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Journal articles on the topic 'Disease-free equilibrium'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Xu, Jinhu, Wenxiong Xu, and Yicang Zhou. "Analysis of a delayed epidemic model with non-monotonic incidence rate and vertical transmission." International Journal of Biomathematics 07, no. 04 (June 25, 2014): 1450041. http://dx.doi.org/10.1142/s1793524514500417.

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A delayed SEIR epidemic model with vertical transmission and non-monotonic incidence is formulated. The equilibria and the threshold of the model have been determined on the bases of the basic reproduction number. The local stability of disease-free equilibrium and endemic equilibrium is established by analyzing the corresponding characteristic equations. By comparison arguments, it is proved that, if R0 < 1, the disease-free equilibrium is globally asymptotically stable. Whereas, the disease-free equilibrium is unstable if R0 > 1. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R0 > 1.

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2

Moghadas, S. M., and A. B. Gumel. "An epidemic model for the transmission dynamics of HIV and another pathogen." ANZIAM Journal 45, no. 2 (October 2003): 181–93. http://dx.doi.org/10.1017/s1446181100013250.

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AbstractA five-dimensional deterministic model is proposed for the dynamics between HIV and another pathogen within a given population. The model exhibits four equilibria: a disease-free equilibrium, an HIV-free equilibrium, a pathogen-free equilibrium and a co-existence equilibrium. The existence and stability of these equilibria are investigated. A competitive finite-difference method is constructed for the solution of the non-linear model. The model predicts the optimal therapy level needed to eradicate both diseases.

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3

LIU, YIPING, and JING-AN CUI. "THE IMPACT OF MEDIA COVERAGE ON THE DYNAMICS OF INFECTIOUS DISEASE." International Journal of Biomathematics 01, no. 01 (March 2008): 65–74. http://dx.doi.org/10.1142/s1793524508000023.

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In this paper, we give a compartment model to discuss the influence of media coverage to the spreading and controlling of infectious disease in a given region. The model exhibits two equilibria: a disease-free and a unique endemic equilibrium. Stability analysis of the models shows that the disease-free equilibrium is globally asymptotically stable if the reproduction number (ℝ0), which depends on parameters, is less than unity. But if ℝ0 > 1, it is shown that a unique endemic equilibrium appears, which is asymptotically stable. On a special case, the endemic equilibrium is globally stable. We discuss the role of media coverage on the spreading based on the theory results.

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4

Chukwu, C. W., and F. Nyabadza. "A Theoretical Model of Listeriosis Driven by Cross Contamination of Ready-to-Eat Food Products." International Journal of Mathematics and Mathematical Sciences 2020 (March 9, 2020): 1–14. http://dx.doi.org/10.1155/2020/9207403.

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Cross contamination that results in food-borne disease outbreaks remains a major problem in processed foods globally. In this paper, a mathematical model that takes into consideration cross contamination of Listeria monocytogenes from a food processing plant environment is formulated using a system of ordinary differential equations. The model has three equilibria: the disease-free equilibrium, Listeria-free equilibrium, and endemic equilibrium points. A contamination threshold ℛwf is determined. Analysis of the model shows that the disease-free equilibrium point is locally stable for ℛwf<1 while the Listeria-free and endemic equilibria are locally stable for ℛwf>1. The time-dependent sensitivity analysis is performed using Latin hypercube sampling to determine model input parameters that significantly affect the severity of the listeriosis. Numerical simulations are carried out, and the results are discussed. The results show that a reduction in the number of contaminated workers and removal of contaminated food products are essential in eliminating the disease in the human population and vice versa. The results have significant public health implications in the management and containment of any listeriosis disease outbreak.

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5

Lu, Jinna, Xiaoguang Zhang, and Rui Xu. "Global stability and Hopf bifurcation of an eco-epidemiological model with time delay." International Journal of Biomathematics 12, no. 06 (August 2019): 1950062. http://dx.doi.org/10.1142/s1793524519500621.

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In this paper, an eco-epidemiological model with time delay representing the gestation period of the predator is investigated. In the model, it is assumed that the predator population suffers a transmissible disease and the infected predators may recover from the disease and become susceptible again. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free and coexistence equilibria are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the coexistence equilibrium, the disease-free equilibrium and the predator-extinct equilibrium of the system, respectively.

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6

Lotfi, El Mehdi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi. "Partial Differential Equations of an Epidemic Model with Spatial Diffusion." International Journal of Partial Differential Equations 2014 (February 10, 2014): 1–6. http://dx.doi.org/10.1155/2014/186437.

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The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

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7

Zhonghua, Zhang, and Suo Yaohong. "Stability and Sensitivity Analysis of a Plant Disease Model with Continuous Cultural Control Strategy." Journal of Applied Mathematics 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/207959.

