airline complex network, using data from openflights.org, to establish a configurable framework for monitoring epidemic [...] Read more.

In light of the current advances in computational epidemics and the need for improved epidemic governance strategies, we propose a novel meta-agent-based model (meta-ABM) constructed using the global airline complex network, using data from openflights.org, to establish a configurable framework for monitoring epidemic dynamics. By integrating our validated SICARQD complex epidemic model with global flights and airport information, we simulate the progression of an airborne epidemic, specifically reproducing the resurgence of COVID-19. In terms of originality, our meta-ABM considers each airport node (i.e., city) as an individual agent-based model assigned to its own independent SICARQD epidemic model. Agents within each airport node engage in probabilistic travel along established flight routes, mirroring real-world mobility patterns. This paper focuses primarily on investigating the effect of mobility restrictions by measuring the total number of cases, the peak infected ratio, and mortality caused by an epidemic outbreak. We analyze the impact of four key restriction policies imposed on the airline network, as follows: no restrictions, reducing flight frequencies, limiting flight distances, and a hybrid policy. Through simulations on scaled population systems of up to 1.36 million agents, our findings indicate that reducing the number of flights leads to a faster and earlier decrease in total infection cases, while restricting maximum flight distances results in a slower and much later decrease, effective only after canceling over 80% of flights. Notably, for practical travel restriction policies (e.g., 25–75% of flights canceled), epidemic control is significantly more effective when limiting flight frequency. This study shows the critical role of reducing global flight frequency as a public health policy to control epidemic spreading in our highly interconnected world. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

34 pages, 1986 KB

Open AccessFeature PaperArticle

Inverse Problem for an Extended Time-Dependent SEIRS Model: Validation with Real-World COVID-19 Data

by Svetozar Margenov, Nedyu Popivanov, Tsvetan Hristov and Veneta Koleva

Mathematics 2026, 14(1), 13; https://doi.org/10.3390/math14010013 - 20 Dec 2025

Viewed by 555

Abstract

This paper introduces a novel SEIRS-type differential model that incorporates significant real-world factors such as vaccination, hospitalization, and vital dynamics. The model is described by a system of nonlinear ordinary differential equations with time-dependent parameters and coefficients. First, fundamental biological properties of the [...] Read more.

This paper introduces a novel SEIRS-type differential model that incorporates significant real-world factors such as vaccination, hospitalization, and vital dynamics. The model is described by a system of nonlinear ordinary differential equations with time-dependent parameters and coefficients. First, fundamental biological properties of the model, including the existence, uniqueness, and non-negativity of its solution, are established. In addition, using official COVID-19 data from Bulgaria, a special inverse problem for the differential model is formulated and investigated through the construction of an appropriate family of time-discrete inverse problems. As a result, the model parameters are identified, and the model is validated using real-world data. The presented numerical experiments confirm that the proposed methodology performs well in real-world applications with actual data. A very good agreement between computed and officially reported data with respect to the

l_{2}

and

l_{\infty}

norms is obtained. The model and its simulation tools are adaptable and can be applied to datasets from other countries, provided suitable epidemiological data are available. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

12 pages, 2973 KB

Open AccessArticle

Epidemic Spreading on Metapopulation Networks with Finite Carrying Capacity

by An-Cai Wu

Mathematics 2025, 13(18), 2994; https://doi.org/10.3390/math13182994 - 17 Sep 2025

Viewed by 557

Abstract

In this study, we formulate and analyze a susceptible–infected–susceptible (SIS) dynamic on metapopulation networks, where each node has a finite carrying capacity and the motion of individuals is modulated by vacant space at the destination. We obtain that the vacancy-dependent mobility pattern results [...] Read more.

In this study, we formulate and analyze a susceptible–infected–susceptible (SIS) dynamic on metapopulation networks, where each node has a finite carrying capacity and the motion of individuals is modulated by vacant space at the destination. We obtain that the vacancy-dependent mobility pattern results in various asymptotic population distributions on heterogeneous metapopulation networks. The resulting population distributions have remarkable impact on the behavior of SIS dynamics. We show that, for the given total number of individuals, higher heterogeneity in population distributions facilitates epidemic spreading in terms of both a smaller epidemic threshold and larger macroscopic incidence. Moreover, we analytically obtain a sufficient condition that the disease-free equilibrium becomes unstable and an endemic state arises. Contrary to the absence of an epidemic threshold in the standard diffusion case without excluded-volume effects, the finite carrying capacity induces a nonzero epidemic threshold under certain conditions in the limit of infinite network sizes with an unbounded maximum degree. Our analytical results agree well with numerical simulations. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

31 pages, 4652 KB

Open AccessArticle

A Delayed Malware Propagation Model Under a Distributed Patching Mechanism: Stability Analysis

by Wei Zhang, Xiaofan Yang and Luxing Yang

Mathematics 2025, 13(14), 2266; https://doi.org/10.3390/math13142266 - 14 Jul 2025

Cited by 3 | Viewed by 953

Abstract

Antivirus (patch) is one of the most powerful tools for defending against malware spread. Distributed patching is superior to its centralized counterpart in terms of significantly lower bandwidth requirement. Under the distributed patching mechanism, a novel malware propagation model with double delays and [...] Read more.

