Deterministic Dynamic Model Research Articles

Electricity Market Stochastic Dynamic Model and Its Mean Stability Analysis

Based on the deterministic dynamic model of electricity market proposed by Alvarado, a stochastic electricity market model, considering the random nature of demand sides, is presented in this paper on the assumption that generator cost function and consumer utility function are quadratic functions. The stochastic electricity market model is a generalization of the deterministic dynamic model. Using the theory of stochastic differential equations, stochastic process theory, and eigenvalue techniques, the determining conditions of the mean stability for this electricity market model under small Gauss type random excitation are provided and testified theoretically. That is, if the demand elasticity of suppliers is nonnegative and the demand elasticity of consumers is negative, then the stochastic electricity market model is mean stable. It implies that the stability can be judged directly by initial data without any computation. Taking deterministic electricity market data combined with small Gauss type random excitation as numerical samples to interpret random phenomena from a statistical perspective, the results indicate the conclusions above are correct, valid, and practical.

Modelling the Effect of DOTS and Isolation on TB Transmission Dynamics

Abstract: A deterministic model for the transmission dynamics of Tuberculosis (TB) under Direct Observation Therapy Strategy (DOTS) and Isolation in Nigeria is developed and rigorously analysed. The model, consisting of mutually-exclusive epidemiological compartments representing the number of undetected, detected and isolated individuals who are treated under DOTS programme and those who developed Multi-drug resistance. The model has a disease free equilibrium (DFE), which is locally asymptotically stable, whenever the maximum of the associated reproduction numbers of the model (denoted by Rc) is less than unity. Furthermore, the model undergoes a backward bifurcation, where the disease-free equilibrium co-exists with a stable endemic equilibrium. Numerical simulations, using epidemiological and demographic data relevant to Nigeria obtained from WHO and USAID [35,36,38], shows that provided the rate at which the undetected individuals with active TB recovered exceeded a critical values, then DOTS, the STOP TB initiative programme of WHO can lead to effective elimination of TB in Nigeria. This suggest that the detection rate plays significant role in the elimination of TB. Furthermore, it is shown that if the progress or rate of individuals who are susceptible to TB is low, it can also lead to elimination of the disease in Nigeria. The results also shows that if the effective contact rate (  ) for TB infection remains below certain critical value (0.187), the disease can be eliminated.

Modelling cholera in periodic environments

We propose a deterministic compartmental model for cholera dynamics in periodic environments. The model incorporates seasonal variation into a general formulation for the incidence (or, force of infection) and the pathogen concentration. The basic reproduction number of the periodic model is derived, based on which a careful analysis is conducted on the epidemic and endemic dynamics of cholera. Several specific examples are presented to demonstrate this general model, and numerical simulation results are used to validate the analytical prediction.

Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination

In this paper, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination is formulated. The effective reproduction number is computed in order to measure the relative impact for individual or combined intervention for effective disease control. The effective reproductive number, R_e is defined as the number of secondary cases that one infected individual will cause through the duration of the infectious period. The disease-free equilibrium is computed and proved to be locally asymptotically stable when R_e<1 and unstable when R_e>1 .It is proved that there exists at least one endemic equilibrium point for all R_e>1. In the absence of disease-induced death, it is proved that the transcritical bifurcation at R_0=1 is supercritical (forward). Sensitivity analysis is performed on the basic reproduction number and it is noted that the most sensitive parameters are the probability of transmission of the disease from an infectious individual to a susceptible individual per contact, β, per capita contact rate ,c, per capita birth rate, π and the progression rate from latent to infectious stage, δ. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.

Modeling Transmission Dynamics ofStreptococcus suiswith Stage Structure and Sensitivity Analysis

Streptococcosis is one of the major infectious and contagious bacterial diseases for swine farm in southern China. The influence of various control measures on the outbreaks and transmission ofS. suisis not currently known. In this study, in order to explore effective control and prevention measures we studied a deterministic dynamic model with stage structure forS. suis. The basic reproduction numberℛ0is identified and global dynamics are completely determined byℛ0. It shows that ifℛ0&lt;1, the disease-free equilibrium is globally stable and the disease dies out, whereas ifℛ0&gt;1, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. The model simulations well agree with new clinical cases and the basic reproduction number of this model is about 1.1333. Some sensitivity analyses ofℛ0in terms of the model parameters are given. Our study demonstrates that combination of vaccination and disinfection of the environment are the useful control strategy forS. suis.

A model of bi-mode transmission dynamics of hepatitis C with optimal control.

