Research

18 pages, 1537 KiB

Open AccessArticle

Impact of Lévy Noise with Infinite Activity on the Dynamics of Measles Epidemics

by Yuqin Song and Peijiang Liu

Fractal Fract. 2023, 7(6), 434; https://doi.org/10.3390/fractalfract7060434 - 27 May 2023

Viewed by 734

This research article investigates the application of Lévy noise to understand the dynamic aspects of measles epidemic modeling and seeks to explain the impact of vaccines on the spread of the disease. After model formulation, the study utilises uniqueness and existence techniques to [...] Read more.

This research article investigates the application of Lévy noise to understand the dynamic aspects of measles epidemic modeling and seeks to explain the impact of vaccines on the spread of the disease. After model formulation, the study utilises uniqueness and existence techniques to derive a positive solution to the underlying stochastic model. The Lyapunov function is used to investigate the stability results associated with the proposed stochastic model. The model’s dynamic characteristics are analyzed in the vicinity of the infection-free and endemic states of the associated ODEs model. The stochastic threshold

R_{s}

that ensures disease’s extinction whenever

R_{s} < 1

is calculated. We utilized data from Pakistan in 2019 to estimate the parameters of the model and conducted simulations to forecast the future behavior of the disease. The results were compared to actual data using standard curve fitting tools. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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14 pages, 358 KiB

Open AccessArticle

First-Passage Times and Optimal Control of Integrated Jump-Diffusion Processes

by Mario Lefebvre

Fractal Fract. 2023, 7(2), 152; https://doi.org/10.3390/fractalfract7020152 - 03 Feb 2023

Cited by 1 | Viewed by 976

Abstract

Let

Y (t)

be a one-dimensional jump-diffusion process and

X (t)

be defined by

d X (t) = ρ [X (t), Y (t)] d t

, where [...] Read more.

Let

Y (t)

be a one-dimensional jump-diffusion process and

X (t)

be defined by

d X (t) = ρ [X (t), Y (t)] d t

, where

ρ (\cdot, \cdot)

is either a strictly positive or negative function. First-passage-time problems for the degenerate two-dimensional process

(X (t), Y (t))

are considered in the case when the process leaves the continuation region at the latest at the moment of the first jump, and the problem of optimally controlling the process is treated as well. A particular problem, in which

ρ [X (t), Y (t)] = Y (t) - X (t)

and

Y (t)

is a standard Brownian motion with jumps, is solved explicitly. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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24 pages, 4710 KiB

Open AccessArticle

Numerical Simulation of Nonlinear Stochastic Analysis for Measles Transmission: A Case Study of a Measles Epidemic in Pakistan

by Bing Guo, Asad Khan and Anwarud Din

Fractal Fract. 2023, 7(2), 130; https://doi.org/10.3390/fractalfract7020130 - 30 Jan 2023

Cited by 4 | Viewed by 1296

Abstract

This paper presents a detailed investigation of a stochastic model that rules the spreading behavior of the measles virus while accounting for the white noises and the influence of immunizations. It is hypothesized that the perturbations of the model are nonlinear, and that [...] Read more.

This paper presents a detailed investigation of a stochastic model that rules the spreading behavior of the measles virus while accounting for the white noises and the influence of immunizations. It is hypothesized that the perturbations of the model are nonlinear, and that a person may lose the resistance after vaccination, implying that vaccination might create temporary protection against the disease. Initially, the deterministic model is formulated, and then it has been expanded to a stochastic system, and it is well-founded that the stochastic model is both theoretically and practically viable by demonstrating that the model has a global solution, which is positive and stochastically confined. Next, we infer adequate criteria for the disease’s elimination and permanence. Furthermore, the presence of a stationary distribution is examined by developing an appropriate Lyapunov function, wherein we noticed that the disease will persist for

R_{0}^{s > 1}

and that the illness will vanish from the community when

R_{0}^{s < 1}

. We tested the model against the accessible data of measles in Pakistan during the first ten months of 2019, using the conventional curve fitting methods and the values of the parameters were calculated accordingly. The values obtained were employed in running the model, and the conceptual findings of the research were evaluated by simulations and conclusions were made. Simulations imply that, in order to fully understand the dynamic behavior of measles epidemic, time-delay must be included in such analyses, and that advancements in every vaccine campaign are inevitable for the control of the disease. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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16 pages, 443 KiB

