Deterministic Dynamic Model Research Articles

Stability Analysis and Optimal Control Strategies of an Echinococcosis Transmission Model

This paper presents a deterministic compartmental model for echinococcosis transmission dynamics. The basic reproduction number of the model determines the existence and stability of the disease-free and disease-endemic equilibrium points. We further formulate the optimal control problem and obtain the necessary conditions to minimize the number of infected individuals and the associated costs. Numerical simulations show that optimal control strategies can significantly reduce the number of infected individuals to lower levels. Environmental disinfection may be essential for the elimination of infections. The results of this study will be beneficial for the prevention and control of echinococcosis in the Ganzi Tibetan Autonomous Prefecture and other areas of China.

Simulating epidemics via the theory of dynamical systems

This paper describes a deterministic dynamical model of the epidemic spreading of disease. We introduce fundamental concepts from nonlinear dynamics and statistical physics, and use computer simulations that can be implemented by advanced undergraduate and by graduate students. It is shown that the model can describe real-world phenomena during an epidemic. Due to its simplicity and flexibility, the model is also a useful tool for research.

Simple but realistic model of contagion aims for education

A fully deterministic dynamical model of the epidemic spreading of disease can teach graduate students as well as aid real-world research during the ongoing pandemic.

Modelling and Qualitative Analysis of the Transmission Dynamics of COVID-19 in Nigeria

The novel coronavirus (COVID-19) pandemic continues despite series of control measures implemented to curtail it. Therefore, it is pertinent to study how the various proposed control measures can be effectively combined in order to stem the alarming spread of the disease and its attendant consequences. In this paper, a deterministic model for the transmission dynamics of the disease, which incorporates the impacts of the various implemented control measures, is presented. Based on the proposed model, the disease’s basic reproduction number (R0) was derived and the equilibrium solutions were determined. It was shown that whenever R0 \(\le\) 1, the model has only the disease-free equilibrium which is globally stable while in circumstances where R0 > 1, there exists an endemic equilibrium which is globally asymptotically stable. When the latter equilibrium state exists, the former becomes unstable. In addition, the model parameters were estimated using Nigeria’s demographic and COVID-19 surveillance data.
 The model is simulated for different scenarios of the disease outbreak and the results suggest that the disease will die out quickly in the population if about half of the population adhere to personal protection, about half of exposed individuals are efficiently traced and about half of symptomatic individuals are promptly isolated and treated.

The Impact of Media Coverage on Obesity

In this paper, we develop a deterministic compartmental model for the obesity dynamics. Contrary to other contributions on this subject we explore the impact of the media on the spreading of this phenomenon in a constant population. Stability analysis shows that the disease-free equilibrium point is globally asymptotically stable when the number of reproduction is less than unity. We also show that the endemic equilibrium point when it exists is globally asymptotically stable under some conditions.

Deep generative model for probabilistic wind speed and wind power estimation at a wind farm

AbstractThis work introduces a novel method to generate probabilistic hub‐height wind speed forecasts aimed at power output prediction. We employ state‐of‐the‐art convolutional variational autoencoders (CVAEs) trained with historical wind speed observations, multivariable outputs (wind speed, direction, temperature, pressure, and humidity) from a numerical weather prediction (NWP) model and spatio‐temporal encodings. After training, we exploit the CVAE data generating capabilities to produce probabilistic forecasts from the same deterministic dynamical NWP model. The resulting probabilistic forecast provides an insight into the uncertainty of the original deterministic input that compensate for errors due to numerical discretization, inaccuracies in initial/boundary conditions and parameterizations and, most importantly wind speed fluctuations due to complex terrain features. To show the performance of the proposed model, we validate the approach with forecasted and observed data for fifteen sites in a wind farm in Awaji Island, Japan, in a challenging zone with complex topography and highly fluctuating wind patterns. We show that the proposed method provides improved wind speed forecasts from both deterministic (reduced root mean square error) and probabilistic (reduced continuous ranked probability score) standpoints. We also use ensemble forecast validation methods to assess the statistical properties of the CVAE and conclude that in zones with rapidly changing wind speed dynamics, the CVAE ensemble performs in a superior way when compared to the analog ensemble method. Finally, we also provide an insight on how the results can be used to obtain reliable wind power estimations.