In this paper, we present a rigorous mathematical analysis of a deterministic model for the transmission dynamics of hepatitis C. The model is suitable for populations where two frequent modes of transmission of hepatitis C virus, namely unsafe blood transfusions and intravenous drug use, are dominant. The susceptible population is divided into two distinct compartments, the intravenous drug users and individuals undergoing unsafe blood transfusions. Individuals belonging to each compartment may develop acute and then possibly chronic infections. Chronically infected individuals may be quarantined. The analysis indicates that the eradication and persistence of the disease is completely determined by the magnitude of basic reproduction number R c. It is shown that for the basic reproduction number R c < 1, the disease-free equilibrium is locally and globally asymptotically stable. For R c > 1, an endemic equilibrium exists and the disease is uniformly persistent. In addition, we present the uncertainty and sensitivity analyses to investigate the influence of different important model parameters on the disease prevalence. When the infected population persists, we have designed a time-dependent optimal quarantine strategy to minimize it. The Pontryagin’s Maximum Principle is used to characterize the optimal control in terms of an optimality system which is solved numerically. Numerical results for the optimal control are compared against the constant controls and their efficiency is discussed.

Probabilistic uncertainty analysis of epidemiological modeling to guide public health intervention policy

Mathematical modeling of disease transmission has provided quantitative predictions for health policy, facilitating the evaluation of epidemiological outcomes and the cost-effectiveness of interventions. However, typical sensitivity analyses of deterministic dynamic infectious disease models focus on model architecture and the relative importance of parameters but neglect parameter uncertainty when reporting model predictions. Consequently, model results that identify point estimates of intervention levels necessary to terminate transmission yield limited insight into the probability of success. We apply probabilistic uncertainty analysis to a dynamic model of influenza transmission and assess global uncertainty in outcome. We illustrate that when parameter uncertainty is not incorporated into outcome estimates, levels of vaccination and treatment predicted to prevent an influenza epidemic will only have an approximately 50% chance of terminating transmission and that sensitivity analysis alone is not sufficient to obtain this information. We demonstrate that accounting for parameter uncertainty yields probabilities of epidemiological outcomes based on the degree to which data support the range of model predictions. Unlike typical sensitivity analyses of dynamic models that only address variation in parameters, the probabilistic uncertainty analysis described here enables modelers to convey the robustness of their predictions to policy makers, extending the power of epidemiological modeling to improve public health.

Evolution of learning strategies in temporally and spatially variable environments: A review of theory

The theoretical literature from 1985 to the present on the evolution of learning strategies in variable environments is reviewed, with the focus on deterministic dynamical models that are amenable to local stability analysis, and on deterministic models yielding evolutionarily stable strategies. Individual learning, unbiased and biased social learning, mixed learning, and learning schedules are considered. A rapidly changing environment or frequent migration in a spatially heterogeneous environment favors individual learning over unbiased social learning. However, results are not so straightforward in the context of learning schedules or when biases in social learning are introduced. The three major methods of modeling temporal environmental change–coevolutionary, two-timescale, and information decay–are compared and shown to sometimes yield contradictory results. The so-called Rogers’ paradox is inherent in the two-timescale method as originally applied to the evolution of pure strategies, but is often eliminated when the other methods are used. Moreover, Rogers’ paradox is not observed for the mixed learning strategies and learning schedules that we review. We believe that further theoretical work is necessary on learning schedules and biased social learning, based on models that are logically consistent and empirically pertinent.

Existence, Uniqueness, and Stability of Slowly Oscillating Periodic Solutions for Delay Differential Equations with Nonnegativity Constraints

Deterministic dynamical system models with delayed feedback and nonnegativity constraints arise in a variety of applications in science and engineering. Under certain conditions oscillatory behavior has been observed and it is of interest to know when this behavior is periodic. Here we consider one-dimensional delay differential equations with nonnegativity constraints as prototypes for such models. We obtain sufficient conditions for the existence of slowly oscillating periodic solutions (SOPS) of such equations when the delay/lag interval is long and the dynamics depend only on the current and delayed state. Under further assumptions, including possibly longer delay intervals and restricting the dynamics to depend only on the delayed state, we prove uniqueness and exponential stability for such solutions. To prove these results, we develop a theory for studying perturbations of these constrained SOPS. We illustrate our results with simple examples of biochemical reaction network models and an Internet rate control model.