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In this paper, a plant disease model with continuous cultural control strategy and time delay is formulated. Then, how the time delay affects the overall disease progression and, mathematically, how the delay affects the dynamics of the model are investigated. By analyzing the transendental characteristic equation, stability conditions related to the time delay are derived for the disease-free equilibrium. Specially, whenR0=1, the Jacobi matrix of the model at the disease-free equilibrium always has a simple zero eigenvalue for allτ≥0. The center manifold reduction and the normal form theory are used to discuss the stability and the steady-state bifurcations of the model near the nonhyperbolic disease-free equilibrium. Then, the sensitivity analysis of the threshold parameterR0and the positive equilibriumE*is carried out in order to determine the relative importance of different factors responsible for disease transmission. Finally, numerical simulations are employed to support the qualitative results.

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8

Khan, Muhammad Altaf, Yasir Khan, Sehra Khan, and Saeed Islam. "Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate." International Journal of Biomathematics 09, no. 05 (June 13, 2016): 1650068. http://dx.doi.org/10.1142/s1793524516500686.

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This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number [Formula: see text], the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if [Formula: see text]. The geometric approach is used to present the global stability of the endemic equilibrium. For [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical results.

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9

Khabouze, Mostafa, Khalid Hattaf, and Noura Yousfi. "Stability Analysis of an Improved HBV Model with CTL Immune Response." International Scholarly Research Notices 2014 (October 29, 2014): 1–8. http://dx.doi.org/10.1155/2014/407272.

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To better understand the dynamics of the hepatitis B virus (HBV) infection, we introduce an improved HBV model with standard incidence function, cytotoxic T lymphocytes (CTL) immune response, and take into account the effect of the export of precursor CTL cells from the thymus and the role of cytolytic and noncytolytic mechanisms. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium.

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10

DAS, PRASENJIT, DEBASIS MUKHERJEE, and A. K. SARKAR. "STUDY OF A CARRIER DEPENDENT INFECTIOUS DISEASE — CHOLERA." Journal of Biological Systems 13, no. 03 (September 2005): 233–44. http://dx.doi.org/10.1142/s0218339005001495.

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This paper analyzes an epidemic model for carrier dependent infectious disease — cholera. Existence criteria of carrier-free equilibrium point and endemic equilibrium point (unique or multiple) are discussed. Some threshold conditions are derived for which disease-free, carrier-free as well as endemic equilibrium become locally stable. Further global stability criteria of the carrier-free equilibrium and endemic equilibrium are achieved. Conditions for survival of all populations are also determined. Lastly numerical simulations are performed to validate the results obtained.

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11

Prawoto, Budi Priyo, Dimas Avian Maulana, and Yuliani Puji Astuti. "The behaviour of measles transmission in three different populations." MATEC Web of Conferences 197 (2018): 01004. http://dx.doi.org/10.1051/matecconf/201819701004.

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SIR Model can be employed to model the transmission of either fatal or non-fatal disease within a closed population based on certain assumptions. In this paper, the behaviour of non-fatal diseases transmission model is observed from three types of population, that is (i) increasing population, (ii) constant population, (iii) decreasing population. This paper acquired two equilibria, i.e, the disease-free equilibrium point [see formula in PDF] and the endemic equilibrium point [see formula in PDF]. At the disease-free equilibrium, the behaviour of the model is stable when [see formula in PDF], while at the endemic equilibrium, its behaviour is stable for any positive parameters α, β, μ, μS, μI, and μR.

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12

Yan, Caijuan, and Jianwen Jia. "Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/109372.

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We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the endemic equilibriumE*of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.

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13

Samanta, Sudip. "Study of an epidemic model with Z-type control." International Journal of Biomathematics 11, no. 07 (October 2018): 1850084. http://dx.doi.org/10.1142/s1793524518500845.

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In the present paper, an epidemic model with Z-type control mechanism has been proposed and analyzed to explore the disease control strategy on an infectious disease outbreak. The uncontrolled model can have a disease-free equilibrium and an endemic equilibrium. The expression of the basic reproduction number and the conditions for the stability of the equilibria are derived. It is also observed that the disease-free equilibrium is globally asymptotically stable if [Formula: see text], whereas the endemic equilibrium is globally asymptotically stable if [Formula: see text]. The model is further improved by considering Z-control mechanism and investigated. Disease can be controlled by using Z-control while the basic reproduction of the uncontrolled system is greater than unity. The positivity conditions of the solutions are derived and the basin of attraction for successful implementation of Z-control mechanism is also investigated. To verify the analytical findings, extensive numerical simulations on the model are carried out.

