Papers by Olusegun Otunuga
International Journal of Statistics and Probability, Aug 11, 2017
In this work, an attempt is made to estimate time varying parameters in a linear stochastic diffe... more In this work, an attempt is made to estimate time varying parameters in a linear stochastic differential equation. By defining m k as the local admissible sample/data observation size at time t k , parameters and state at time t k are estimated using past data on interval [t k−m k +1 , t k ]. We show that the parameter estimates at each time t k converge in probability to the true value of the parameters being estimated. A numerical simulation is presented by applying the local lagged adapted generalized method of moments (LLGMM) method to the stochastic differential models governing prices of energy commodities and stock price processes.
Results in physics, Sep 1, 2021
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time-dependent probability density function for general stochastic logistic population model with harvesting effort
Physica D: Nonlinear Phenomena, Jul 1, 2021
Abstract We derive and analyze the time-dependent probability density function for the number of ... more Abstract We derive and analyze the time-dependent probability density function for the number of individuals in a population at a given time in a general logistic population model with harvesting effort using the Fokker–Planck equation. The time-dependent probability density function (obtained as the unique principal solution of the Fokker–Planck equation corresponding to certain initial value and boundary conditions) is used to describe how the distribution of the population process changes with time. We assume the environment is randomly varying and the population is subject to a continuous spectrum of disturbances, with fluctuations in the intrinsic growth rate and the harvesting effort. The randomness is expressed as independent white noise processes. The effect of changes in the intrinsic growth rate, harvesting effort, and noise intensities on the distribution is investigated. In addition, conditions for the existence of optimal harvesting policy are obtained using properties of the time-dependent probability density function. The results obtained in this work are validated using population and published parameters.
Employing basic economic principles, we systematically develop both deterministic and stochastic ... more Employing basic economic principles, we systematically develop both deterministic and stochastic dynamic models for the log-spot price process of energy commodity. Furthermore, treating a diffusion coefficient parameter in the non-seasonal log-spot price dynamic system as a stochastic volatility functional of log-spot price, an interconnected system of stochastic model for log-spot price, expected log-spot price and hereditary volatility process is developed. By outlining the risk-neutral dynamics and pricing, sufficient conditions are given to guarantee that the risk-neutral dynamic model is equivalent to the developed model. Furthermore, it is shown that the expectation of the square of volatility under the risk-neutral measure is a deterministic continuous-time delay differential equation. The presented oscillatory and non-oscillatory results exhibit the hereditary effects on the mean-square volatility process. Using a numerical scheme, a time-series model is developed to estimate the system parameters by applying the Least Square optimization and Maximum Likelihood techniques. In fact, the developed time-series model includes the extended GARCH model as a special case.
Supply and demand in the World oil market are balanced through responses to price movement with c... more Supply and demand in the World oil market are balanced through responses to price movement with considerable complexity in the evolution of underlying supply-demand expectation process. In order to be able to understand the price balancing process, it is important to know the economic forces and the behavior of energy commodity spot price processes. The relationship between the different energy sources and its utility together with uncertainty also play a role in many important energy issues. The qualitative and quantitative behavior of energy commodities in which the trend in price of one commodity coincides with the trend in price of other commodities, have always raised the questions regarding their interactions. Moreover, if there is any interaction, then one would like to know the extent of influence on each other. In this work, we undertake the study to shed a light on the above highlighted processes and issues. The presented study systematically deals with the development of stochastic dynamic models and mathematical, statistical and computational analysis of energy commodity spot price and interaction processes. Below we list the main components of the research carried out in this dissertation. (1) Employing basic economic principles, interconnected deterministic and stochastic models of linear log-spot and expected log-spot price processes coupled with non-linear volatility process are initiated. (2) Closed form solutions of the models are analyzed. (3) Introducing a change of probability measure, a risk-neutral interconnected stochastic model is derived. (4) Furthermore, under the risk-neutral measure, expectation of the square of volatility is reduced to a continuoustime deterministic delay differential equation. ( ) The by-product of this exhibits the hereditary effects on the mean-square volatility process. (6) Using a numerical scheme, a time-series model is developed and utilized to estimate the state and parameters of the dynamic model. In fact, the developed time-series model includes the extended GARCH model as special case. ( ) Using the Henry Hub natural gas data set, the usefulness of the linear interconnected stochastic models is outlined. (8) Using natural and basic economic ideas, interconnected deterministic and stochastic models in (1) are extended to non-linear log-spot price, expected log-spot price and volatility processes. ( ) The presented extended models are validated. (10) Closed form solution and risk-neutral models of (8) are outlined. (11) To exhibit the usefulness of the non-linear interconnected stochastic model, to increase the efficiency and to reduce the magnitude of error, it was essential to develop a modified version of extended Kalman filtering approach. The modified approach exhibits the reduction of magnitude of error. Furthermore, Henry Hub natural gas data set is used to show the advantages of the non-linear interconnected stochastic model. ( ) Parameter and state estimation problems of continuous time non-linear stochastic dynamic process is motivated to initiate an alternative innovative approach. This led to introduce the concept of statistic processes, namely, local sample mean and sample variance. ( ) Then it led to the development of an interconnected discrete-time dynamic system of local statistic processes and ( ) its mathematical model. ( ) This paved the way for developing an innovative approach referred as Local Lagged adapted Generalized Method of Moments (LLGMM). This approach exhibits the balance between model specification and model prescription of continuous time dynamic processes. ( ) In addition, it motivated to initiate conceptual computational state and parameter estimation and simulation schemes that generates a mean square suboptimal procedure. (17) The usefulness of this approach is illustrated by applying this technique to four energy commodity data sets, the U. S. Treasury Bill Yield Interest Rate and the U.S. Eurocurrency Exchange Rate data sets for state and parameter estimation problems. (18) Moreover, the forecasting and confidence-interval problems are also investigated. ( ) The non-linear interconnected stochastic model ( ) was further extended to multivariate interconnected energy commodities and sources with and without external random intervention processes. (20) Moreover, it was essential to extend the interconnected discrete-time dynamic system of local sample mean and variance processes to multivariate discrete-time dynamic system. (21) Extending the LLGMM approach in (15) to a multivariate interconnected stochastic dynamic model under intervention process, the parameters in the multivariate model are estimated. These estimated parameters help in analyzing the short and long term relationship between the energy commodities. These developed results are applied to the Henry Hub natural gas, crude oil and coal data sets.
