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16 pages, 2255 KiB

Open AccessArticle

Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

by Ardak Kashkynbayev and Fathalla A. Rihan

Mathematics 2021, 9(15), 1829; https://doi.org/10.3390/math9151829 - 03 Aug 2021

Cited by 11 | Viewed by 2187

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if [...] Read more.

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if

R_{0} < 1

, the disease-free steady-state is locally and globally asymptotically stable. However, for

R_{0} > 1

, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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24 pages, 613 KiB

Open AccessArticle

Dynamical Strategy to Control the Accuracy of the Nonlinear Bio-Mathematical Model of Malaria Infection

by Samad Noeiaghdam and Sanda Micula

Mathematics 2021, 9(9), 1031; https://doi.org/10.3390/math9091031 - 02 May 2021

Cited by 15 | Viewed by 1484

Abstract

This study focuses on solving the nonlinear bio-mathematical model of malaria infection. For this aim, the HATM is applied since it performs better than other methods. The convergence theorem is proven to show the capabilities of this method. Instead of applying the FPA, [...] Read more.

This study focuses on solving the nonlinear bio-mathematical model of malaria infection. For this aim, the HATM is applied since it performs better than other methods. The convergence theorem is proven to show the capabilities of this method. Instead of applying the FPA, the CESTAC method and the CADNA library are used, which are based on the DSA. Applying this method, we will be able to control the accuracy of the results obtained from the HATM. Also the optimal results and the numerical instabilities of the HATM can be obtained. In the CESTAC method, instead of applying the traditional absolute error to show the accuracy, we use a novel condition and the CESTAC main theorem allows us to do that. Plotting several ℏ-curves the regions of convergence are demonstrated. The numerical approximations are obtained based on both arithmetics. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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22 pages, 709 KiB

Open AccessArticle

Dynamics of a Two Prey and One Predator System with Indirect Effect

by Renato Colucci, Érika Diz-Pita and M. Victoria Otero-Espinar

Mathematics 2021, 9(4), 436; https://doi.org/10.3390/math9040436 - 22 Feb 2021

Cited by 2 | Viewed by 2121

Abstract

We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study [...] Read more.

We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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13 pages, 2512 KiB

Open AccessArticle

Modeling the Dependence of Immunodominance on T Cell Dynamics in Prime-Boost Vaccines

by Clemente Fernandez Arias, Miguel Angel Herrero, Federica Bertocchini, Francisco Javier Acosta and Cristina Fernandez-Arias

Mathematics 2021, 9(1), 28; https://doi.org/10.3390/math9010028 - 24 Dec 2020

Viewed by 2011

Abstract

The protection induced by vaccines against infectious diseases such as malaria, dengue or hepatitis relies on a the creation of immune memory by T cells, key components of the human immune system. The induction of a strong T cell response leading to long [...] Read more.

The protection induced by vaccines against infectious diseases such as malaria, dengue or hepatitis relies on a the creation of immune memory by T cells, key components of the human immune system. The induction of a strong T cell response leading to long lasting memory can be improved by using prime-boost (PB) vaccines, which consist in successive inoculations of appropriate vectors carrying target antigens that can be recognized by specific T cell clones. A problem faced by PB vaccines is the fact that T cell response is often biased towards a few clones that can identify only a small set of antigens, out of the many that could be displayed by the pathogen. This phenomenon, known as immunodominance, can significantly compromise the effectiveness of vaccination. In this work we will use mathematical modeling to better understand the role of T cell population dynamics in the onset of immunodominance in PB vaccines. In particular, we will use mathematical analysis and simulations to compare single-dose vaccines with PB ones, both for homologous (where the same antigen is used in every shot) and heterologous protocols (in which different antigens are used at each step). Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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22 pages, 3660 KiB

Open AccessArticle

Estimation of Synaptic Activity during Neuronal Oscillations

by Catalina Vich, Rafel Prohens, Antonio E. Teruel and Antoni Guillamon

Mathematics 2020, 8(12), 2153; https://doi.org/10.3390/math8122153 - 03 Dec 2020

Cited by 2 | Viewed by 1846

Abstract

In the study of brain connectivity, an accessible and convenient way to unveil local functional structures is to infer the time trace of synaptic conductances received by a neuron by using exclusively information about its membrane potential (or voltage). Mathematically speaking, it constitutes [...] Read more.