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14

Ozair, Muhammad, Abid Ali Lashari, Il Hyo Jung, Young Il Seo, and Byul Nim Kim. "Stability Analysis of a Vector-Borne Disease with Variable Human Population." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/293293.

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A mathematical model of a vector-borne disease involving variable human population is analyzed. The varying population size includes a term for disease-related deaths. Equilibria and stability are determined for the system of ordinary differential equations. IfR0≤1, the disease-“free” equilibrium is globally asymptotically stable and the disease always dies out. IfR0>1, a unique “endemic” equilibrium is globally asymptotically stable in the interior of feasible region and the disease persists at the “endemic” level. Our theoretical results are sustained by numerical simulations.

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15

Pambudi, Adiluhung Setya, Fatmawati Fatmawati, and Windarto Windarto. "Analisis Kontrol Optimal Model Matematikan Penyebaran Penyakit Mosaic pada Tanaman Jarak Pagar." Contemporary Mathematics and Applications (ConMathA) 1, no. 2 (January 15, 2020): 104. http://dx.doi.org/10.20473/conmatha.v1i2.17386.

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Mosaic disease is an infectious disease that attacks Jatropha curcas caused by Begomoviruses. Mosaic disease can be transmitted through the bite of a whitefly as a vector. In this paper, we studied a mathematical model of mosaic disease spreading of Jatropha curcas with awareness effect. We also studied the effect of prevention and extermination strategies as optimal control variables. Based on the results of the model analysis, we found two equilibriums namely the mosaic-free equilibrium and the endemic equilibrium. The stability of equilibriums and the existence of endemic equilibrium depend on basic reproduction number ( ). When , the spread of mosaic disease does not occur in the population, while when , the spread of mosaic disease occurs in the population. Furthermore, we determined existence of the optimal control variable by Pontryagin's Maximum Principle method. Simulation results show that prevention and extermination have a significant effect in eliminating mosaic disease.

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16

Aligaz, Achamyelesh A., and Justin M. W. Munganga. "MODELLING THE TRANSMISSION DYNAMICS OF CONTAGIOUS BOVINE PLEUROPNEUMONIA IN THE PRESENCE OF ANTIBIOTIC TREATMENT WITH LIMITED MEDICAL SUPPLY." Mathematical Modelling and Analysis 26, no. 1 (January 18, 2021): 1–20. http://dx.doi.org/10.3846/mma.2021.11795.

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We present and analyze a mathematical model of the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) in the presence of antibiotic treatment with limited medical supply. We use a saturated treatment function to model the effect of delayed treatment. We prove that there exist one disease free equilibrium and at most two endemic equilibrium solutions. A backward bifurcation occurs for small values of delay constant such that two endemic equilibriums exist if Rt (R*t,1); where, Rt is the treatment reproduction number and R*t is a threshold such that the disease dies out if and persists in the population if Rt > R*t. However, when a backward bifurcation occurs, a disease free system may easily be shifted to an epidemic. The bifurcation turns forward when the delay constant increases; thus, the disease free equilibrium becomes globally asymptotically stable if Rt < 1, and there exist unique and globally asymptotically stable endemic equilibrium if Rt > 1. However, the amount of maximal medical resource required to control the disease increases as the value of the delay constant increases. Thus, antibiotic treatment with limited medical supply setting would not successfully control CBPP unless we avoid any delayed treatment, improve the efficacy and availability of medical resources or it is given along with vaccination.

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17

Hove-Musekwa, S. D., and F. Nyabadza. "The Dynamics of an HIV/AIDS Model with Screened Disease Carriers." Computational and Mathematical Methods in Medicine 10, no. 4 (2009): 287–305. http://dx.doi.org/10.1080/17486700802653917.

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The presence of carriers usually complicates the dynamics and prevention of a disease. They are not recognized as disease cases themselves unless they are screened and they usually spread the infection without them being aware. We argue that this has been one of the major causes of the spread of human immunodeficiency virus (HIV). We propose, in this paper, a model for the heterogeneous transmission of HIV/acquired immunodeficiency syndrome in the presence of disease carriers. The model allows us to assess the role of screening, as an intervention program that can slow the epidemic. A threshold value ψ*, for the screening rate is obtained. It is shown numerically that if 80% or more of the carrier population is screened, the epidemic can be contained. The qualitative analysis is done in terms of the model reproduction numberR. The model has two equilibria, the disease free equilibrium and a unique endemic equilibrium. The disease free equilibrium is globally stable ofR < 1 and the endemic equilibrium is is locally stable forR > 1. A detailed discussion of the model reproduction number is given and numerical simulations are done to show the role of some of the important model parameters.

