Deterministic Nonlinear Model Research Articles

A mathematical study of dynamical model for Japanese encephalitis-dengue co-infection using JE vaccine

A nonlinear deterministic mathematical model has been described with the co-infection dynamics of Japanese encephalitis (JE) and dengue disease, incorporating the JE vaccine. Basic mathematical results have been discussed along with their stabilities. By using center manifold theory, the model undergoes a backwards bifurcation phenomenon and this has occurred when the basic reproduction number is smaller than unity. Sensitivity analysis has been performed to determine which parameters significantly affect disease dynamics. The effects of these parameters on the transmission of disease were investigated using a numerical simulation. According to the findings, we ascertain that JE-dengue co-infection can be managed with the use of JE vaccine, also minimising JE transmission rate rapidly.

Effect of Intervention of Vaccination and Treatment on the Transmission Dynamics of HBV Disease: A Mathematical Model Analysis

Hepatitis B is a globally infectious disease. It is pretty contagious and can be transmitted by blood or bodily fluids, through things like sharing razors and toothbrushes. It has been called the silent killer because it is asymptomatic, one might have the virus but not know until it manifests itself until much later. Since people do not give attention, it will develop into cirrhosis and hepatocellular carcinoma that leads to liver transplantation and death. This nature of HBV disease motivated us to perform this work. Mathematical modeling of HBV transmission is an interesting research area. In this paper, we present characteristics of HBV virus transmission in the form of a mathematical model. We proposed and analyzed a compartmental nonlinear deterministic mathematical model SEACTR for transmission dynamics and control of hepatitis B virus disease. In this model, we used force infection which takes the contact rate of susceptible population and transmission probability into account. We proved that the solution of the considered dynamical system is positive and bounded. The model is studied qualitatively using the stability theory of differential equations and the effective reproductive number which represents the epidemic indicator is obtained from the largest eigenvalue of the next‐generation matrix. Both local and global asymptotic stability conditions for disease‐free and endemic equilibria are determined. The sensitivity index shows that the transfer rate from exposed class to acute infective class and transfer rate from exposed class to chronic infective class are the most dominant parameters contributing to the transmission of HBV. On the one hand, the vaccination rate and treatment rate are the parameters that suppress the transmission of the disease the most, and enhancing the vaccination rate for newborns and treatment for chronically infected individuals is very effective to stop the transmission of HBV. The combined efforts of vaccination, effective treatment, and interruption of transmission make elimination of the infection plausible and may eventually lead to the eradication of the virus.

Solving a single period inventory model with fuzzy inequality

The purpose of this paper is to present a fuzzy chance-constrained single period inventory model (FCCSPIM) in which the fuzziness appears in the space constraint and objective function is crisp. Here the partial order relation exists in between a random variable and a real number. That means the probability of the event is discussed under vague data. Our approach for the solution process uses mostly fuzzy Zimmermann technique to convert the FCCSPIM into a proper deterministic equivalent. Then the resulting nonlinear deterministic model is solved by using LINGO software. The result indicate that the fuzzy programming approach is effective for the inventory problem. The applications of an optimisation model under uncertainty are used to solve day to day problems. Many methods were developed by using tools of mathematics, probability theory and stochastic process. Here, one new approach of fuzzy programming technique is introduced to obtain a deterministic form.

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.

Modeling the Impact of Screening on the Transmission Dynamics of Human Papillomavirus with Optimal Control

In this study, a nonlinear deterministic mathematical model of Human Papillomavirus was formulated. The model is studied qualitatively using the stability theory of differential equations. The model is analyzed qualitatively for validating the existence and stability of disease ¬free and endemic equilibrium points using a basic reproduction number that governs the disease transmission. It's observed that the model exhibits a backward bifurcation and the sensitivity analysis is performed. The optimal control problem is designed by applying Pontryagin maximum principle with three control strategies viz. prevention strategy, treatment strategy, and screening strategy. Numerical results of the optimal control model reveal that a combination of prevention, screening, and treatment is the most effective strategy to wipe out the disease in the community.

Modelling bursts and chaos regularization in credit risk with a deterministic nonlinear model

In this paper, we suggest a deterministic approach for modelling credit risk time series even in distressed periods (including COVID-19). We examine the Moody’s Seasoned Aaa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant Maturity as well as the ICE BofA US High Yield Index Option-Adjusted Spread and we find that the proposed model could fit well the alternation between periods of low and high volatility. This result is compared to the ARIMA-EGARCH model to determine how a chaotic deterministic model stands with respect to a stochastic model expressly designed for handling moving average, autoregression, cointegration and heteroscedastic volatility. According to recent literature, we find that both models give comparable results.

