Mathematics

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21 pages, 3167 KiB

Open AccessArticle

A New Mathematical Model of COVID-19 with Quarantine and Vaccination

by Ihtisham Ul Haq, Numan Ullah, Nigar Ali and Kottakkaran Sooppy Nisar

Mathematics 2023, 11(1), 142; https://doi.org/10.3390/math11010142 - 28 Dec 2022

Cited by 16 | Viewed by 4392

A mathematical model revealing the transmission mechanism of COVID-19 is produced and theoretically examined, which has helped us address the disease dynamics and treatment measures, such as vaccination for susceptible patients. The mathematical model containing the whole population was partitioned into six different [...] Read more.

A mathematical model revealing the transmission mechanism of COVID-19 is produced and theoretically examined, which has helped us address the disease dynamics and treatment measures, such as vaccination for susceptible patients. The mathematical model containing the whole population was partitioned into six different compartments, represented by the SVEIQR model. Important properties of the model, such as the nonnegativity of solutions and their boundedness, are established. Furthermore, we calculated the basic reproduction number, which is an important parameter in infection models. The disease-free equilibrium solution of the model was determined to be locally and globally asymptotically stable. When the basic reproduction number

R_{0}

is less than one, the disease-free equilibrium point is locally asymptotically stable. To discover the approximative solution to the model, a general numerical approach based on the Haar collocation technique was developed. Using some real data, the sensitivity analysis of

R_{0}

was shown. We simulated the approximate results for various values of the quarantine and vaccination populations using Matlab to show the transmission dynamics of the Coronavirus-19 disease through graphs. The validation of the results by the Simulink software and numerical methods shows that our model and adopted methodology are appropriate and accurate and could be used for further predictions for COVID-19. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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17 pages, 746 KiB

Open AccessArticle

Is the Increased Transmissibility of SARS-CoV-2 Variants Driven by within or Outside-Host Processes?

by Yehuda Arav, Eyal Fattal and Ziv Klausner

Mathematics 2022, 10(19), 3422; https://doi.org/10.3390/math10193422 - 20 Sep 2022

Cited by 1 | Viewed by 1944

Abstract

Understanding the factors that increase the transmissibility of the recently emerging variants of SARS-CoV-2 can aid in mitigating the COVID-19 pandemic. Enhanced transmissibility could result from genetic variations that improve how the virus operates within the host or its environmental survival. Variants with [...] Read more.

Understanding the factors that increase the transmissibility of the recently emerging variants of SARS-CoV-2 can aid in mitigating the COVID-19 pandemic. Enhanced transmissibility could result from genetic variations that improve how the virus operates within the host or its environmental survival. Variants with enhanced within-host behavior are either more contagious (leading infected individuals to shed more virus copies) or more infective (requiring fewer virus copies to infect). Variants with improved outside-host processes exhibit higher stability on surfaces and in the air. While previous studies focus on a specific attribute, we investigated the contribution of both within-host and outside-host processes to the overall transmission between two individuals. We used a hybrid deterministic-continuous and stochastic-jump mathematical model. The model accounts for two distinct dynamic regimes: fast-discrete actions of the individuals and slow-continuous environmental virus degradation processes. This model produces a detailed description of the transmission mechanisms, in contrast to most-viral transmission models that deal with large populations and are thus compelled to provide an overly simplified description of person-to-person transmission. We based our analysis on the available data of the Alpha, Epsilon, Delta, and Omicron variants on the household secondary attack rate (hSAR). The increased hSAR associated with the recent SARS-CoV-2 variants can only be attributed to within-host processes. Specifically, the Delta variant is more contagious, while the Alpha, Epsilon, and Omicron variants are more infective. The model also predicts that genetic variations have a minimal effect on the serial interval distribution, the distribution of the period between the symptoms’ onset in an infector–infectee pair. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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14 pages, 1335 KiB

Open AccessArticle

A (2+1)-Dimensional Fractional-Order Epidemic Model with Pulse Jumps for Omicron COVID-19 Transmission and Its Numerical Simulation

by Wen-Jing Zhu, Shou-Feng Shen and Wen-Xiu Ma

Mathematics 2022, 10(14), 2517; https://doi.org/10.3390/math10142517 - 20 Jul 2022

Cited by 2 | Viewed by 1497

Abstract

In this paper, we would like to propose a (2+1)-dimensional fractional-order epidemic model with pulse jumps to describe the spread of the Omicron variant of COVID-19. The problem of identifying the involved parameters in the proposed model is reduced to a minimization problem [...] Read more.

In this paper, we would like to propose a (2+1)-dimensional fractional-order epidemic model with pulse jumps to describe the spread of the Omicron variant of COVID-19. The problem of identifying the involved parameters in the proposed model is reduced to a minimization problem of a quadratic objective function, based on the reported data. Moreover, we perform numerical simulation to study the effect of the parameters in diverse fractional-order cases. The number of undiscovered cases can be calculated precisely to assess the severity of the outbreak. The results by numerical simulation show that the degree of accuracy is higher than the classical epidemic models. The regular testing protocol is very important to find the undiscovered cases in the beginning of the outbreak. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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14 pages, 281 KiB

Open AccessArticle

The Stability of Functional Equations with a New Direct Method

by Dongwen Zhang, Qi Liu, John Michael Rassias and Yongjin Li

Mathematics 2022, 10(7), 1188; https://doi.org/10.3390/math10071188 - 5 Apr 2022

Cited by 2 | Viewed by 2229

Abstract

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form [...] Read more.