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18

Agusto, F. B., J. Cook, P. D. Shelton, and M. G. Wickers. "Mathematical Model of MDR-TB and XDR-TB with Isolation and Lost to Follow-Up." Abstract and Applied Analysis 2015 (2015): 1–21. http://dx.doi.org/10.1155/2015/828461.

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We present a deterministic model with isolation and lost to follow-up for the transmission dynamics of three strains ofMycobacterium tuberculosis(TB), namely, the drug sensitive, multi-drug-resistant (MDR), and extensively-drug-resistant (XDR) TB strains. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the model has locally asymptotically stable (LAS) disease-free equilibrium when the associated reproduction number is less than unity. Furthermore, the model undergoes in the presence of disease reinfection the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the disease-free equilibrium is globally asymptotically stable (GAS) in the absence of disease reinfection. The result of the global sensitivity analysis indicates that the dominant parameters are the disease progression rate, the recovery rate, the infectivity parameter, the isolation rate, the rate of lost to follow-up, and fraction of fast progression rates. Our results also show that increase in isolation rate leads to a decrease in the total number of individuals who are lost to follow-up.

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19

Andrawus, J., F. Y. Eguda, I. G. Usman, S. I. Maiwa, I. M. Dibal, T. G. Urum, and G. H. Anka. "A Mathematical Model of a Tuberculosis Transmission Dynamics Incorporating First and Second Line Treatment." Journal of Applied Sciences and Environmental Management 24, no. 5 (June 24, 2020): 917–22. http://dx.doi.org/10.4314/jasem.v24i5.29.

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This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

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20

Yusuf, Tunde Tajudeen. "On Global Stability of Disease-Free Equilibrium in Epidemiological Models." European Journal of Mathematics and Statistics 2, no. 3 (July 14, 2021): 37–42. http://dx.doi.org/10.24018/ejmath.2021.2.3.21.

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This paper considers the problem of constructing appropriate Lyapunov function for establishing the global stability of a disease-free equilibrium in epidemiological models. A generalised algorithm is proposed and it is tested for some selected epidemiological models. Experience from the application of the algorithm on test examples shows that the algorithm is easy to use, less cumbersome, and yielded the desired result, particularly in models with homogeneous population. Thus, the proposed algorithm provides a direct approach for establishing global stability of disease-free equilibrium.

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21

Ge, Shao Ting, Gong You Tang, Xue Yang, Qi Lei Xu, Hao Yu, and Pei Dong Wang. "Stability Analysis of Computer Virus Model System in Networks." Applied Mechanics and Materials 278-280 (January 2013): 2033–38. http://dx.doi.org/10.4028/www.scientific.net/amm.278-280.2033.

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This paper considers stability analysis of a discrete-time computer virus model in networks. The disease-free equilibrium and the disease equilibrium are first derived from the mathematical model. Then the sufficient condition of stability for the disease-free equilibrium is obtained by the first Lyapunov method. And the sufficient conditions of stability for the disease equilibrium are given by disc theorem. Simulation results demonstrate the effectiveness of the stability conditions.

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22

Wang, Zizi, and Zhiming Guo. "Dynamical Behavior of a New Epidemiological Model." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/854528.

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A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain timeτ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive numberR0is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided thatR0≤1; ifR0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the timeτis also addressed.

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Naim, Mouhcine, and Fouad Lahmidi. "Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response." Discrete Dynamics in Nature and Society 2020 (May 26, 2020): 1–11. http://dx.doi.org/10.1155/2020/5362716.

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The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.

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Sun, Bei, Xue Zhang, and Marco Tosato. "Effects of Coinfection on the Dynamics of Two Pathogens in a Tick-Host Infection Model." Complexity 2020 (May 28, 2020): 1–14. http://dx.doi.org/10.1155/2020/5615173.

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As both ticks and hosts may carry one or more pathogens, the phenomenon of coinfection of multiple tick-borne diseases becomes highly relevant and plays a key role in tick-borne disease transmission. In this paper, we propose a coinfection model involving two tick-borne diseases in a tick-host population and calculate the basic reproduction numbers at the disease-free equilibrium and two boundary equilibria. To explore the impact of coinfection, we also derive the invasion reproduction numbers which indicate the potential of a pathogen to persist when another pathogen already exists in tick and host populations. Then, we obtain the global stability of the system at the disease-free equilibrium and the boundary equilibrium, respectively, and further demonstrate the existence conditions for uniform persistence of the two diseases. The final numerical simulations mainly verify the theoretical results of coinfection.