In the study of brain connectivity, an accessible and convenient way to unveil local functional structures is to infer the time trace of synaptic conductances received by a neuron by using exclusively information about its membrane potential (or voltage). Mathematically speaking, it constitutes a challenging inverse problem: it consists in inferring time-dependent parameters (synaptic conductances) departing from the solutions of a dynamical system that models the neuron’s membrane voltage. Several solutions have been proposed to perform these estimations when the neuron fluctuates mildly within the subthreshold regime, but very few methods exist for the spiking regime as large amplitude oscillations (revealing the activation of complex nonlinear dynamics) hinder the adaptability of subthreshold-based computational strategies (mostly linear). In a previous work, we presented a mathematical proof-of-concept that exploits the analytical knowledge of the period function of the model. Inspired by the relevance of the period function, in this paper we generalize it by providing a computational strategy that can potentially adapt to a variety of models as well as to experimental data. We base our proposal on the frequency versus synaptic conductance curve (

f - g_{s y n}

), derived from an analytical study of a base model, to infer the actual synaptic conductance from the interspike intervals of the recorded voltage trace. Our results show that, when the conductances do not change abruptly on a time-scale smaller than the mean interspike interval, the time course of the synaptic conductances is well estimated. When no base model can be cast to the data, our strategy can be applied provided that a suitable

f - g_{s y n}

table can be experimentally constructed. Altogether, this work opens new avenues to unveil local brain connectivity in spiking (nonlinear) regimes. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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14 pages, 427 KiB

Open AccessArticle

Mathematical Modeling of Japanese Encephalitis under Aquatic Environmental Effects

by Faïçal Ndaïrou, Iván Area and Delfim F. M. Torres

Mathematics 2020, 8(11), 1880; https://doi.org/10.3390/math8111880 - 30 Oct 2020

Cited by 4 | Viewed by 2633

Abstract

We propose a mathematical model for the spread of Japanese encephalitis with emphasis on the environmental effects on the aquatic phase of mosquitoes. The model is shown to be biologically well-posed and to have a biologically and ecologically meaningful disease-free equilibrium point. Local [...] Read more.

We propose a mathematical model for the spread of Japanese encephalitis with emphasis on the environmental effects on the aquatic phase of mosquitoes. The model is shown to be biologically well-posed and to have a biologically and ecologically meaningful disease-free equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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11 pages, 1926 KiB

Open AccessArticle

Analytical Estimation of Temperature in Living Tissues Using the TPL Bioheat Model with Experimental Verification

by Aatef Hobiny, Faris Alzahrani and Ibrahim Abbas

Mathematics 2020, 8(7), 1188; https://doi.org/10.3390/math8071188 - 19 Jul 2020

Cited by 24 | Viewed by 2522

Abstract

The aim of this study is to propose the analytical method associated with Laplace transforms and experimental verification to estimate thermal damages and temperature due to laser irradiation by utilizing measurement information of skin surface. The thermal damages to the tissues are totally [...] Read more.

The aim of this study is to propose the analytical method associated with Laplace transforms and experimental verification to estimate thermal damages and temperature due to laser irradiation by utilizing measurement information of skin surface. The thermal damages to the tissues are totally estimated by denatured protein ranges using the formulations of Arrhenius. By using Laplace transformations, the exact solution of all physical variables is obtained. Numerical results for the temperature and thermal damage are presented graphically. Furthermore, the comparisons between the numerical calculations with experimental verification show that the three-phase lag bioheat mathematical model is an efficient tool for estimating the bioheat transfer in skin tissue. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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14 pages, 290 KiB

Open AccessArticle

On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

by Maoan Han, Jaume Llibre and Yun Tian

Mathematics 2020, 8(7), 1137; https://doi.org/10.3390/math8071137 - 12 Jul 2020

Cited by 4 | Viewed by 1843

Abstract

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least [...] Read more.

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

Review

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50 pages, 612 KiB

Open AccessReview

A Review on a Class of Second Order Nonlinear Singular BVPs

by Amit K. Verma, Biswajit Pandit, Lajja Verma and Ravi P. Agarwal

Mathematics 2020, 8(7), 1045; https://doi.org/10.3390/math8071045 - 28 Jun 2020

Cited by 22 | Viewed by 3828

Abstract

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their [...] Read more.

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most cases. Since the exact solution does not exist, it is natural to look for the existence of the analytical solution and numerical solution. In this survey, we focus on both aspects of nonlinear singular boundary value problems (SBVPs) and cover different analytical and numerical techniques which are developed to deal with a class of nonlinear singular differential equations

- {(p (x) y^{'} (x))}^{'} = q (x) f (x, y, p y^{'})

for

x \in (0, b)

, subject to suitable initial and boundary conditions. The monotone iterative technique has also been briefed as it gained a lot of attention during the last two decades and it has been merged with most of the other existing techniques. A list of SBVPs is also provided which will be of great help to researchers working in this area. Full article

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

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