Mathematical Analysis and Optimal Control of Giving up the Smoking Model

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokers P , the occasional smokers L , the chain smokers S , the temporarily quit smokers Q T , and the permanently quit smokers Q P . Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.

Transmission of SARS-CoV-2 considering shared chairs in outpatient dialysis: a real-world case-control study

Background SARS-CoV-2 can remain transiently viable on surfaces. We examined if use of shared chairs in outpatient hemodialysis associates with a risk for indirect patient-to-patient transmission of SARS-CoV-2. Methods We used data from adults treated at 2,600 hemodialysis facilities in United States between February 1st and June 8th, 2020. We performed a retrospective case-control study matching each SARS-CoV-2 positive patient (case) to a non-SARS-CoV-2 patient (control) treated in the same dialysis shift. Cases and controls were matched on age, sex, race, facility, shift date, and treatment count. For each case-control pair, we traced backward 14 days to assess possible prior exposure from a ‘shedding’ SARS-CoV-2 positive patient who sat in the same chair immediately before the case or control. Conditional logistic regression models tested whether chair exposure after a shedding SARS-CoV-2 positive patient conferred a higher risk of SARS-CoV-2 infection to the immediate subsequent patient. Results Among 170,234 hemodialysis patients, 4,782 (2.8 %) tested positive for SARS-CoV-2 (mean age 64 years, 44 % female). Most facilities (68.5 %) had 0 to 1 positive SARS-CoV-2 patient. We matched 2,379 SARS-CoV-2 positive cases to 2,379 non-SARS-CoV-2 controls; 1.30 % (95 %CI 0.90 %, 1.87 %) of cases and 1.39 % (95 %CI 0.97 %, 1.97 %) of controls were exposed to a chair previously sat in by a shedding SARS-CoV-2 patient. Transmission risk among cases was not significantly different from controls (OR = 0.94; 95 %CI 0.57 to 1.54; p = 0.80). Results remained consistent in adjusted and sensitivity analyses. Conclusions The risk of indirect patient-to-patient transmission of SARS-CoV-2 infection from dialysis chairs appears to be low.

Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems

In this paper, we discuss a class of Caputo fractional evolution equations on Banach space with feedback control constraint whose value is non-convex closed in the control space. First, we prove the existence of solutions for the system with feedback control whose values are the extreme points of the convexified constraint that belongs to the original one. Secondly, we study the topological properties of the sets of admissible “state-control” pair for the original system with various feedback control constraints and the relations between them. Moreover, we obtain necessary and sufficient conditions for the solution set of original systems to be closed. In the end, an example is given to illustrate the applications of our main results.

Optimal Control of Chlamydia Model with Vaccination

This paper presents an SVEIRT epidemiological model in the human population with Chlamydia trachomatis. The model incorporated the vaccination class and investigated the role played by some control strategies in the dynamics of the disease (Chlamydia tracomatis). The reproduction number which helps in determining the rate of spread of the disease, was calculated using the method proosed by van den Driessche and Watmough. The local and global stability of the equilibrium points where established, where it was observed that the model is locally asymptotically stable if the reproduction number is less than unity, and globally stable if a certain threshold value is greater than unity or the re-infection rate is zero. The effect of the re-infection rate on the global stability suggests the exhibition of the phenomenon of backward bifurcation of the model. The backward bifurcation of the system was later studied, and it shows that backward bifurcation will occur if the value of the bifurcation parameter a is positive. The optimal control of the model shows the effect of different strategies in the transmission dynamicsof the disease and the cost effectivenes of each control pair. It was observed that the treatment and control effort gives the most cost effective combinations and at the same time the highest rate of disease avertion when compared to other stratagies. Sensitivity analysis of the parameters as shown in model, shows parameters that have high impact on the chosen classes.

A personalised approach for identifying disease-relevant pathways in heterogeneous diseases

Numerous time-course gene expression datasets have been curated for studying the biological dynamics that drive disease progression; and nearly as many methods have been proposed to analyse them. However, barely any method exists that can appropriately model time-course data and at the same time account for heterogeneity that entails many complex diseases. Most methods manage to fulfil either one of those qualities, but not both. The lack of appropriate methods hinders our capability of understanding the disease process and pursuing preventive or curative treatments. Here, we present a method that models time-course data in a personalised manner, i.e. for each case-control pair individually, using Gaussian processes in order to identify differentially expressed genes (DEGs); and combines the lists of DEGs on a pathway-level using a permutation-based empirical hypothesis testing in order to overcome gene-level variability and inconsistencies prevalent to heterogeneous datasets from complex diseases. Our method can be applied to study the time-course dynamics as well as specific time-windows of heterogeneous diseases. We apply our personalised approach on two longitudinal type 1 diabetes (T1D) datasets to determine perturbations that take place during early prognosis of the disease as well as in time-windows before seroconversion and clinical onset of T1D. By comparing to non-personalised methods, we demonstrate that our approach is biologically motivated and can reveal more insights into progression of heterogeneous diseases. With its robust capabilities of identifying immunologically interesting and disease-relevant pathways, our approach could be useful for predicting certain events in the progression of heterogeneous diseases and even biomarker identification.AvailabilityThe implemented code of our personalised approach will be available online upon publication.