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LIU, LILI, XINZHI REN, and XIANNING LIU. "DYNAMICAL BEHAVIORS OF AN INFLUENZA EPIDEMIC MODEL WITH VIRUS MUTATION." Journal of Biological Systems 26, no. 03 (September 2018): 455–72. http://dx.doi.org/10.1142/s0218339018500201.

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Influenza virus mutates frequently. This poses immense challenges to prevent the spread of influenza. This paper aims to investigate an influenza epidemic model in which influenza virus can mutate into a mutant influenza virus. We find a threshold condition that determines the stability of the disease-free equilibrium. Two equilibria may also exist: mutant-dominant equilibrium and endemic equilibrium. We show that the mutant-dominant equilibrium is globally asymptotically stable under some biological feasible conditions. Furthermore, the influenza is endemic in the sense of permanence if and only if the endemic equilibrium exists. Numerical simulations are also performed to illustrate theoretical results and demonstrate the effects of disease-induced death on the dynamics of the model.

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Khan, Muhammad Altaf, Yasir Khan, Taj Wali Khan, and Saeed Islam. "Dynamical system of a SEIQV epidemic model with nonlinear generalized incidence rate arising in biology." International Journal of Biomathematics 10, no. 07 (September 21, 2017): 1750096. http://dx.doi.org/10.1142/s1793524517500966.

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In this paper, a dynamical system of a SEIQV mathematical model with nonlinear generalized incidence arising in biology is investigated. The stability of the disease-free and endemic equilibrium is discussed. The basic reproduction number of the model is obtained. We found that the disease-free and endemic equilibrium is stable locally as well as globally asymptotically stable. For [Formula: see text], the disease-free equilibrium is stable both locally and globally and for [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, some numerical results are presented.

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Wang, Xiaoyan, Junyuan Yang, and Fengqin Zhang. "Dynamic of a TB-HIV Coinfection Epidemic Model with Latent Age." Journal of Applied Mathematics 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/429567.

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A coepidemic arises when the spread of one infectious disease stimulates the spread of another infectious disease. Recently, this has happened with human immunodeficiency virus (HIV) and tuberculosis (TB). The density of individuals infected with latent tuberculosis is structured by age since latency. The host population is divided into five subclasses of susceptibles, latent TB, active TB (without HIV), HIV infectives (without TB), and coinfection class (infected by both TB and HIV). The model exhibits three boundary equilibria, namely, disease free equilibrium, TB dominated equilibrium, and HIV dominated equilibrium. We discuss the local or global stabilities of boundary equilibria. We prove the persistence of our model. Our simple model of two synergistic infectious disease epidemics illustrates the importance of including the effects of each disease on the transmission and progression of the other disease. We simulate the dynamic behaviors of our model and give medicine explanations.

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Khan, Muhammad Altaf, Saeed Islam, Sher Afzal Khan, and Gul Zaman. "Global Stability of Vector-Host Disease with Variable Population Size." BioMed Research International 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/710917.

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The paper presents the vector-host disease with a variability in population. We assume, the disease is fatal and for some cases the infected individuals become susceptible. We first show the local and global stability of the disease-free equilibrium, for the case when, the disease free-equilibrium of the model is both locally as well as globally stable. For , the disease persistence occurs. The endemic equilibrium is locally as well as globally asymptotically stable for . Numerical results are presented for the justifications of theoratical results.

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Wang, Jinghai. "Analysis of an SEIS Epidemic Model with a Changing Delitescence." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/318150.

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An SEIS epidemic model with a changing delitescence is studied. The disease-free equilibrium and the endemic equilibrium of the model are studied as well. It is shown that the disease-free equilibrium is globally stable under suitable conditions. Moreover, we also show that the unique endemic equilibrium of the system is globally asymptotically stable under certain conditions.

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Hu, Zhixing, Shanshan Yin, and Hui Wang. "Stability and Hopf Bifurcation of a Vector-Borne Disease Model with Saturated Infection Rate and Reinfection." Computational and Mathematical Methods in Medicine 2019 (June 9, 2019): 1–17. http://dx.doi.org/10.1155/2019/1352698.

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This paper established a delayed vector-borne disease model with saturated infection rate and cure rate. First of all, according to the basic reproductive number R0, we determined the disease-free equilibrium E0 and the endemic equilibrium E1. Through the analysis of the characteristic equation, we consider the stability of two equilibriums. Furthermore, the effect on the stability of the endemic equilibrium E1 by delay was studied, the existence of Hopf bifurcations of this system in E1 was analyzed, and the length of delay to preserve stability was estimated. The direction and stability of the Hopf bifurcation were also been determined. Finally, we performed some numerical simulation to illustrate our main results.

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Harianto, Joko. "Local Stability Analysis of an SVIR Epidemic Model." CAUCHY 5, no. 1 (November 30, 2017): 20. http://dx.doi.org/10.18860/ca.v5i1.4388.

