iosr-Jm   Volume-15 ~ Issue-6 ~ Nov – Dec 2019

Abstract: In this article, we have analyzed international concerns about the environment, in particular the current United Nations policy, indicating what Brazil is developing in order to achieve a healthy environment, not polluting and achieving the results to reach a environment with acceptable concentrations of pollutants. Here we present a model through a general periodic differential equation system that simulates the elimination of pollution. The theorem that reduces this system to a system where the linear part matrix has constant coefficients is demonstrated; necessary and sufficient conditions are given on the future behavior of the trajectories; An example is introduced that concretely shows the theory seen in the article..

Key Word: Environment, mathematical model, pollution.

[1]. BatistaE, SánchezS, LacortM, FerreiraR, RibeiroZ e RuizA. I." MathematicalModelingofPollutionElimination"Journal of Scientific and EngineeringResearch, 5(5):234-240, 2018.

[2]. BatistaE, LacortM, FerreiraR, RibeiroZand RuizA. I. "Normal Form Combined In Pollution Elimination Model.Journal of Research in Applied Mathematics Volume 5 pp: 21-282. (2018).
[3]. Boudonov, M. Study of ordinary differential equations. Habana University. Evolutionary dictation. Year 1998.
[4]. Cachapuz, A. The teachingofscience for excellence in learning. In: Oak, A (Org.). New MethodologiesofEducation. Porto: Porto Publisher, 1995.
[5]. CNN. "Greta ThunbergDonald Trump injury". Journal J. R. Monday 23 September 2019.

Abstract: In this paper, we study that two Banach spaces are isometrically iso- morphic if Hausdorff distance between them measures zero.

Key Word: Convex set, Banach Space, Hausdorff Distance, Hausdorff Metric Space.

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Abstract: we have established the existence and uniqueness of the solutions for a general utility function. In our work we have established existence and uniqueness of the solution of the renewal equation arising in dynamic programming. We have proved it in a different method using dilation principle. In the present model we have considered a dynamic model of renewal equation.

Keywords: Dynamic programming, multistage allocation, fixed point, renewal equation, dilation principle

[1]. Alexander Shapiro, Analysis of stochastic dual programming method, European journal of operational research,209(2011),63-72.
[2]. Dmitry Ivanov, Boris sokolov, A multi-structural framework for adaptive supply chain planning and operations control with structure dynamics considerations, European journal of operational research, 200(2010),409-420.
[3]. Daniel Schunk, Behavioral heterogeneity in dynamic search situations, Theory and experimental evidence, Journal of Economic Dynamics and Control,33(2009),1719-1738.
[4]. R. Bhardwaj, S. S. Rajput R. N. Yadava ,some fixed point theorems in complete metric Spaces , International J. of Math. Sci.and Engg. Appls., 2 (2007),193-198.
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Abstract: Epidemic modeling is an important theoretical approach for investigating the transmission dynamics of infectious diseases. It formulates mathematical models to describe the mechanisms of disease transmissions and dynamics of infectious agents and then informs the health control practitioners the likely impact of the control methods. In this paper we investigate the spread of an infectious disease in a human population structured into n-patches. The population is initially fully susceptible until an infectious individual is introduced in one of the patches. The interaction between patches is dominated by movement of individuals between patches and also the migration........

Key Word: Metapopulation, Basic reproduction number, Epidemic Control, Target reproduction number, migration, Lipchitz continuity.

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[2]. Mutuguta John Wanjau, Rotich Titus and Chepkwony Isaac, Mathematical Modeling of the Transmission Dynamics of Measles under the Effect of Vaccination, IOSR Journal of Mathematics, Vol. 15 Issue 4, Ser. II. pp 10-19, ISSN: 2278-5728, 2019.
[3]. Knipl Diana. A new approach for designing disease intervention strategies in metapopulation models. Journal of Biological Dynamics, 10(1):71–94, 2016.
[4]. O. Diekmann, J. A. P. Heesterbeek, and M. G. Roberts. The construction of the next generation matrices for compartmental epidemic models. Journal of Royal Society Interface, 7:873–885, 2010.
[5]. Augustino Isdory, Eunice W. Mureithi, and David J. T. Sumpter. The impact of human mobility on HIV transmission in Kenya. PLoS ONE, 10(11):1–21, 2015.

Abstract: In this research work, we formulated a mathematical model for the transmission dynamics of tuberculosis, a case study of Ika Christian Hospital, Ankpa L.G.A, Kogi State, Nigeria. The model which adopts a standard incidence formulation incorporates treatment and vaccination as control strategies.The Disease Free Equilibrium (DFE) statewas determined, which was shown to be locally asymptotically stable. The basic reproduction number of the model was determined using the next generation matrix approach. The Endemic Equilibrium (EE) state of the model wasalsoestablished and proved to be locally asymptotically stable using the trace and determinant........

Key Word: Tuberculosis, model, heterogeneous, transmission

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Abstract: Linear Operators on invariant spaces and between Banach spaces we define a semi norm vanishing on the subspace of operators having the alternate signs Banach-Saks property. In particular, the estimates show that the alternate signsinvariant spaces and Banach-Saks property are inherited from a space of an interpolation pair (𝐴0,𝐴1) tothe real interpolation spaces 𝐴𝜃,𝑝. Finally, examples are given to support our results.

Key Word: invariant spaces ,Banach-Saks , Lions-Peetre

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Abstract: In this study it is proposes and analyzed a compartmental nonlinear deterministic mathematical model for the human Papilloma virus epidemic together with the inclusion of optimal control strategies in a community with varying population. The model is studied qualitatively using stability theory of differential equations. The basic reproductive number that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined. It is observed that the model exhibits a backward bifurcation and the sensitivity analysis is performed. The optimal control problem is designed by applying Pontryagin maximum principle with three control strategies viz. prevention strategy, treatment strategy and screening strategy. Numerical results of the optimal control model reveal that a combination of prevention, screening and treatment is the most effective strategy to eradicate the disease from the community.

Key Word: Reproductive Number, Stability Analysis, Bifurcation, Optimal Control, Numerical Simulation

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Abstract: An epidemic model with protection and treatment is investigated for typhoid disease that can be transmitted through infected individuals. In this study, we used a deterministic compartmental model for assessing the effect of protection and treatment on controlling the transmission dynamics of typhoid fever in the community. Stability theory of differential equations is used to study the qualitative behavior of the system. The basic reproduction number that represents the epidemic indicator is obtained by using the next generation matrix. Both the local stability and global stability conditions for disease free equilibrium is established. The endemic equilibrium was determined and the model exhibits a forward trans-critical bifurcation. Numerical simulation of the model showed that an increase in protection and treatment leads to low disease prevalence in a population.

Key Word: Mathematical model, Typhoid fever, Basic reproduction number, Protection.

[1]. Mushayabasa, S., Bhunu, C.P. and Ngarakana-Gwasira, E.T. (2013) Mathematical Analysis of a Typhoid Model with Carriers,
Direct and Indirect Disease Transmission. International Journal of Mathematical Sciences and Engineering Applications, 7, 79-90.
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dynamics of typhoid in South Asia: A mathematical modeling study. PLoS Negl Dis. 2014;8(1):1-12.