Point-kinetics neutron noise modeling and analysis via probabilistic finite state automata

This paper addresses two issues for safety & performance analysis of nuclear reactors; these issues are: (i) Modeling of the neutron noise process, which is constructed as stochastic differential equations by augmenting deterministic dynamic models of point kinetics, thermal hydraulics, and mechanical vibrations in the reactor core, and (ii) Time-series analysis of nuclear reactor internals for early-stage anomaly detection by making use of the neutron noise model. The model is used to generate an ensemble of time series of neutron noise under both normal and anomalous operating conditions, where an anomaly in the postulated probability distribution of neutron noise is caused by abnormal vibration of the fuel assembly. A symbolic time series analysis (STSA) tool, called probabilisitc finite state automata (PFSA), has been selected to detect the anomalies, if any. The PFSA method symbolizes a time-series and then encodes the resulting symbol string into a state transition probability matrix, which is both stochastic and ergodic by construction. The discrimination function for anomaly detection is built upon the residual error of the principal (left) eigenvector of the state transition probability matrix, which is also the state probability vector of the PFSA. The accuracy of the proposed analysis and its capability for online anomaly detection is demonstrated by simulation on a low-order dynamic model of a 2,500 MWt PWR of the Three-Mile-Island type.

Mathematical modeling for COVID-19 transmission dynamics: A case study in Ethiopia

In this paper, we proposed a nonlinear deterministic mathematical model for the transmission dynamics of COVID-19. First, we analyzed the system properties such as boundedness of the solutions, existence of disease-free and endemic equilibria, local and global stability of equilibrium points. Besides, we computed the basic reproduction number R0 and studied its normalized sensitivity for model parameters to identify the most influencing parameter. The local stability of the disease-free equilibrium point is also verified via the help of the Jacobian matrix and Routh Hurwitz criteria. Moreover, the global stability of the disease-free equilibrium point is proved by using the approach of Castillo-Chavez and Song. We also proved the existence of the forward bifurcation using the center manifold theory. Then the model is fitted with COVID-19 infected cases reported from March 13, 2020, to July 31, 2021, in Ethiopia. The values of model parameters are then estimated from the data reported using the least square method together with the fminsearch function in the MATLAB optimization toolbox. Finally, different simulation cases were performed using PYTHON software to compare with analytical results. The simulation results suggest that the spread of COVID-19 can be managed via minimizing the contact rate of infected and increasing the quarantine of exposed individuals.

Transmission dynamic and backward bifurcation of Middle Eastern respiratory syndrome coronavirus

Middle East respiratory syndrome coronavirus (MERS-CoV) remains an emerging disease threat with regular human cases on the Arabian Peninsula driven by recurring camels to human transmission events. In this paper, we present a new deterministic model for the transmission dynamics of (MERS-CoV). In order to do this, we develop a model formulation and analyze the stability of the proposed model. The stability conditions are obtained in term of R0, we find those conditions for which the model become stable. We discuss basic reproductive number R0 along with sensitivity analysis to show the impact of every epidemic parameter. We show that the proposed model exhibits the phenomena of backward bifurcation. Finally, we show the numerical simulation of our proposed model for supporting our analytical work. The aim of this work is to show via mathematical model the transmission of MERS-CoV between humans and camels, which are suspected to be the primary source of infection.

Mathematical modeling of the dynamics of maize streak virus disease (MSVD)

<abstract><p>In this paper, a deterministic ordinary differential equations model for the transmission dynamics of maize streak virus disease (MSVD) in maize plants is proposed and analyzed qualitatively. Using the next generation matrix approach, the basic reproduction number, $ \mathcal{R}_{0} $ with respect to the MSVD free equilibrium is found to be $ 4.7065\approx5 $. The conditions for local stability of the disease-free equilibrium and endemic equilibrium points were established. From the results, the disease-free equilibrium point was found to be unstable whenever $ \mathcal{R}_{0} > 1 $ and the endemic equilibrium point was found to be locally asymptotically stable whenever $ \mathcal{R}_{0} > 1 $. The sensitivity indices of various parameters with respect to the MSVD eradication or spreading were determined. It was found that $ b\; , \beta_{{1}}\; , \beta_{{11}}\; , $ and $ \beta_{{2}} $ are the parameters that are directly related to $ \mathcal{R}_{0} $, and $ H_{2}, \mu, \mu_{{1}} $ and $ \gamma $ are inversely related to $ \mathcal{R}_{0} $. Numerical simulation was performed and displayed graphically to justify the analytical results.</p></abstract>

Mathematical model and stability analysis on the transmission dynamics of skin sores.

In this study, a non-linear deterministic model for the transmission dynamics of skin sores (impetigo) disease is developed and analysed by the help of stability of differential equations. Some basic properties of the model including existence and positivity as well as boundedness of the solutions of the model are investigated. The disease-free and endemic equilibrium were investigated, as well as the basic reproduction number, R0, also calculated using the next-generation matrix approach. When R0 < 1, the model's stability analysis reveals that the system is asymptotically stable at disease-free critical point globally as well as locally. If R0 > 1, the system is asymptotically stable at disease-endemic equilibrium both locally and globally. The long-term behaviour of the skin sores model's steady-state solution in a population is investigated using numerical simulations of the model.

Optimal Control Techniques for the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics

From the studies of HIV/AIDS transmission and treatment dynamicsusing mathematical modeling, literature reviews have shown that attention hadnot been given to the behavioral attitude of screen-aware infectives not readyto receive treatment, HIV-aware infectives that initiated treatment buttruncated only to resume treatment later (therapy abuse) and those onconsistent treatment protocols. Moreso, following the non-outright eradicationof the deadly HI-virus, recommendations have been geared towards exploringoptimal control theory for the maximization of healthy uninfected CD4+T-cells. Therefore, this present investigation seeks and formulated an optimalcontrol 6-Dimensional deterministic mathematical dynamic model, which accountedfor the Role of Antiretroviral Therapy (ART) abuse in the treatment dynamics ofthe HIV/AIDS epidemic. The materials and methods for this model are constitutedby a set of 6-Dimensional varying subpopulations interacting with concentratedHI-viral load. Interactions are investigated using bilinear control functions(condom use and ART) with empirically generated data. The model assumed adeterministic approach and was formulated using the fundamental theory ofdifferential equations. Theoretical optimal predictions explored classicalnumerical methods with optimal control techniques (Pontryagin's maximumprinciple in conjunction with Hessian matrix) as a basis. Numerical simulationswere conducted using in-built Runge-Kutta of the order of precision 4 in aMathcad surface. Following the derived model for both off-optimal control andonset-optimal control functions and model optimal control pair as well as model optimality system, resultsof simulations indicated that at off-optimal control function, near zeropopulation extinction was observed. From the application of optimal controlfunctions under optimal control techniques, there exists tremendousrejuvenation of susceptible populations vindicated by a reduction in the rateof ART abuse under a minimal proportion of bilinear control functions. Thestudy concluded that adopting optimal control techniques for the investigationof the role of ART abuse in HIV/AIDS treatment yield highly significantrecovery of healthy CD4+ T-Cells at minimal systemic cost whencompared with off-optimal control outcome. Therefore, the study not onlyaffirmed the vital concept of optimal control strategy but also, instituted theviability of the model. Thus, this model can be extensively used in Bio-systemand applied mathematics.

Optimal control analysis of Taenia saginata bovine cysticercosis and human taeniasis

Bovine cysticercosis and human taeniasis are neglected food-borne diseases that pose challenge to food safety, human health and livelihood of rural livestock farmers. In this paper, we have formulated and analyzed a deterministic model for transmission dynamics and control of taeniasis and cysticercosis in humans and cattle respectively. The analysis shows that both the disease free equilibrium (DFE) and endemic equilibrium (EE) exist. To study the dynamics of the diseases, we derived the basic reproduction number R0 by next generation matrix method which shows whether the diseases die or persist in humans and cattle. The diseases clear if R0<1 and persist when R0>1. The normalized forward sensitivity index is used to derive sensitive indices of model parameters. Sensitivity analysis results indicate that human's and cattle's recruitment rates, infection rate of cattle from contaminated environment, probability of humans to acquire taeniasis due to consumption of infected meat, defecation rate of humans with taeniasis and the consumption rate of raw or undercooked infected meat are the most positive sensitive parameters whereas the natural death rates for humans, cattle, Taenia saginata eggs and the proportion of unconsumed infected meat are the most negative sensitive parameters in diseases’ transmission. These results suggest that control measures such as improving meat cooking, meat inspection and treatment of infected humans will be effective for controlling taeniasis and cysticercosis in humans and cattle respectively. The optimal control theory is applied by considering three time dependent controls which are improved meat cooking, vaccination of cattle, and treatment of humans with taeniasis when they are implemented in combination. The Pontryagin's maximum principle is adopted to find the necessary conditions for existence of the optimal controls. The Runge Kutta order four forward-backward sweep method is implemented in Matlab to solve the optimal control problem. The results indicate that a strategy which focuses on improving meat cooking and treatment of humans with taeniasis is the optimal strategy for diseases’ control.

Probabilistic Forecasts of Arctic Sea Ice Thickness

In recent decades, warming temperatures have caused sharp reductions in the volume of sea ice in the Arctic Ocean. Predicting changes in Arctic sea ice thickness is vital in a changing Arctic for making decisions about shipping and resource management in the region. We propose a statistical spatio-temporal two-stage model for sea ice thickness and use it to generate probabilistic forecasts up to three months into the future. Our approach combines a contour model to predict the ice-covered region with a Gaussian random field to model ice thickness conditional on the ice-covered region. Using the most complete estimates of sea ice thickness currently available, we apply our method to forecast Arctic sea ice thickness. Point predictions and prediction intervals from our model offer comparable accuracy and improved calibration compared with existing forecasts. We show that existing forecasts produced by ensembles of deterministic dynamic models can have large errors and poor calibration. We also show that our statistical model can generate good forecasts of aggregate quantities such as overall and regional sea ice volume. Supplementary materials accompanying this paper appear on-line.

Estimating vaccination threshold and impact in the 2017–2019 hepatitis A virus outbreak among persons experiencing homelessness or who use drugs in Louisville, Kentucky, United States

BackgroundBetween September 2017 and June 2019, an outbreak of hepatitis A virus (HAV) occurred in Louisville, Kentucky, resulting in 501 cases and 6 deaths, predominantly among persons who experience homelessness or who use drugs (PEH/PWUD). The critical vaccination threshold (Vc) required to achieve herd immunity in this population is unknown. We investigated Vc and vaccination impact using epidemic modeling. MethodsTo determine which population subgroups had high infection risks, we employed a technique based on comparing the proportion of cases arising before and after the epidemic peak, across subgroups. We also developed a dynamic deterministic model of HAV transmission among PEH/PWUD to estimate the basic reproduction number (R0), herd immunity threshold, Vc and the effect of timing of the vaccination intervention on epidemic and economic outcomes. ResultsOf the 501 confirmed or probable cases, 385 (76.8%) were among PEH/PWUD. Among PEH/PWUD and within the general population, homelessness was a significant risk factor for infection in the initial stages of the outbreak (odds ratios for homeless versus not homeless: 2.62; 95% confidence interval (CI): 1.62–4.25 for PEH/PWUD and 2.39; 95% CI: 1.51–3.78 for all detected cases). Our estimate for R0 ranges between 2.85 and 3.54, corresponding to an estimate of 69% (95% CI: 65–72) for herd immunity threshold and 76% (95% CI: 72%-80%) for Vc, assuming a vaccine with 90% efficacy. The observed vaccination program was estimated to have averted 30 hospitalizations (95% CI: 19–43), associated with over US$490 000 (95% CI: $310 000–700 000) in hospitalization cost. Greater impact was observed with earlier and faster vaccination implementation. ConclusionsVaccination coverage of at least 77% is likely required to prevent outbreaks of HAV among PEH/PWUD in Louisville, assuming a 90% vaccine efficacy. Proactive hepatitis A vaccination programs among PEH/PWUD will maximize health and economic benefits of these programs and reduce the likelihood of another outbreak.

Deterministic and stochastic effects in spreading dynamics: A case study of bovine viral diarrhea.

Bovine viral diarrhea (BVD) is a disease in cattle with complex transmission dynamics that causes substantial economic losses and affects animal welfare. The infection can be transient or persistent. The mostly asymptomatic persistently infected hosts are the main source for transmission of the virus. This characteristic makes it difficult to control the spreading of BVD. We develop a deterministic compartmental model for the spreading dynamics of BVD within a herd and derive the basic reproduction number. This epidemiological quantity indicates that identification and removal of persistently infected animals is a successful control strategy if the transmission rate of transiently infected animals is small. Removing persistently infected animals from the herd at birth results in recurrent outbreaks with decreasing peak prevalence. We propose a stochastic version of the compartmental model that includes stochasticity in the transmission parameters. This stochasticity leads to sustained oscillations in cases where the deterministic model predicts oscillations with decreasing amplitude. The results provide useful information for the design of control strategies.

Modeling the dynamics of Lassa fever in Nigeria

Lassa fever is a zoonotic disease spread by infected rodents known as multimammate rats. The disease has posed a significant and major health challenge in West African countries, including Nigeria. To have a deeper understanding of Lassa fever epidemiology in Nigeria, we present a deterministic dynamical model to study its dynamical transmission behavior in the population. To mimic the disease’s biological history, we divide the population into two groups: humans and rodents. We established the quantity known as reproduction number {mathcal {R}}_{0}. The results show that if {mathcal {R}}_{0} <1 then the system is stable, otherwise it is unstable. The model fitting was performed using the nonlinear least square method on cumulative reported cases from Nigeria between 2018 and 2020 to obtain the best fit that describes the dynamics of this disease in Nigeria. In addition, sensitivity analysis was performed, and the numerical solution of the system was derived using an iterative scheme, the fifth-order Runge–Kutta method. Using different numeric values for each parameter, we investigate the effect of all highest sensitivity indices’ parameters on the population of infected humans and infected rodents. Our findings indicate that any control strategies and methods that reduce rodent populations and the risk of transmission from rodents to humans and rodents would aid in the population’s control of Lassa fever.

Exploring the future of the Coral Sea micronekton

Ecosystem models forced by future climate simulations outputs from the Coupled Model Intercomparison Project Phase 5 (CMIP5) simulate a substantial decline of tropical marine animal biomass over the course of the 21st century. Regional projections are however far more uncertain because of well-known biases common to most CMIP5 historical simulations that propagate within the food web. Moreover, the model outputs for high trophic levels marine fauna suffer from lack of validation based on in situ data. In this study, we implement a “bias-mitigation” strategy to reduce the physical oceanography and biogeochemical biases simulated by three CMIP5 models under the future RCP8.5 scenario. We force two very different micronekton models with these “bias-mitigated” outputs to infer the future micronekton changes in the Coral Sea: a 3-D deterministic population dynamics model; and a 3-D statistical model based on in situ hydro-acoustic data. These two models forecast a consistent pattern of micronekton abundance changes in the epipelagic layer (0–150 m) by 2100 for three different climate forcing used, with a marked decrease south of 22°S and a smaller increase further north mostly related to temperature and chlorophyll changes. In contrast, changes in the vertical patterns of micronekton predicted by the two models considerably differ in the upper mesopelagic layers (150–450 m) and lower mesopelagic layer (450–1000 m), highlighting the structural sensitivity in model type. Since micronekton are prey of all larger marine predators, those discrepancies in vertical structures of micronekton may hamper our potential to predict how top predators may evolve in the future.

Consideration of streamflow forecast uncertainty in the development of short-term hydropower station optimal operation schemes: A novel approach based on mean-variance theory

The uncertainty of streamflow forecast will cause the ineffectiveness on the short-term hydropower station optimal operation (SHSOO) and lead to the risk of producing less hydropower than planned. Traditionally, stochastic optimal theory represented by the stochastic dynamic programming model (SDPM) is commonly used for SHSOO, which is focus on the uncertainty during a single lead time and lack of an overall consideration of multiple independent lead times of differing length. Thus, this paper proposes a short-term hydropower station optimal operation model based on mean-variance theory (MV-SHSOOM) for reducing operational risk. The proposed multi-step model considers the forecasting uncertainty of different lead times as an integral component to guide a station’s optimal operation. Specifically, the forecast uncertainty was integrated by the joint probability distribution of forecast error using a multivariate hydrologic uncertainty processor (HUP) at first. Then, possible observed streamflow sequences were obtained using the joint probability distribution of forecast error and taken as input to the short-term optimal operation model of the hydropower station. Finally, the mean-variance theory was applied to construct the optimal operation model for balancing the power generation and operational risk. The proposed model was applied to the Jinxi hydropower station, China, to determine its performance. The results show that: 1) the dependent structure among the forecast errors at different lead times is significant in terms of Spearman correlation, and the joint probability distribution of forecast error can be effectively described by a multivariate HUP; 2) the proposed MV-SHSOOM is superior to the traditional deterministic dynamic programming model (DDPM) and SDPM, and can more effectively reduce operational risk while increasing power generation.

Functional approach to deterministic finite-state automata systems dynamic modeling

Research aim is to study the functional approach to modeling the deterministic finite-state automata system which is not confined to the elements communication topology and the heterogeneity of the algorithm types. Relevance. The substantial part of engineering systems applied in the mining industry may be described through the finite-state automaton model. They include the mine conveyor systems, shaft signal systems, processing facilities control systems, etc. Such model makes it possible to shorten the time spent on control software development and carry out algorithm analysis, debug, and testing effectively. There are a lot of effective approaches and tools to solve the problem of finite-state automata dynamic modeling, each of which has its own advantages and disadvantages. Methodology. In this article, the methodology of finite-state automata systems modeling is considered as applied to mine conveyor systems. Results. Final-state automata (FSA) models have been developed together with the conditions for FSA systems dynamic modeling as applied to mine conveyor systems. Conclusions. The considered approach to modeling, which involves functors and applicative functors 122 "Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal". No. 2. 2021 ISSN 0536-1028 for structure composition and its operational dynamics study, as well as the possibility to mathematically prove the model’s properties, makes the approach a good alternative when choosing tools for systems models development.