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In this paper, we present an SVIR epidemic model with deadly deseases. Initially the basic formulation of the model is presented. Two equilibrium point exists for the system; disease free and endemic equilibrium. The local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. If the value of R0 less than one then the desease free equilibrium is locally stable, and if its exceeds, the endemic equilibrium is locally stable. The numerical results are presented for illustration.

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Khan, Muhammad Altaf, Yasir Khan, Qaiser Badshah, and Saeed Islam. "Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination." International Journal of Biomathematics 08, no. 06 (October 15, 2015): 1550082. http://dx.doi.org/10.1142/s1793524515500825.

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In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

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Bessey, K., M. Mavis, J. Rebaza, and J. Zhang. "Global Stability Analysis of a General Model of Zika Virus." Nonautonomous Dynamical Systems 6, no. 1 (March 1, 2019): 18–34. http://dx.doi.org/10.1515/msds-2019-0002.

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AbstractMathematical models of Zika virus dynamics are relatively new, and they mostly focus on either vector and horizontal, or vector and vertical transmission only. In this work,we first revisit a recent model that considers vector and vertical transmission, and we provide an alternative proof on the global stability of the disease-free equilibrium point. Then, a new and general model is presented which includes vector, horizontal and vertical transmission. For this new model, existence of both a disease-free and an endemic equilibrium is studied. Using matrix and graph-theoretic methods, appropriate Lyapunov functions are constructed and results on the global stability properties of both equilibria are established.

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Keng Deng and Yixiang Wu. "Dynamics of a susceptible–infected–susceptible epidemic reaction–diffusion model." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 5 (July 19, 2016): 929–46. http://dx.doi.org/10.1017/s0308210515000864.

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We study a susceptible–infected–susceptible reaction–diffusion model with spatially heterogeneous disease transmission and recovery rates. A basic reproduction number is defined for the model. We first prove that there exists a unique endemic equilibrium if . We then consider the global attractivity of the disease-free equilibrium and the endemic equilibrium for two cases. If the disease transmission and recovery rates are constants or the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals, we show that the disease-free equilibrium is globally attractive if , while the endemic equilibrium is globally attractive if .

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Oumarou, Abba Mahamane, and Saley Bisso. "Modelling and Simulating a Transmission of COVID-19 Disease: Niger Republic Case." European Journal of Pure and Applied Mathematics 13, no. 3 (July 31, 2020): 549–66. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3727.

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This paper focuses on the dynamics of spreads of a coronavirus disease (Covid-19).Through this paper, we study the impact of a contact rate in the transmission of the disease. We determine the basic reproductive number R0, by using the next generation matrix method. We also determine the Disease Free Equilibrium and Endemic Equilibrium points of our model. We prove that the Disease Free Equilibrium is asymptotically stable if R0 < 1 and unstable if R0 > 1. The asymptotical stability of Endemic Equilibrium is also establish. Numerical simulations are made to show the impact of contact rate in the spread of disease.

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Wang, Juan, Xue-Zhi Li, and Souvik Bhattacharya. "The backward bifurcation of a model for malaria infection." International Journal of Biomathematics 11, no. 02 (February 2018): 1850018. http://dx.doi.org/10.1142/s1793524518500183.

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In this paper, an epidemic model of a vector-borne disease, namely, malaria, is considered. The explicit expression of the basic reproduction number is obtained, the local and global asymptotical stability of the disease-free equilibrium is proved under certain conditions. It is shown that the model exhibits the phenomenon of backward bifurcation where the stable disease-free equilibrium coexists with a stable endemic equilibrium. Further, it is proved that the unique endemic equilibrium is globally asymptotically stable under certain conditions.

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Ouaro, Stanislas, and Ali Traoré. "On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination." International Journal of Differential Equations 2018 (2018): 1–11. http://dx.doi.org/10.1155/2018/4168061.

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We study a vector-borne disease with age of vaccination. A nonlinear incidence rate including mass action and saturating incidence as special cases is considered. The global dynamics of the equilibria are investigated and we show that if the basic reproduction number is less than 1, then the disease-free equilibrium is globally asymptotically stable; that is, the disease dies out, while if the basic reproduction number is larger than 1, then the endemic equilibrium is globally asymptotically stable, which means that the disease persists in the population. Using the basic reproduction number, we derive a vaccination coverage rate that is required for disease control and elimination.

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Lozano-Ochoa, Enrique, Jorge Fernando Camacho, and Cruz Vargas-De-León. "Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion." Discrete Dynamics in Nature and Society 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/1084769.