Oil and economic development: Libya in the post-Gaddafi era

Libya experienced traumatic political and economic upheaval during 2011 arising from an eight-month-long civil war that cost thousands of lives, resulted in major economic dysfunction, destroyed part of the country's infrastructure, almost halted oil production, the country's major source of revenue generation and exports, as well as destroyed part of the sector's support infrastructure. While the civil war resulted in the ending of 42years under Muammar Gaddafi rule, the economic legacy as represented by the costs of reconstruction efforts is enormous. While the freeing up of tens of billions of dollars of frozen assets may be the key to the country's short-term rehabilitation, longer-term reconstruction, growth and stability will fundamentally depend upon rehabilitating the country's oil sector. Interestingly, this rehabilitation will also have a wider global impact.This paper uses a deterministic dynamic macroeconomic model to analyse the effects upon key macroeconomic variables of a recovery in Libyan oil production to levels that existed prior to the revolution. Model simulation results indicate that additional oil revenue brings about: an increase in government revenue, increased government spending in the domestic economy, increased foreign asset stocks and increased output and wages in the non-oil sector. However, increased oil revenue may also produce adverse consequences, particularly upon the non-oil trade balance, arising from a loss of competitiveness of non-oil tradable goods induced by an appreciation of the real exchange rate and increased imports stimulated by increased real income. Model simulation results also suggest that investment-stimulating policy measures by the government produce the most substantive longer-term benefits for the economy.

Monetary Policy as an Optimal Control Problem

This paper analyses the monetary policy of a central bank in a simple deterministic and continuous dynamic non-linear New-Keynesian model with an active central bank conducting monetary policy within inflation targeting framework. To meet this purpose, first we derive two differential equations capturing the dynamics in the economy: the dynamic IS curve representing the commodity market and the Phillips curve capturing the connection between the real and nominal sectors of the economy in a continuous form. By introducing a quadratic loss function commonly used in New Keynesian Economics we get optimal control problem which solution will be analysed with the use of fuzzy control. Then we introduce a modified form of the Taylor rule and analyse the solution of the same differential equations capturing the dynamics of the economy using Taylor rule instead of loss function. The comparison of the solutions of both models will be demonstrated in examples in which the main characteristic of dynamics of production and inflation are displayed.

Qualitative dynamics of lowly- and highly-pathogenic avian influenza strains

A new deterministic model for the transmission dynamics of the lowly- and highly-pathogenic avian influenza (LPAI and HPAI) strains is designed and rigorously analyzed. The model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. It is shown that the re-infection of birds infected with the LPAI strain causes the backward bifurcation phenomenon. In the absence of such re-infection, the disease-free equilibrium of the model is globally-asymptotically stable when the associated reproduction number is less than unity. Using non-linear Lyapunov functions of Goh–Volterra type, the LPAI-only and HPAI-only boundary equilibria of the model are shown to be globally-asymptotically stable when they exist. A special case of the model is shown to have a continuum of co-existence equilibria whenever the associated reproduction numbers of the two strains are equal and exceed unity. Furthermore, numerical simulations of the model suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity.

Herd immunity effect of the HPV vaccination program in Australia under different assumptions regarding natural immunity against re-infection

Deterministic dynamic compartmental transmission models (DDCTMs) of human papillomavirus (HPV) transmission have been used in a number of studies to estimate the potential impact of HPV vaccination programs. In most cases, the models were built under the assumption that an individual who cleared HPV infection develops (life-long) natural immunity against re-infection with the same HPV type (this is known as SIR scenario). This assumption was also made by two Australian modelling studies evaluating the impact of the National HPV Vaccination Program to assist in the health-economic assessment of male vaccination. An alternative view denying natural immunity after clearance (SIS scenario) was only presented in one study, although neither scenario has been supported by strong evidence. Some recent findings, however, provide arguments in favour of SIS.We developed HPV transmission models implementing life-time (SIR), limited, and non-existent (SIS) natural immunity. For each model we estimated the herd immunity effect of the ongoing Australian HPV vaccination program and its extension to cover males. Given the Australian setting, we aimed to clarify the extent to which the choice of model structure would influence estimation of this effect. A statistically robust and efficient calibration methodology was applied to ensure credibility of our results. We observed that for non-SIR models the herd immunity effect measured in relative reductions in HPV prevalence in the unvaccinated population was much more pronounced than for the SIR model. For example, with vaccine efficacy of 95% for females and 90% for males, the reductions for HPV-16 were 3% in females and 28% in males for the SIR model, and at least 30% (females) and 60% (males) for non-SIR models.The magnitude of these differences implies that evaluations of the impact of vaccination programs using DDCTMs should incorporate several model structures until our understanding of natural immunity is improved.