Estimation uncertainty for some common seismic fragility curve fitting methods

This technical note illustrates and makes available some simple procedures to assess the estimation uncertainty for the parameters of seismic fragility curves. The considered fragility fitting methods refer to the lognormal assumption and are supposed to be based on the results of multi-stripe dynamic analysis of a deterministic non-linear structural model, so that the uncertainty in the fragility parameters arises from the so-called record-to-record variability. The discussed procedures are based on the statistics approach of resampling with substitution, which is commonly referred to as bootstrap. It is also briefly discussed how the estimation uncertainty depends on the maximum value of the probability of failure given seismic intensity that is observed from structural analysis. This work may aid earthquake engineering practice because, both the curve fitting and estimation uncertainty algorithms are implemented in a major update of an application-ready software tool made available at https://www.reluis.it/it/progettazione/software/r2r-eu.html.

Modeling and Sensitivity Analysis of Dynamical Transmission of Lassa Fever

Lassa fever is an acute viral disease which occurs across West Africa regions. In view of this a non-linear deterministic model was considered to study dynamics spread of Lassa fever. Existence and uniqueness of the model were determined and the basic reproduction number of the model was computed using next-generation matrix, and sensitivity analysis of the model was performed and shows that the most sensitive parameters to the dynamical spread of Lassa fever are human birth rate bH, followed by average number of female partners acquired by susceptible male c2, and transmission rate resulting from sexual interaction between infected male and susceptible female h2 have the highest values. It concludes that average number of female partners acquired by susceptible male and sexual transmission rate between the infected male and susceptible female should be reduced. Keywords: Existence and Uniqueness, disease free equilibrium, basic reproduction numbers and sensitivity analysis.

IMPACT OF TEMPERATURE VARIABILITY ON SIRS MALARIA MODEL

In this paper, we proposed and analyzed a nonlinear deterministic model for the impact of temperature variability on the epidemics of the malaria. The model analysis showed that all solutions of the systems are positive and bounded with initial conditions in a certain set. Thus, the model is proved to be both epidemiologically meaningful and mathematically well-posed. Using the next-generation matrix approach, the basic reproduction number with respect to the disease-free equilibrium (DFE) point is obtained. The local stability of the equilibria points is shown using the Routh–Hurwitz criterion. The global stability of the equilibria points is performed using the Lyapunov function. Also, we proved that if the basic reproduction number is less than one, the DFE is locally and globally asymptotically stable. But, if the basic reproduction number is greater than one, the unique endemic equilibrium exists, locally and globally asymptotically stable. The sensitivity analysis of the parameters is also described. Moreover, we used the method implemented by the center manifold theorem to identify that the model exhibits forward and backward bifurcations. From our analytical results, we confirmed that the variation of temperature plays a significant role on the transmission of malaria. Lastly, numerical simulations are demonstrated to enhance the analytical results of the model.

A unifying nonlinear probabilistic epidemic model in space and time

Covid-19 epidemic dramatically relaunched the importance of mathematical modelling in supporting governments decisions to slow down the disease propagation. On the other hand, it remains a challenging task for mathematical modelling. The interplay between different models could be a key element in the modelling strategies. Here we propose a continuous space-time non-linear probabilistic model from which we can derive many of the existing models both deterministic and stochastic as for example SI, SIR, SIR stochastic, continuous-time stochastic models, discrete stochastic models, Fisher–Kolmogorov model. A partial analogy with the statistical interpretation of quantum mechanics provides an interpretation of the model. Epidemic forecasting is one of its possible applications; in principle, the model can be used in order to locate those regions of space where the infection probability is going to increase. The connection between non-linear probabilistic and non-linear deterministic models is analyzed. In particular, it is shown that the Fisher–Kolmogorov equation is connected to linear probabilistic models. On the other hand, a generalized version of the Fisher–Kolmogorov equation is derived from the non-linear probabilistic model and is shown to be characterized by a non-homogeneous time-dependent diffusion coefficient (anomalous diffusion) which encodes information about the non-linearity of the probabilistic model.