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We study an epidemiological mathematical model formulated in terms of an ODE system taking into account both social and nonsocial contagion risks of obesity. Analyzing first the case in which the model presents only the effect due to social contagion and using qualitative methods of the stability analysis, we prove that such system has at the most three equilibrium points, one disease-free equilibrium and two endemic equilibria, and also that it has no periodic orbits. Particularly, we found that when considering R0 (the basic reproductive number) as a parameter, the system exhibits a backward bifurcation: the disease-free equilibrium is stable when R0<1 and unstable when R0>1, whereas the two endemic equilibria appear from R0⁎ (a specific positive value reached by R0 and less than unity), one being asymptotically stable and the other unstable, but for R0>1 values, only the former remains inside the feasible region. On the other hand, considering social and nonsocial contagion and following the same methodology, we found that the dynamic of the model is simpler than that described above: it has a unique endemic equilibrium point that is globally asymptotically stable.

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Wang, Jing Hai. "Equilibriums of an SIS Epidemic Model." Applied Mechanics and Materials 678 (October 2014): 103–6. http://dx.doi.org/10.4028/www.scientific.net/amm.678.103.

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An SIS epidemic model with nonlinear incidence rate is considered. At the same time we decide that the birth rate is equal to death rate. We get the basic reproduction number . We analyze the existences of the equilibriums of the system. There are some endemic equilibriums and one disease-free equilibrium of the system.

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Harianto, Joko, and Titik Suparwati. "SVIR Epidemic Model with Non Constant Population." CAUCHY 5, no. 3 (December 5, 2018): 102. http://dx.doi.org/10.18860/ca.v5i3.5511.

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In this article, we present an SVIR epidemic model with deadly deseases and non constant population. We only discuss the local stability analysis of the model. Initially the basic formulation of the model is presented. Two equilibrium point exists for the system; disease free and endemic equilibrium point. The local stability of the disease free and endemic equilibrium exists when the basic reproduction number less or greater than unity, respectively. If the value of R0 less than one then the desease free equilibrium point is locally asymptotically stable, and if its exceeds, the endemic equilibrium point is locally asymptotically stable. The numerical results are presented for illustration.

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Zhang, Wenjing, and Pei Yu. "Hopf and Generalized Hopf Bifurcations in a Recurrent Autoimmune Disease Model." International Journal of Bifurcation and Chaos 26, no. 05 (May 2016): 1650079. http://dx.doi.org/10.1142/s0218127416500796.

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This paper is concerned with bifurcation and stability in an autoimmune model, which was established to study an important phenomenon — blips arising from such models. This model has two equilibrium solutions, disease-free equilibrium and disease equilibrium. The positivity of the solutions of the model and the global stability of the disease-free equilibrium have been proved. In this paper, we particularly focus on Hopf bifurcation which occurs from the disease equilibrium. We present a detailed study on the use of center manifold theory and normal form theory, and derive the normal form associated with Hopf bifurcation, from which the approximate amplitude of the bifurcating limit cycles and their stability conditions are obtained. Particular attention is also paid to the bifurcation of multiple limit cycles arising from generalized Hopf bifurcation, which may yield bistable phenomenon involving equilibrium and oscillating motion. This result may explain some complex dynamical behavior in real biological systems. Numerical simulations are compared with the analytical predictions to show a very good agreement.

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Kamgang, Jean Claude, and Gauthier Sallet. "Global asymptotic stability for the disease free equilibrium for epidemiological models." Comptes Rendus Mathematique 341, no. 7 (October 2005): 433–38. http://dx.doi.org/10.1016/j.crma.2005.07.015.

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Zheng, Lifei, Xiuxiang Yang, and Liang Zhang. "On global stability analysis for SEIRS models in epidemiology with nonlinear incidence rate function." International Journal of Biomathematics 10, no. 02 (January 18, 2017): 1750019. http://dx.doi.org/10.1142/s179352451750019x.

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We study an SEIRS epidemic model with an isolation and nonlinear incidence rate function. We have obtained a threshold value [Formula: see text] and shown that there is only a disease-free equilibrium point, when [Formula: see text] and an endemic equilibrium point if [Formula: see text]. We have shown that both disease-free and endemic equilibrium point are globally stable.

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Ozair, Muhammad. "Analysis of Pine Wilt Disease Model with Nonlinear Incidence and Horizontal Transmission." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/204241.

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The deterministic pine wilt model with vital dynamics to determine the equilibria and their stability by considering nonlinear incidence rates with horizontal transmission is analyzed. The complete global analysis for the equilibria of the model is discussed. The explicit formula for the reproductive number is obtained and it is shown that the “disease-free” equilibrium always exists and is globally asymptotically stable wheneverR0≤1. Furthermore, the disease persists at an “endemic” level when the reproductive number exceeds unity.