Managing Transportation Facilities in Design–Build–Finance–Operate Partnerships

In recent years, transportation planning has been challenged by an increasing need for infrastructure development, a shortfall of revenue from the public sector, and political trending toward deregulation of transportation infrastructure development. These factors have led to increased interest in privatization of transportation infrastructure and the development of public–private partnerships, such as design–build–finance–operate. Although the overall goal of a transportation infrastructure project is to provide safe, reliable transportation systems for the public, parties involved in public–private partnerships take different roles and responsibilities. The public sector leads in laying out the terms and standards to regulate the obligations between the state departments of transportation and private entities. The private sector makes capital investment to provide agreed-upon services as well as to assume various investment risks, including project operational and financial risks. Toll-pricing strategies are a key component for the public sector in regulating the operation of a public–private partnership facility and for the private sector in controlling investment risks. This study investigated the applicability of deterministic dynamic optimization models for determining toll-pricing strategies that can help improve mobility, secure the public interest, and attract investment from the private sector. A case study of a design–build–finance–operate project was completed. Results showed that the proposed model provides a useful tool to assist both the public and private sectors in making more informed decisions, including study of optimal strategies to seek investment return and determination of the predefined contract regulations.

Optimal control for forest management and conservation analysis in dehesa ecosystems

This paper presents a deterministic finite time horizon dynamic optimisation model aimed to determine optimal paths for artificial plantations and natural regeneration of two main tree species in dehesa multiple use ecosystems, holm oak (Quercus ilex L.) and cork oak (Q. suber L.). Whilst dehesa forest sustainability problems associated to exhaustive use of grazing resources have been indirectly approached by European Union authorities, providing support for artificial plantations over treeless land, no mention is made to natural regeneration techniques. In this sense, the formulated model allows for natural regeneration of already established ageing stands as a complement or even a substitute of actual reforestation practices. The proposed methodology is neither designed to determine optimal rotation of tree species nor optimal decorticating or pruning cycles of cork oaks and holm oaks, respectively. Instead, this information enters the model exogenously through knowledge of region specific silvicultural cycles for those commercially relevant tree species, and the optimisation program acts as an optimal land use allocator and thus a practical tool for policy analysis purposes. In addition to existing cost benefit analysis applications in dehesa ecosystems, the presented model allows in one side efficient evaluation of long term management dynamics —thus oak woodlands sustainability can be tested for sufficiently large time horizons—, and in the other, management decisions, instead of being forced through predefined scenarios, correspond to the optimal actions a decision agent would take from the complete set of feasible possibilities given actual land use and tree age distributions.

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

A new deterministic model for the transmission dynamics of two strains of influenza is designed and used to qualitatively assess the role of cross-immunity on the transmission process. It is shown that incomplete cross-immunity could induce the phenomenon of backward bifurcation when the associated reproduction number is less than unity. The model undergoes competitive exclusion (where Strain i drives out Strain j to extinction whenever R0i>1>R0j;i,j=1,2,i≠j). For the case where infection with one strain confers complete immunity against infection with the other strain, it is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the reproduction number is less than unity. In the absence of cross-immunity, the model can have a continuum of co-existence endemic equilibria (which is shown to be globally-asymptotically stable for a special case). When infection with one strain confers incomplete immunity against the other, numerical simulations of the model show that the two strains co-exist, with Strain i dominating (but not driving out Strain j), whenever R0i>R0j>1.

Topological field theory of dynamical systems

Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively stable due to what is generically known as non-renormalization theorems. As a result, all (equilibrium) dynamical models are divided into three major categories: Markovian models with unbroken Q-symmetry, chaotic models with Q-symmetry spontaneously broken on the mean-field level by, e.g., fractal invariant sets (e.g., strange attractors), and intermittent or self-organized critical (SOC) models with Q-symmetry dynamically broken by the condensation of instanton-antiinstanton configurations (earthquakes, avalanches, etc.) SOC is a full-dimensional phase separating chaos and Markovian dynamics. In the deterministic limit, however, antiinstantons disappear and SOC collapses into the "edge of chaos." Goldstone theorem stands behind spatio-temporal self-similarity of Q-broken phases known under such names as algebraic statistics of avalanches, 1/f noise, sensitivity to initial conditions, etc. Other fundamental differences of Q-broken phases is that they can be effectively viewed as quantum dynamics and that they must also have time-reversal symmetry spontaneously broken. Q-symmetry breaking in non-equilibrium situations (quenches, Barkhausen effect, etc.) is also briefly discussed.

Modelling the Role of Diagnosis, Treatment, and Health Education on Multidrug-Resistant Tuberculosis Dynamics

Tuberculosis, an airborne disease affecting almost a third of the world’s population remains one of the major public health burdens globally, and the resurgence of multidrug-resistant tuberculosis in some parts of sub-Saharan Africa calls for concern. To gain insight into its qualitative dynamics at the population level, mathematical modeling which require as inputs key demographic and epidemiological information can fill in gaps where field and lab data are not readily available. A deterministic model for the transmission dynamics of multi-drug resistant tuberculosis to assess the impact of diagnosis, treatment, and health education is formulated. The model assumes that exposed individuals develop active tuberculosis due to endogenous activation and exogenous re-infection. Treatment is offered to all infected individuals except those latently infected with multi-drug resistant tuberculosis. Qualitative analysis using the theory of dynamical systems shows that, in addition to the disease-free equilibrium, there exists a unique dominant locally asymptotically stable equilibrium corresponding to each strain. Numerical simulations suggest that, at the current level of control strategies (with Malawi as a case study), the drug-sensitive tuberculosis can be completely eliminated from the population, thereby reducing multi-drug resistant tuberculosis.

Post-hoc selection of dynamic causal models

Dynamic causal modelling (DCM) was originally proposed as a hypothesis driven procedure in which a small number of neurobiologically motivated models are compared. Model comparison in this context usually proceeds by individually fitting each model to data and then approximating the corresponding model evidence with a free energy bound. However, a recent trend has emerged for comparing very large numbers of models in a more exploratory manner. This led Friston and Penny (2011) to propose a post-hoc approximation to the model evidence, which is computed by optimising only the largest (full) model of a set of models. The evidence for any (reduced) submodel is then obtained using a generalisation of the Savage-Dickey density ratio (Dickey, 1971). The benefit of this post-hoc approach is a huge reduction in the computational time required for model fitting. This is because only a single model is fitted to data, allowing a potentially huge model space to be searched relatively quickly. In this paper, we explore the relationship between the free energy bound and post-hoc approximations to the model evidence in the context of deterministic (bilinear) dynamic causal models (DCMs) for functional magnetic resonance imaging data.

Analysis and modeling of resistive switching statistics

The resistive random access memory (RRAM), based on the reversible switching between different resistance states, is a promising candidate for next-generation nonvolatile memories. One of the most important challenges to foster the practical application of RRAM is the control of the statistical variation of switching parameters to gain low variability and high reliability. In this work, starting from the well-known percolation model of dielectric breakdown (BD), we establish a framework of analysis and modeling of the resistive switching statistics in RRAM devices, which are based on the formation and disconnection of a conducting filament (CF). One key aspect of our proposal is the relation between the CF resistance and the switching statistics. Hence, establishing the correlation between SET and RESET switching variables and the initial resistance of the device in the OFF and ON states, respectively, is a fundamental issue. Our modeling approach to the switching statistics is fully analytical and contains two main elements: (i) a geometrical cell-based description of the CF and (ii) a deterministic model for the switching dynamics. Both ingredients might be slightly different for the SET and RESET processes, for the type of switching (bipolar or unipolar), and for the kind of considered resistive structure (oxide-based, conductive bridge, etc.). However, the basic structure of our approach is thought to be useful for all the cases and should provide a framework for the physics-based understanding of the switching mechanisms and the associated statistics, for the trustful estimation of RRAM performance, and for the successful forecast of reliability. As a first application example, we start by considering the case of the RESET statistics of NiO-based RRAM structures. In particular, we statistically analyze the RESET transitions of a statistically significant number of switching cycles of Pt/NiO/W devices. In the RESET transition, the ON-state resistance (RON) is a key parameter to describe the initial state of the CF. Hence, we subdivide the statistical samples (obtained mainly in a single device) in several RON ranges so as to study the switching statistics as a function of RON. In this regard, we have found that the experimental data can be nicely fit to a Weibull model in all the resistance ranges. Moreover, the distributions significantly change with RON. This change might be even more significant than the device-to-device related variations and, hence, mostly determine the overall statistics of switching parameters. In particular, we have found that the Weibull slopes of both VRESET, IRESET, and PRESET cumulative distributions increase linearly with 1/RON, i.e., they increase with the area of the CF. On the other hand, while the scale factor of the VRESET distribution (V63%) is roughly independent of RON, the scale factor of the distribution of IRESET and PRESET (I63% and P63%) linearly increases with 1/RON. Upon a direct analogy with the cell-based analytical percolation model of oxide BD, two simple geometrical cell-based models, the single path model with variable width and the multiple parallel path model, are proposed to address the RESET statistics. In the limit where these two geometrical models coincide, we incorporate a deterministic model for the RESET switching dynamics based on the self-accelerated thermal dissolution of the CF. With these two ingredients, the complete physics-based model for the RESET statistics is constructed. This analytical model is shown to nicely account for the experimental results with remarkable agreement.