International Journal of Stochastic Analysis, Jan 23, 2017
We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuat... more We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic (0) and stochastic (R 0) basic reproductive number. Contrary to some author's remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., 0 < 1), epidemic can still grow initially (if R 0 > 1) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise.
Acta Biotheoretica
In this work, we study and analyze the aggregate death counts of COVID-19 reported by the United ... more In this work, we study and analyze the aggregate death counts of COVID-19 reported by the United States Centers for Disease Control and Prevention (CDC) for the fifty states in the United States. To do this, we derive a stochastic model describing the cumulative number of deaths reported daily by CDC from the first time Covid-19 death is recorded to June 20, 2021 in the United States, and provide a forecast for the death cases. The stochastic model derived in this work performs better than existing deterministic logistic models because it is able to capture irregularities in the sample path of the aggregate death counts. The probability distribution of the aggregate death counts is derived, analyzed, and used to estimate the count's per capita initial growth rate, carrying capacity, and the expected value for each given day as at the time this research is conducted. Using this distribution, we estimate the expected first passage time when the aggregate death count is slowing down. Our result shows that the expected aggregate death count is slowing down in all states as at the time this analysis is conducted (June 2021). A formula for predicting the end of Covid-19 deaths is derived. The daily expected death count for each states is plotted as a function of time. The probability density function for the current day, together with the forecast and its confidence interval for the next four days, and the root mean square error for our simulation results are estimated.
Vaccine breakthrough and rebound infections modeling: Analysis for the United States and the ten U.S. HHS regions
Infectious Disease Modelling
International Journal of Dynamics and Control
In this work, we examine the impact of certain preventive measures for effective measles control.... more In this work, we examine the impact of certain preventive measures for effective measles control. To do this, a mathematical model for the dynamics of measles transmission is developed and analyzed. A suitable Lyapunov function is used to establish the global stability of the equilibrium points. Our analysis shows that the disease-free equilibrium is globally stable, with the measles dying out on the long run because the reproduction number R 0 ≤ 1. The condition for the global stability of the endemic equilibrium is also derived and analyzed. Our findings show that when R 0 > 1, the endemic equilibrium is globally stable in the required feasible region. In this situation, measles will spread across the populace. A numerical simulation was performed to demonstrate and support the theoretical findings. The results suggest that lowering the effective contact with an infected person and increasing the rate of vaccinating susceptible people with high-efficacy vaccines will lower the prevalence of measles in the population.
Nowcasting of the Short-run Euro-Dollar Exchange Rate with Economic Fundamentals and Time-varying Parameters
Finance Research Letters
PLOS ONE
In this work, an innovative multi-strain SV EAIR epidemic model is developed for the study of the... more In this work, an innovative multi-strain SV EAIR epidemic model is developed for the study of the spread of a multi-strain infectious disease in a population infected by mutations of the disease. The population is assumed to be completely susceptible to n different variants of the disease, and those who are vaccinated and recovered from a specific strain k (k ≤ n) are immune to previous and present strains j = 1, 2, ⋯, k, but can still be infected by newer emerging strains j = k + 1, k + 2, ⋯, n. The model is designed to simulate the emergence and dissemination of viral strains. All the equilibrium points of the system are calculated and the conditions for existence and global stability of these points are investigated and used to answer the question as to whether it is possible for the population to have an endemic with more than one strain. An interesting result that shows that a strain with a reproduction number greater than one can still die out on the long run if a newer emergi...