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Bankuru, Sri Vibhaav, Samuel Kossol, William Hou, Parsa Mahmoudi, Jan Rychtář, and Dewey Taylor. "A game-theoretic model of Monkeypox to assess vaccination strategies." PeerJ 8 (June 22, 2020): e9272. http://dx.doi.org/10.7717/peerj.9272.

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Monkeypox (MPX) is a zoonotic disease similar to smallpox. Its fatality rate is about 11% and it is endemic to the Central and West African countries. In this paper, we analyze a compartmental model of MPX dynamics. Our goal is to see whether MPX can be controlled and eradicated by voluntary vaccinations. We show that there are three equilibria—disease free, fully endemic and previously neglected semi-endemic (with disease existing only among humans). The existence of semi-endemic equilibrium has severe implications should the MPX virus mutate to increased viral fitness in humans. We find that MPX is controllable and can be eradicated in a semi-endemic equilibrium by vaccination. However, in a fully endemic equilibrium, MPX cannot be eradicated by vaccination alone.

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Shi, Xiangyun, and Guohua Song. "Analysis of the Mathematical Model for the Spread of Pine Wilt Disease." Journal of Applied Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/184054.

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This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction numberℛ0and the other valueℛcwhich is larger thanℛ0. Ifℛ0andℛcare both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.

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Bernoussi, Amine, Abdelilah Kaddar, and Said Asserda. "Global Stability of a Delayed SIRI Epidemic Model with Nonlinear Incidence." International Journal of Engineering Mathematics 2014 (December 7, 2014): 1–6. http://dx.doi.org/10.1155/2014/487589.

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In this paper we propose the global dynamics of an SIRI epidemic model with latency and a general nonlinear incidence function. The model is based on the susceptible-infective-recovered (SIR) compartmental structure with relapse (SIRI). Sufficient conditions for the global stability of equilibria (the disease-free equilibrium and the endemic equilibrium) are obtained by means of Lyapunov-LaSalle theorem. Also some numerical simulations are given to illustrate this result.

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Bornaa, Christopher Saaha, Baba Seidu, and Yakubu Ibrahim Seini. "Modeling the impact of early interventions on the transmission dynamics of coronavirus infection." F1000Research 10 (June 30, 2021): 518. http://dx.doi.org/10.12688/f1000research.54268.1.

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A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

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Bornaa, Christopher Saaha, Baba Seidu, and Yakubu Ibrahim Seini. "Modeling the impact of early interventions on the transmission dynamics of coronavirus infection." F1000Research 10 (August 17, 2021): 518. http://dx.doi.org/10.12688/f1000research.54268.2.

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A deterministic model is proposed to describe the transmission dynamics of coronavirus infection with early interventions. Epidemiological studies have employed modeling to unravel knowledge that transformed the lives of families, communities, nations and the entire globe. The study established the stability of both disease free and endemic equilibria. Stability occurs when the reproduction number, R0, is less than unity for both disease free and endemic equilibrium points. The global stability of the disease-free equilibrium point of the model is established whenever the basic reproduction number R0 is less than or equal to unity. The reproduction number is also shown to be directly related to the transmission probability (β), rate at which latently infected individuals join the infected class (δ) and rate of recruitment (Λ). It is inversely related to natural death rate (μ), rate of early treatment (τ1), rate of hospitalization of infected individuals (θ) and Covid-induced death rate (σ). The analytical results established are confirmed by numerical simulation of the model.

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Welker, Jonathan Shane, and Maia Martcheva. "A novel multi-scale immuno-epidemiological model of visceral leishmaniasis in dogs." BIOMATH 8, no. 1 (January 23, 2019): 1901026. http://dx.doi.org/10.11145/j.biomath.2019.01.026.

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Leishmaniasis is a neglected and emerging disease prevalent in Mediterranean and tropical climates. As such, the study and development of new models are of increasing importance. We introduce a new immuno-epidemiological model of visceral leishmaniasis in dogs. The within-host system is based on previously collected and published data, showing the movement and proliferation of the parasite in the skin and the bone-marrow, as well as the IgG response. The between-host system structures the infected individuals in time-since-infection and is of vector-host type. The within-host system has a parasite-free equilibrium and at least one endemic equilibrium, consistent with the fact that infected dogs do not recover without treatment. We compute the basic reproduction number R0 of the immuno-epidemiological model and provide the existence and stability results of the population-level disease-free equilibrium. Additionally, we prove existence of an unique endemic equilibrium when R0 > 1, and evidence of backward bifurcation and existence of multiple endemic equilibria when R0 < 1.

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