DETERMINISTIC MODEL OF THE DYNAMICS STRAIGHTENING OF FIBRES DURING CARDING ON A ROLLER MACHINE

The article proposes a simple nonlinear deterministic model of the dynamics of fibre straightening in the working areas “drum – drum”, “drum – roller”, “roller – roller”. The model allows you to predict the value of the average straightness of the fibers in the mass that repeatedly passes through the working area of the machine. The model includes only one parameter that determines the efficiency of fibre straightening during carding. The dependence of the average straightness of the fibres on the number of repeated passes of the fibre through the working area at different values of the carding efficiency is obtained.

Transmission dynamics model of Tuberculosis with optimal control strategies in Haramaya district, Ethiopia

In this study, we use a compartmental nonlinear deterministic mathematical model to investigate the effect of different optimal control strategies in controlling Tuberculosis (TB) disease transmission in the community. We employ stability theory of differential equations to investigate the qualitative behavior of the model by obtaining the basic reproduction number and determining the local stability conditions for the disease-free and endemic equilibria. We consider three control strategies representing distancing, case finding, and treatment efforts and numerically compare the levels of exposed and infectious populations with and without control strategies. The results suggest that combination of all controls is the best strategy to eradicate TB disease from the community at an optimal level with minimum cost of interventions.

Modeling and optimal control analysis of malaria epidemic in the presence of temperature variability

In this paper, we propose and analyze a nonlinear deterministic malaria disease model for the impact of temperature variability on malaria epidemics. Firstly, we analyzed the invariant region and the positivity solution of the model. The basic reproduction number with respect to disease free-equilibrium is calculated by the next-generation matrix method. The local stability and global stability of the equilibrium points are shown using the Routh–Hurwitz criterion and the Lyapunov function, respectively. A disease-free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one and endemic equilibrium exists otherwise. Moreover, we have shown the sensitivity analysis of the basic reproduction number and the model exhibits forward and backward bifurcation. Secondly, we apply the optimal control theory to describe the model with incorporates three controls, namely using treated bed nets, treatment of infected with anti-malaria drugs and for vector killing using insecticide spray strategy. Pontraygin’s maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the simulation result of optimal control problem and analysis of cost-effectiveness show that a combination of using treated bed nets and treatment is the most effective and least-cost strategy to prevent the malaria disease.

Modeling the Effect of Public Awareness in Local Dialects in the Spread of Corona Virus in Nigeria

To study Corona virus, a non-linear deterministic mathematical model is formulated and analyzed to study a measure which will be efficient and sufficient to control corona virus combining the already existing control measures. In this model, all individuals are susceptible having the tendency of being infected if control measures are not implemented. The disease-free equilibrium point, feasible region of the model’s solutions was carried out and proves the systems uniqueness and existence. Numerical solutions of this model indicate that the combination of already existing control measures of corona virus and public awareness in local dialects) is an efficient method to control the spread of corona virus.

Modelling the dynamics of antimicrobial-resistant typhoid infection with environmental transmission

The emergence of treatment-induced acquired resistance is considered one of the most significant challenges in managing typhoid infection. A deterministic nonlinear mathematical model describing the transmission dynamics of antimicrobial-resistant typhoid infection is developed and analyzed. The basic reproduction number is computed for the model. The condition for the existence of possible equilibria and their stability has been discussed. The sensitivity analysis shows that the direct transmission rates of two strains and the consumption rate of bacteria in environmental transmission strongly impacts the basic reproduction number R0. The parameter p, related to treatment-induced acquired resistance, does not affect R0, but numerical simulations reveal that it enhances the load of resistant strain in a co-existence equilibrium state. Although the resistant strain has lower transmissibility than the sensitive strain, the resistant strain alone can persist in a community due to re-infection. The study suggests that access to safe drinking water combined with improved sanitation and hygiene practices can reduce the emergence and global spread of antimicrobial-resistant S. Typhi strains.

Simulated annealing method for molecular constants calculation: the electronic transition A2Σ+–X 2Π of N2O+ radical 000–000 and 001–001 band cases

The Simulated Annealing (SA) method is here first adopted to retrieve the molecular constants used to fit wavenumbers into a model, from a vibrational band of the electronic transition A of the NO radical, 000–000, starting from a set of molecular constants obtained in a previous work, and varying their values within a range in order to mimic an error in each one parameter used. The SA calculations have been carried out following the proposal of Vanderbilt and Louie for performing a minimisation of an objective function where the parameters are continuous variables whose values can vary. Through SA, the 280 experimentally wavenumbers from the previous work were reduced using a Hamiltonian model and new molecular constants obtained. A comparison among the results obtained through a nonlinear deterministic model and the retrieved values obtained using the SA shows that the simulated annealing can be considered as a very strong method for molecular constants determination. Second, intending to test the SA method in a direct fit to obtain the molecular constants, the 142 wavenumbers from the 001–001 band A electronic transition of the NO were studied in this work and the molecular constants analysed.