Open AccessArticle

On the Estimation of the Persistence Exponent for a Fractionally Integrated Brownian Motion by Numerical Simulations

by Mario Abundo and Enrica Pirozzi

Fractal Fract. 2023, 7(2), 107; https://doi.org/10.3390/fractalfract7020107 - 20 Jan 2023

Cited by 1 | Viewed by 837

Abstract

For a fractionally integrated Brownian motion (FIBM) of order

α \in (0, 1],

X_{α} (t),

we investigate the decaying rate of

P (τ_{S}^{α} > t)

as

t \to + \infty,

[...] Read more.

For a fractionally integrated Brownian motion (FIBM) of order

α \in (0, 1],

X_{α} (t),

we investigate the decaying rate of

P (τ_{S}^{α} > t)

as

t \to + \infty,

where

τ_{S}^{α} = inf {t > 0 : X_{α} (t) \geq S}

is the first-passage time (FPT) of

X_{α} (t)

through the barrier

S > 0 .

Precisely, we study the so-called persistent exponent

θ = θ (α)

of the FPT tail, such that

P (τ_{S}^{α} > t) = t^{- θ + o (1)},

as

t \to + \infty,

and by means of numerical simulation of long enough trajectories of the process

X_{α} (t),

we are able to estimate

θ (α)

and to show that it is a non-increasing function of

α \in (0, 1],

with

1 / 4 \leq θ (α) \leq 1 / 2 .

In particular, we are able to validate numerically a new conjecture about the analytical expression of the function

θ = θ (α),

for

α \in (0, 1] .

Such a numerical validation is carried out in two ways: in the first one, we estimate

θ (α),

by using the simulated FPT density, obtained for any

α \in (0, 1];

in the second one, we estimate the persistent exponent by directly calculating

P ({max}_{0 \leq s \leq t} X_{α} (s) < 1) .

Both ways confirm our conclusions within the limit of numerical approximation. Finally, we investigate the self-similarity property of

X_{α} (t)

and we find the upper bound of its covariance function. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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14 pages, 1067 KiB

Open AccessArticle

Approximating the First Passage Time Density of Diffusion Processes with State-Dependent Jumps

by Giuseppe D’Onofrio and Alessandro Lanteri

Fractal Fract. 2023, 7(1), 30; https://doi.org/10.3390/fractalfract7010030 - 28 Dec 2022

Cited by 2 | Viewed by 1246

Abstract

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating [...] Read more.

We study the problem of the first passage time through a constant boundary for a jump diffusion process whose infinitesimal generator is a nonlocal Jacobi operator. Due to the lack of analytical results, we address the problem using a discretization scheme for simulating the trajectories of jump diffusion processes with state-dependent jumps in both frequency and amplitude. We obtain numerical approximations on their first passage time probability density functions and results for the qualitative behavior of other statistics of this random variable. Finally, we provide two examples of application of the method for different choices of the distribution involved in the mechanism of generation of the jumps. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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23 pages, 667 KiB

Open AccessArticle

On the Absorbing Problems for Wiener, Ornstein–Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

by Virginia Giorno and Amelia G. Nobile

Fractal Fract. 2023, 7(1), 11; https://doi.org/10.3390/fractalfract7010011 - 24 Dec 2022

Cited by 1 | Viewed by 939

Abstract

For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the [...] Read more.

For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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26 pages, 1450 KiB

Open AccessArticle

Ergodic Stationary Distribution and Threshold Dynamics of a Stochastic Nonautonomous SIAM Epidemic Model with Media Coverage and Markov Chain

by Chao Liu, Peng Chen and Lora Cheung

Fractal Fract. 2022, 6(12), 699; https://doi.org/10.3390/fractalfract6120699 - 26 Nov 2022

Viewed by 856

Abstract

A stochastic nonautonomous SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic model with Markov chain and nonlinear noise perturbations has been constructed, which is used to research the hybrid dynamic impacts of media coverage and Lévy jumps on infectious disease transmission. The uniform upper [...] Read more.