Mathematical Analysis of Transfusion—Transmitted Malaria Model with Optimal Control

An SIRS (Susceptible–Infected–Removed-Susceptible) mathematical model for the transmission dynamics of the Transfusion–Transmitted Malaria (TTM) model with optimal control pair u1(t) and u2(t) was developed and studied in this research work. The model Transfusion–Transmitted Malaria disease–free equilibrium and endemic equilibriums points were determined. The model exhibited two equilibriums; disease-free and endemic equilibrium. It is shown that the disease–free equilibrium was locally asymptotically stable if the associated basic reproduction numbers R0 is less than unity while the disease persists if R0 is greater than unity. The global stability of the Transfusion–Transmitted Malaria model at the disease-free equilibrium was established using the comparison method. The optimality system was derived and an optimal control model of blood screening and drug treatment for the Transfusion–Transmitted Malaria model was investigated. Conditions for the optimal control were considered using Pontryagin’s Maximum Principle and solved numerically using the Forward and Backward Finite Difference Method (FBDM). Numerical results obtained are in perfect agreement with our analytical results.

An underactuated drift-free left-invariant control system on the Lie group ISO(3,1)

The purpose of our paper is to study a class of left-invariant, drift-free optimal control problem on the Lie group ISO(3,1). The left-invariant, drift-free optimal control problems involves finding a trajectory-control pair on ISO(3,1), which minimize a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on G*, where the maximum principle yields the optimal control. We use energy methods (Arnold’s method, in this case) to give sufficient conditions fornonlinear stability of the equilibrium states. Around this equilibrium states we might be able to find the periodical orbits using Moser's theorem, as future work. For the some unstable equilibrium states, a quadratic control is considered in order to stabilize the dynamics.

Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids

We model the transmission of the hepatitis B virus (HBV) by six differential equations that represent the reactions between HBV with DNA-containing capsids, the hepatocytes, the antibodies and the cytotoxic T-lymphocyte (CTL) cells. The intracellular delay and treatment are integrated into the model. The existence of the optimal control pair is supported and the characterization of this pair is given by the Pontryagin’s minimum principle. Note that one of them describes the effectiveness of medical treatment in restraining viral production, while the second stands for the success of drug treatment in blocking new infections. Using the finite difference approximation, the optimality system is derived and solved numerically. Finally, the numerical simulations are illustrated in order to determine the role of optimal treatment in preventing viral replication.

Mathematical Analysis of Transfusion – Transmitted Malaria Model with Optimal Control

An SIR (Susceptible – Infected – Removed) mathematical model for the transmission dynamics of the Transfusion –Transmitted Malaria (TTM) model with optimal control pair and was developed and studied in this research work. The model Transfusion –Transmitted Malaria disease – free equilibrium and endemic equilibriums points were determined. The model exhibited two equilibriums; disease-free and endemic equilibrium. It was shown that the disease – free equilibrium was locally asymptotically stable if the associated basic reproduction numbers  is less than unity while the disease persists if  is greater than unity. The global stability of the Transfusion –Transmitted Malaria model at the disease – free equilibrium was established using the comparison method. The optimality system was derived and an optimal control model of blood screening and drug treatment for the Transfusion –Transmitted Malaria model was investigated. Conditions for the optimal control were considered using Pontryagin’s Maximum Principle and solved numerically using the Forward and Backward Finite Difference Method (FBDM). Numerical results obtained are in perfect agreement with our analytical results.

Is the matched extreme case–control design more powerful than the nested case–control design?

For time-to-event data, the study sample is commonly selected using the nested case–control design in which controls are selected at the event time of each case. An alternative sampling strategy is to sample all controls at the same (pre-specified) time, which can either be at the last event time or further out in time. Such controls are the long-term survivors and may therefore constitute a more ‘extreme’ comparison group and be more informative than controls from the nested case–control design. We investigate this potential information gain by comparing the power of various ‘extreme’ case–control designs with that of the nested case–control design using simulation studies. We derive an expression for the theoretical average information in a nested and extreme case–control pair for the situation of a single binary exposure. Comparisons reveal that the efficiency of the extreme case–control design increases when the controls are sampled further out in time. In an application to a study of dementia, we identified Apolipoprotein E as a risk factor using a 1:1 extreme case–control design, which provided a hazard ratio estimate with a smaller standard error than that of a 2:1 nested case–control design.