Antivirus (patch) is one of the most powerful tools for defending against malware spread. Distributed patching is superior to its centralized counterpart in terms of significantly lower bandwidth requirement. Under the distributed patching mechanism, a novel malware propagation model with double delays and double saturation effects is proposed. The basic properties of the model are discussed. A pair of thresholds, i.e., the first threshold

R_{0}

and the second threshold

R_{1}

, are determined. It is shown that (a) the model admits no malware-endemic equilibrium if

R_{0} \leq 1

, (b) the model admits a unique patch-free malware-endemic equilibrium and admits no patch-endemic malware-endemic equilibrium if

1 < R_{0} \leq R_{1}

, and (c) the model admits a unique patch-free malware-endemic equilibrium and a unique patch-endemic malware-endemic equilibrium if

R_{0} > R_{1}

. A criterion for the global asymptotic stability of the malware-free equilibrium is given. A pair of criteria for the local asymptotic stability of the patch-free malware-endemic equilibrium are presented. A pair of criteria for the local asymptotic stability of the patch-endemic malware-endemic equilibrium are derived. Using cybersecurity terms, these theoretical outcomes have the following explanations: (a) In the case where the first threshold can be kept below unity, the malware can be eradicated through distributed patching. (b) In the case where the first threshold can only be kept between unity and the second threshold, the patches may fail completely, and the malware cannot be eradicated through distributed patching. (c) In the case where the first threshold cannot be kept below the second threshold, the patches may work permanently, but the malware cannot be eradicated through distributed patching. The influence of the delays and the saturation effects on malware propagation is examined experimentally. The relevant conclusions reveal the way the delays and saturation effects modulate these outcomes. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

15 pages, 1447 KB

Open AccessArticle

The Waxing and Waning of Fear Influence the Control of Vector-Borne Diseases

by Jing Jiao

Mathematics 2025, 13(5), 879; https://doi.org/10.3390/math13050879 - 6 Mar 2025

Viewed by 1099

Abstract

One major challenge in preventing infectious diseases comes from human control behaviors. In the context of vector-borne diseases (VBDs), I explored how the waxing and waning of a human psychological emotion—fear—can generate diverse control actions, which, in turn, influence disease dynamics. Fear may [...] Read more.

One major challenge in preventing infectious diseases comes from human control behaviors. In the context of vector-borne diseases (VBDs), I explored how the waxing and waning of a human psychological emotion—fear—can generate diverse control actions, which, in turn, influence disease dynamics. Fear may diminish over time after being triggered but can also be reinforced when new triggers emerge. By integrating fear dynamics into a generic Ross–MacDonald model tailored for the Zika virus, I found that an increase in initial fear can enhance control efforts, thereby reducing the number of infected individuals and deaths. Once initial fear becomes strong enough to deplete the mosquito population, any further increase in fear no longer impacts disease dynamics. When initial fear is at an intermediate level, the increase in disease caused by greater decay in fear can be counterbalanced by increasing the frequency of fear triggers. Interestingly, when the control period is short and initial fear is at an intermediate level, increasing the frequency of fear reinforcement can lead to a “hydra effect”, which increases disease transmission. These findings help explain variations in human control efforts and provide insights for developing more effective disease control strategies that account for the fear dynamics of local communities. This work also contributes to advancing the theory at the intersection of human behavior, disease ecology, and epidemiology. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

23 pages, 400 KB

Open AccessFeature PaperArticle

Qualitative Analysis of a COVID-19 Mathematical Model with a Discrete Time Delay

by Abraham J. Arenas, Gilberto González-Parra and Miguel Saenz Saenz

Mathematics 2025, 13(1), 120; https://doi.org/10.3390/math13010120 - 31 Dec 2024

Viewed by 1350

Abstract

The aim of this paper is to investigate the qualitative behavior of a mathematical model of the COVID-19 pandemic. The constructed SAIRS-type mathematical model is based on nonlinear delay differential equations. The discrete-time delay is introduced in the model in order to take [...] Read more.

The aim of this paper is to investigate the qualitative behavior of a mathematical model of the COVID-19 pandemic. The constructed SAIRS-type mathematical model is based on nonlinear delay differential equations. The discrete-time delay is introduced in the model in order to take into account the latent stage where the individuals already have the virus but cannot yet infect others. This aspect is a crucial part of this work since other models assume exponential transition for this stage, which can be unrealistic. We study the qualitative dynamics of the model by performing global and local stability analysis. We compute the basic reproduction number

R_{0}^{d}

, which depends on the time delay and determines the stability of the two steady states. We also compare the qualitative dynamics of the delayed model with the model without time delay. For global stability, we design two suitable Lyapunov functions that show that under some scenarios the disease persists whenever

R_{0}^{d} > 1

. Otherwise, the solution approaches the disease-free equilibrium point. We present a few numerical examples that support the theoretical analysis and the methodology. Finally, a discussion about the main results and future directions of research is presented. Full article

(This article belongs to the Special Issue Mathematical Epidemiological Models: Classical and Interdisciplinary Applications)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Mathematical Epidemiological Models: Classical and Interdisciplinary Applications

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (8 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI