• Kyungpook Mathematical Journal
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• v.60 no.2
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• pp.319-334
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• 2020

We propose a mathematical model with Holling type II functional response, to study the dynamics of vaccination. In order to make our model more realistic, we have incorporated the recruitment of infected individuals as a continuous process. We have assumed that vaccination cannot be perfect and there is always a possibility of re-infection. We have obtained the existence of a disease free and endemic equilibrium point, when the recruitment of infective is not considered and also obtained the existence of at least one endemic equilibrium point when recruitment of infective is considered. We have proved that if R_v < 1, disease free equilibrium is locally asymptotically stable, which leads to the elimination of the disease from the population. The persistence of the model has also been established. Numerical simulations have been done to establish the results obtained.

• Journal of applied mathematics & informatics
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• v.41 no.5
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• pp.1017-1035
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• 2023

In this paper, we construct a monkeypox model which is similar to smallpox infection. It is caused by a monkeypox virus which is related to Poxviridae family. It will occur mostly in West African communities and in remote Central. We develop a system of differential equations for an SEIR (Suspected, Exposed, Infected and Recovered) model and analyze the outbreak of monkeypox disease and its effect on United States(US) population. We establish theorems on asymptotical stability conditions for endemic equilibrium and disease-free equilibrium. The basic reproduction number R₀ has been determined using next generation matrix. We expect that this study will be effective at controlling monkeypox spread in United States. Our goal is to see whether monkeypox can be controlled and destroyed by smallpox vaccination. We find that monkeypox is controllable and can be fully destroyed in disease free state by vaccination. However, in the endemic state, monkeypox cannot be destroyed by vaccination alone.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.987-997
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• 2017

A deterministic model for the spread of pine wilt disease with asymptomatic carrier trees in the host pine population is designed and rigorously analyzed. We have taken four different classes for the trees, namely susceptible, exposed, asymptomatic carrier and infected, and two different classes for the vector population, namely susceptible and infected. A complete global analysis of the model is given, which reveals that the global dynamics of the disease is completely determined by the associated basic reproduction number, denoted by $\mathcal{R}_0$. If $\mathcal{R}_0$ is less than one, the disease-free equilibrium is globally asymptotically stable, and in such a case, the endemic equilibrium does not exist. If $\mathcal{R}_0$ is greater than one, the disease persists and the unique endemic equilibrium is globally asymptotically stable.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.1
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• pp.1-16
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• 2010

An epidemic model with Classical Kermack-Mckendrick incidence rate under a limited resource for treatment is proposed to understand the effect of the capacity for treatment. We have assumed that treatment function is strictly increasing function of infective individuals and becomes constant when the number of infective is very large. Existence and stability of the disease free and endemic equilibrium are investigated, boundedness of the solutions are shown. Even in this simple version of the model, backward bifurcation and multiple epidemic steady states can be observed with some sets of parameter values. Hopf-bifurcation analyses are given and numerical examples are provided to help understanding.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.685-699
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• 2019

This research is focused on a continuous epidemic model of transmission of Plasmodium vivax malaria with a time delay. The model is represented as a system of ordinary differential equations with delay. There are two equilibria, which are the disease-free state and the endemic equilibrium, depending on the basic reproduction number, $R_0$, which is calculated and decreases with the time delay. Moreover, the disease-free equilibrium is locally asymptotically stable if $R_0<1$. If $R_0>1$, a unique endemic steady state exists and is locally stable. Furthermore, Hopf bifurcation is applied to determine the conditions for periodic solutions.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.45-67
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• 2021

In this paper, a mathematical dynamical system involving both deterministic (with or without delay) and stochastic "SIR" epidemic model with nonlinear incidence rate in a continuous reactor is considered. A profound qualitative analysis is given. It is proved that, for both deterministic models, if ��d > 1, then the endemic equilibrium is globally asymptotically stable. However, if ��d ≤ 1, then the disease-free equilibrium is globally asymptotically stable. Concerning the stochastic model, the Feller's test combined with the canonical probability method were used in order to conclude on the long-time dynamics of the stochastic model. The results improve and extend the results obtained for the deterministic model in its both forms. It is proved that if ��s > 1, the disease is stochastically permanent with full probability. However, if ��s ≤ 1, then the disease dies out with full probability. Finally, some numerical tests are done in order to validate the obtained results.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1893-1912
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• 2017

In this paper, we aim to construct a nonstandard finite difference (NSFD) scheme to solve numerically a mathematical model for cholera epidemic dynamics. We first show that if the basic reproduction number is less than unity, the disease-free equilibrium (DFE) is locally asymptotically stable. Moreover, we mainly establish the global stability analysis of the DFE and endemic equilibrium by using suitable Lyapunov functionals regardless of the time step size. Finally, numerical simulations with different time step sizes and initial conditions are carried out and comparisons are made with other well-known methods to illustrate the main theoretical results.

• The Pure and Applied Mathematics
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• v.30 no.2
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• pp.203-220
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• 2023

A new modification of the SIS epidemic model incorporating the adaptive host behavior is proposed. Unlike the common situation in most epidemic models, this system has two disease-free equilibrium points, and we were able to prove that as the basic reproduction number approaches the threshold of 1, these two points merge and a Bogdanov-Takens bifurcation of codimension three occurs. The occurrence of this bifurcation is a sign of the complexity of the dynamics of the system near the value 1 of basic reproduction number. Both local and global stability of disease-free and endemic equilibrium point are studied.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.529-546
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• 2011

A deterministic tuberculosis model for theoretically assessing the potential impact of the combined effects of case detection in the presence of treatment is formulated. The qualitative features of its equilibria are analyzed and it is found that the disease-free equilibrium may not be globally asymptotically stable when the reproduction number is less than unity. This disease threshold number is further used to assess the impact of active TB case finding alone and in conjunction with treatment. A critical threshold parameter ${\Theta}$ say for which case detection will have a positive impact is derived. Using the Centre Manifold theory, the model may exhibit the phenomenon of backward bifurcation (coexistence of a locally stable endemic equilibrium with a stable disease-free equilibrium) when the reproduction number is less than unity. It is shown that the possibility of backward bifurcation occurring decreases with increase case detection. Graphical representations suggest that increase in case finding accompanied by treatment of detected TB cases, result in a marked decrease of TB cases (both latent and active TB).

• Animal Systematics, Evolution and Diversity
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• v.33 no.2
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• pp.136-139
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• 2017

The genus Iksookimia (Actinopterygii: Cypriniformes: Cobitidae) is a bottom-dwelling freshwater loaches, which are well-known as their endemism and high geographic variation. However, population genetic relationships among Iksookimia spp. have remained unclear due to a shortage of genetic markers that can be applied generally in the genus. Here, we developed high-resolving microsatellite markers using I. koreensis and I. longicorpa as representatives of Iksookimia species because of their wide distribution range and phylogenetic position. Ten of polymorphic microsatellite loci were isolated from Iksookimia koreensis and were successfully cross-amplified in I. longicorpa. The mean number of observed alleles per locus was about 10.4 (range, 2-17) for I. koreensis and about 13.2 (range, 2-24) for I. longicorpa. The loci, IK03 and IK08, deviated from the Hardy-Weinberg equilibrium in I. koreensis, after applying the Bonferroni correction. The microsatellite markers obtained in the present study will be useful to evaluate population genetic structure and to establish conservation strategies for I. koreensis and related Iksookimia species.