A dynamical model for HIV-typhoid co-infection with typhoid vaccine

Individuals living with HIV/AIDS are significantly at higher risk of infection with Salmonella Typhi. A deterministic nonlinear mathematical model that describes the co-infection dynamics of HIV and typhoid incorporating typhoid vaccine and treatment has been developed. The basic reproduction number for the co-infection model is computed. The co-infection model exhibits four steady states, namely, the disease-free, HIV alone endemic, typhoid alone endemic, and the co-infection endemic states. The local stability of the disease-free state has been investigated. The co-infection endemic state, if it exists, is found to be locally stable. The minimum vaccination rate that eliminates the typhoid infection is determined. Sensitivity analysis has been performed to ascertain model parameters that have a strong impact on the disease dynamics. It is demonstrated that the co-infection basic reproduction number can be reduced to below unity by simultaneous preventive measures, thereby eliminating both diseases. Numerical simulation of the co-infection model is carried out to examine the effects of parameters on disease spread. The study suggests that the two diseases need to be managed simultaneously for effective control of the co-infection.

Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: A case study.

Life style of people almost in every country has been changed with arrival of corona virus. Under the drastic influence of the virus, mathematicians, statisticians, epidemiologists, microbiologists, environmentalists, health providers, and government officials started searching for strategies including mathematical modeling, lock-down, face masks, isolation, quarantine, and social distancing. With quarantine and isolation being the most effective tools, we have formulated a new nonlinear deterministic model based upon ordinary differential equations containing six compartments (susceptible exposed quarantined infected isolated and recovered ). The model is found to have positively invariant region whereas equilibrium points of the model are investigated for their local stability with respect to the basic reproductive number . The computed value of proves endemic level of the epidemic. Using nonlinear least-squares method and real prevalence of COVID-19 cases in Pakistan, best parameters are obtained and their sensitivity is analyzed. Various simulations are presented to appreciate quarantined and isolated strategies if applied sensibly.

Investigate future population projection of Bangladesh with the help of Malthusian model, Sharpe-lotka model and Gurtin Mac-Camy model

A nonlinear deterministic model of age-dependent population growth is formulated in terms of joint integral equations in the total population size function and the total birth rate function. Existence of a unique solution of the model equation is established using Banach fixed point theorem. Then we investigate the total population size of Malthusian model, Sharpe-Lotka linear model and Gurtin Mac-Camy model for some specific age group and compare our investigated solutions with the total population of Bangladesh by using numerical method “Newton divided difference formula”. In our present study, we also investigate future population projection of Bangladesh with the help of Malthusian model, Sharpe-Lotka model and Gurtin Mac-Camy model.

A non-linear deterministic model of action selection in the basal ganglia to simulate motor fluctuations in Parkinson's disease.

Motor fluctuations and dyskinesias are severe complications of Parkinson's disease (PD), especially evident at its advanced stage, under long-term levodopa therapy. Despite their strong clinical prevalence, the neural origin of these motor symptoms is still a subject of intense debate. In this work, a non-linear deterministic neurocomputational model of the basal ganglia (BG), inspired by biology, is used to provide more insights into possible neural mechanisms at the basis of motor complications in PD. In particular, the model is used to simulate the finger tapping task. The model describes the main neural pathways involved in the BG to select actions [the direct or Go, the indirect or NoGo, and the hyperdirect pathways via the action of the sub-thalamic nucleus (STN)]. A sensitivity analysis is performed on some crucial model parameters (the dopamine level, the strength of the STN mechanism, and the strength of competition among different actions in the motor cortex) at different levels of synapses, reflecting major or minor motor training. Depending on model parameters, results show that the model can reproduce a variety of clinically relevant motor patterns, including normokinesia, bradykinesia, several attempts before movement, freezing, repetition, and also irregular fluctuations. Motor symptoms are, especially, evident at low or high dopamine levels, with excessive strength of the STN and with weak competition among alternative actions. Moreover, these symptoms worsen if the synapses are subject to insufficient learning. The model may help improve the comprehension of motor complications in PD and, ultimately, may contribute to the treatment design.