A stochastic nonautonomous SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic model with Markov chain and nonlinear noise perturbations has been constructed, which is used to research the hybrid dynamic impacts of media coverage and Lévy jumps on infectious disease transmission. The uniform upper bound and lower bound of the positive solution are studied. Based on defining suitable random Lyapunov functions, we researched the existence of a nontrival positive T-periodic solution. Sufficient conditions are derived to discuss the exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, the persistence in the average sense and extinction of infectious disease are investigated using stochastic analysis techniques. Finally, numerical simulations are utilized to provide evidence for the dynamical properties of the stochastic nonautonomous SIAM. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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20 pages, 936 KiB

Open AccessArticle

Modelling the Frequency of Interarrival Times and Rainfall Depths with the Poisson Hurwitz-Lerch Zeta Distribution

by Carmelo Agnese, Giorgio Baiamonte, Elvira Di Nardo, Stefano Ferraris and Tommaso Martini

Fractal Fract. 2022, 6(9), 509; https://doi.org/10.3390/fractalfract6090509 - 11 Sep 2022

Viewed by 1218

Abstract

The Poisson-stopped sum of the Hurwitz–Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz–Lerch zeta distribution and the [...] Read more.

The Poisson-stopped sum of the Hurwitz–Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz–Lerch zeta distribution and the one inflated Hurwitz–Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz–Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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15 pages, 2090 KiB

Open AccessArticle

A Model Based on Fractional Brownian Motion for Temperature Fluctuation in the Campi Flegrei Caldera

by Antonio Di Crescenzo, Barbara Martinucci and Verdiana Mustaro

Fractal Fract. 2022, 6(8), 421; https://doi.org/10.3390/fractalfract6080421 - 30 Jul 2022

Cited by 4 | Viewed by 1252

Abstract

The aim of this research is to identify an efficient model to describe the fluctuations around the trend of the soil temperatures monitored in the volcanic caldera of the Campi Flegrei area in Naples (Italy). This study focuses on the data concerning the [...] Read more.

The aim of this research is to identify an efficient model to describe the fluctuations around the trend of the soil temperatures monitored in the volcanic caldera of the Campi Flegrei area in Naples (Italy). This study focuses on the data concerning the temperatures in the mentioned area through a seven-year period. The research is initially finalized to identify the deterministic component of the model given by the seasonal trend of the temperatures, which is obtained through an adapted regression method on the time series. Subsequently, the stochastic component from the time series is tested to represent a fractional Brownian motion (fBm). An estimation based on the periodogram of the data is used to estabilish that the data series follows an fBm motion rather than fractional Gaussian noise. An estimation of the Hurst exponent H of the process is also obtained. Finally, an inference test based on the detrended moving average of the data is adopted in order to assess the hypothesis that the time series follows a suitably estimated fBm. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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14 pages, 396 KiB

Open AccessArticle

Threshold Dynamics and the Density Function of the Stochastic Coronavirus Epidemic Model

by Jianguo Sun, Miaomiao Gao and Daqing Jiang

Fractal Fract. 2022, 6(5), 245; https://doi.org/10.3390/fractalfract6050245 - 29 Apr 2022

Cited by 4 | Viewed by 1281

Abstract

Since November 2019, each country in the world has been affected by COVID-19, which has claimed more than four million lives. As an infectious disease, COVID-19 has a stronger transmission power and faster propagation speed. In fact, environmental noise is an inevitable important [...] Read more.

Since November 2019, each country in the world has been affected by COVID-19, which has claimed more than four million lives. As an infectious disease, COVID-19 has a stronger transmission power and faster propagation speed. In fact, environmental noise is an inevitable important factor in the real world. This paper mainly gives a new random infectious disease system under infection rate environmental noise. We give the existence and uniqueness of the solution of the system and discuss the ergodic stationary distribution and the extinction conditions of the system. The probability density function of the stochastic system is studied. Some digital simulations are used to demonstrate the probability density function and the extinction of the system. Full article

(This article belongs to the Special Issue Stochastic Modeling in Biological System)

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