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form

∥ f (x) - α f (k^{n} (x)) - β f (k^{n + 1} (x)) ∥ ⩽ u (x)

where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

20 pages, 3862 KiB

Open AccessArticle

Investigation of the Stochastic Modeling of COVID-19 with Environmental Noise from the Analytical and Numerical Point of View

by Shah Hussain, Elissa Nadia Madi, Hasib Khan, Sina Etemad, Shahram Rezapour, Thanin Sitthiwirattham and Nichaphat Patanarapeelert

Mathematics 2021, 9(23), 3122; https://doi.org/10.3390/math9233122 - 3 Dec 2021

Cited by 27 | Viewed by 2865

Abstract

In this article, we propose a novel mathematical model for the spread of COVID-19 involving environmental white noise. The new stochastic model was studied for the existence and persistence of the disease, as well as the extinction of the disease. We noticed that [...] Read more.

In this article, we propose a novel mathematical model for the spread of COVID-19 involving environmental white noise. The new stochastic model was studied for the existence and persistence of the disease, as well as the extinction of the disease. We noticed that the existence and extinction of the disease are dependent on

R_{0}

(the reproduction number). Then, a numerical scheme was developed for the computational analysis of the model; with the existing values of the parameters in the literature, we obtained the related simulations, which gave us more realistic numerical data for the future prediction. The mentioned stochastic model was analyzed for different values of

σ_{1}, σ_{2}

and

β_{1}, β_{2}

, and both the stochastic and the deterministic models were compared for the future prediction of the spread of COVID-19. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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13 pages, 378 KiB

Open AccessArticle

Existence and Uniqueness of Nontrivial Periodic Solutions to a Discrete Switching Model

by Lijie Chang, Yantao Shi and Bo Zheng

Mathematics 2021, 9(19), 2377; https://doi.org/10.3390/math9192377 - 25 Sep 2021

Cited by 1 | Viewed by 1554

Abstract

To control the spread of mosquito-borne diseases, one goal of the World Mosquito Program’s Wolbachia release method is to replace wild vector mosquitoes with Wolbachia-infected ones, whose capability of transmitting diseases has been greatly reduced owing to the Wolbachia infection. In this [...] Read more.

To control the spread of mosquito-borne diseases, one goal of the World Mosquito Program’s Wolbachia release method is to replace wild vector mosquitoes with Wolbachia-infected ones, whose capability of transmitting diseases has been greatly reduced owing to the Wolbachia infection. In this paper, we propose a discrete switching model which characterizes a release strategy including an impulsive and periodic release, where Wolbachia-infected males are released with the release ratio

α_{1}

during the first N generations, and the release ratio is

α_{2}

from the

(N + 1)

-th generation to the T-th generation. Sufficient conditions on the release ratios

α_{1}

and

α_{2}

are obtained to guarantee the existence and uniqueness of nontrivial periodic solutions to the discrete switching model. We aim to provide new methods to count the exact numbers of periodic solutions to discrete switching models. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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20 pages, 323 KiB

Open AccessArticle

Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

by Hongpeng Guo and Zhiming Guo

Mathematics 2021, 9(16), 1999; https://doi.org/10.3390/math9161999 - 20 Aug 2021

Cited by 2 | Viewed by 1749

Abstract

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling [...] Read more.

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed

c > c^{*}

, where

c^{*} > 0

is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

20 pages, 372 KiB

Open AccessArticle

Existence Results for p₁(x,·) and p₂(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

by Weichun Bu, Tianqing An, Guoju Ye and Chengwen Jiao

Mathematics 2021, 9(16), 1973; https://doi.org/10.3390/math9161973 - 18 Aug 2021

Cited by 2 | Viewed by 1731

Abstract

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable

s (x, \cdot)

-order fractional

p_{1} (x, \cdot)

and

p_{2} (x, \cdot)

-Laplacian. Assuming some reasonable conditions [...] Read more.

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable

s (x, \cdot)

-order fractional

p_{1} (x, \cdot)

and

p_{2} (x, \cdot)

-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

20 pages, 2215 KiB

Open AccessArticle

Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator

by Shaohong Wang and Zhan Zhou

Mathematics 2021, 9(14), 1691; https://doi.org/10.3390/math9141691 - 19 Jul 2021

Cited by 2 | Viewed by 1865

Abstract

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical [...] Read more.

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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19 pages, 324 KiB

Open AccessArticle

Solving a System of Nonlinear Integral Equations via Common Fixed Point Theorems on Bicomplex Partial Metric Space

by Zhaohui Gu, Gunaseelan Mani, Arul Joseph Gnanaprakasam and Yongjin Li

Mathematics 2021, 9(14), 1584; https://doi.org/10.3390/math9141584 - 6 Jul 2021

Cited by 11 | Viewed by 1640

Abstract

In this paper, we introduce the notion of bicomplex partial metric space and prove some common fixed point theorems. The presented results generalize and expand some of the literature’s well-known results. An example and application on bicomplex partial metric space is given. Full article

(This article belongs to the Special Issue Functional Differential Equations and Epidemiological Modelling)

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