We present a mathematical analysis of the transmission of certain diseases using a stochastic sus... more We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external fluctuations in the transmission, treatment and recovery rates. We assume external fluctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By defining RT,n and ℛT,n as the basic deterministic and stochastic reproduction numbers, respectively, in stage n of infection and treatment, we show mathematically that as the intensity of the noise in the transmission, treatment and recovery rates increases, the number of secondary cases of infection increases. The global stability of the disease-free and endemic equilibrium for the determini...
Employing basic economic principles, we systematically develop both deterministic and stochastic ... more Employing basic economic principles, we systematically develop both deterministic and stochastic dynamic models for the log-spot price process of energy commodity. Furthermore, treating a diffusion coefficient parameter in the non-seasonal log-spot price dynamic system as a stochastic volatility functional of log-spot price, an interconnected system of stochastic model for log-spot price, expected log-spot price and hereditary volatility process is developed. By outlining the risk-neutral dynamics and pricing, sufficient conditions are given to guarantee that the risk-neutral dynamic model is equivalent to the developed model. Furthermore, it is shown that the expectation of the square of volatility under the risk-neutral measure is a deterministic continuous-time delay differential equation. The presented oscillatory and non-oscillatory results exhibit the hereditary effects on the mean-square volatility process. Using a numerical scheme, a time-series model is developed to estimat...
Time-dependent probability distribution for number of infection in a stochastic SIS model: case study COVID-19
Chaos, Solitons & Fractals, 2021
We derive the time-dependent probability distribution of the number of infected individuals at a ... more We derive the time-dependent probability distribution of the number of infected individuals at a given time in a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model. The mean, variance, skewness and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. Our analysis reveals that the time-dependent probability density function exists if the basic reproduction number is greater than one. It converges to the Dirac delta function on the long run (entirely concentrated on zero) as the basic reproduction number tends to one from above. The result is applied using published COVID-19 parameters and also applied to analyze the probability distribution of the aggregate number of COVID-19 cases in the United States for the period: January 22, 2020-March 23, 2021. Findings show that the distribution shifts concentration to the right until it concentrates entirely on the carrying infection capacity as the infection growth rate increases or the recovery rate reduces. The disease eradication and disease persistence thresholds are calculated.
Results in Physics, 2021
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal of Energy, 2020
In this work, we examine the relationship between different energy commodity spot prices. To do t... more In this work, we examine the relationship between different energy commodity spot prices. To do this, multivariate stochastic models with and without external random interventions describing the price of energy commodities are developed. Random intervention process is described by a continuous jump process. The developed mathematical model is utilized to examine the relationship between energy commodity prices. The time-varying parameters in the stochastic model are estimated using the recently developed parameter identification technique called local lagged adapted generalized method of moment (LLGMM). The LLGMM method provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. The usefulness of the LLGMM approach is illustrated by applying to energy commodity data sets for state and parameter estimation problems. Moreover, the forecasting and confidence interval problems are also investigated (U.S. Patent Pend...
Infectious Disease Modelling, 2019
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Heliyon, 2019
We study the effects of external fluctuations in the transmission rate of certain diseases and ho... more We study the effects of external fluctuations in the transmission rate of certain diseases and how these affect the distribution of the number of infected individuals over time. To do this, we introduce random noise in the transmission rate in a deterministic SIS model and study how the number of infections changes over time. The objective of this work is to derive and analyze the closed form probability distribution of the number of infections at a given time in the resulting stochastic SIS epidemic model. Using the Fokker-Planck equation, we reduce the differential equation governing the number of infections to a generalized Laguerre differential equation. The properties of the distribution, together with the effect of noise intensity, are analyzed. The distribution is demonstrated using parameter values relevant to the transmission dynamics of influenza in the United States.
Mathematical Biosciences, 2018
In this work, we derive and analyze a 2n+1-dimensional deterministic differential equation modeli... more In this work, we derive and analyze a 2n+1-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R 0,n , R t,n and R t,n as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R 0,n , R t,n and R t,n . We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if R t,n = 1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if R t,n < 1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing R t,n > 1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation.
Journal of Mathematical Biology, 2019
A recent parameter identification technique, the local lagged adapted generalized method of momen... more A recent parameter identification technique, the local lagged adapted generalized method of moments, is used to identify the time-dependent disease transmission rate and time-dependent noise for the stochastic susceptible, exposed, infectious, temporarily immune, susceptible disease model (S E I RS) with vital rates. The stochasticity appears in the model due to fluctuations in the time-dependent transmission rate of the disease. All other parameter values are assumed to be fixed, known constants. The method is demonstrated with US influenza data from the 2004-2005 through 2016-2017 influenza seasons. The transmission rate and noise intensity stochastically work together to generate the yearly peaks in infections. The local lagged adapted generalized method of moments is tested for forecasting ability. Forecasts are made for the 2016-2017 influenza season and for infection data in year 2017. The forecast method qualitatively matches a single influenza season. Confidence intervals are given for possible future infectious levels. Keywords Compartment disease model • Stochastic disease model • Local lagged adapted generalized method of moments • Time-dependent transmission rate Mathematics Subject Classification 60H10 • 60P10 • 92D30
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Papers by Olusegun Otunuga