Differential Equation Mathematical Model Research Articles

To modulate or to skip: De-escalating PARP inhibitor maintenance therapy in ovarian cancer using adaptive therapy

Toxicity and emerging drug resistance pose important challenges in poly-adenosine ribose polymerase inhibitor (PARPi) maintenance therapy of ovarian cancer. We propose that adaptive therapy, which dynamically reduces treatment based on the tumor dynamics, might alleviate both issues. Utilizing invitro time-lapse microscopy and stepwise model selection, we calibrate and validate a differential equation mathematical model, which we leverage to test different plausible adaptive treatment schedules. Our model indicates that adjusting the dosage, rather than skipping treatments, is more effective at reducing drug use while maintaining efficacy due to a delay in cell kill and a diminishing dose-response relationship. Invivo pilot experiments confirm this conclusion. Although our focus is toxicity mitigation, reducing drug use may also delay resistance. This study enhances our understanding of PARPi treatment scheduling and illustrates the first steps in developing adaptive therapies for new treatment settings. A record of this paper's transparent peer review process is included in the supplemental information.

Dynamic analysis of delayed vaccination process along with impact of retrial queues

Abstract An unprecedented and precise time-scheduled rollout for the vaccine is needed for an effective vaccination process. This study is based on the development of a novel mathematical model considering a delay in vaccination due to the inability to book a slot in one go for a system. Two models are proposed which involve a delay differential equation mathematical model whose dynamical analysis is done to show how the delay in vaccination can destabilize the system. Further, this delay led to the formulation of a queuing model that accounts for the need to retry for the vaccination at a certain rate as delay in vaccination can have negative repercussions. The transition rates from one stage to another follow an exponential distribution. The transient state probabilities of the model are acquired by applying the Runge-Kutta method and hence performance indices are also obtained. These performance measures include the expected number of people in various states. Finally, numerical analysis is also provided to validate both models. Our results would specifically focus on what happens if the delay time increases or if the retrial rate increases (delay time decreases). The results reveal that a delay in being vaccinated by the first dose (i.e., 80 days) leads to an unstable system whereas there exists a delay simultaneously in getting vaccinated by both doses that destabilize the system early (i.e., 80 and 120 days for dose one and two, respectively). The system destabilizes faster in the presence of a delay for slot booking for both doses as compared to one dose delay. Further, the numerical results of queuing models show that if the retrial rate increases in this delay time to book the slots, it not only increases in the vaccinated class but also increases the recovered population.

Dual solutions of magnetite/cobalt/manganese-zinc-aqueous nano-ferrofluids from a stretching sheet with magnetic induction effects: MHD stagnation flow computation and analysis

A theoretical study is presented for the steady magnetohydrodynamic (MHD) boundary layer stagnation point flow of a nano-ferrofluid along a linearly moving stretching sheet, as a simulation of functional magnetic materials processing. Due to having imerging applications in heat transfer, the nano-ferrofluids draw the attention which comprise an aqueous base fluid doped with a variety of magnetic nanoparticles, i.e. magnetite (Fe3O4), cobalt ferrite (CoFe2O4) and Manganese-zinc (Mn-Zn) ferrite. A partial differential equation mathematical model is developed for mass, momentum, magnetic field continuity (induction), and energy with appropriate wall and free stream boundary conditions. Following similarity transformations, the dimensionless resultant nonlinear ordinary differential boundary value problem is solved numerically using the robust bvp4c function in MATLAB which features very efficient 4th order optimized Runge–Kutta quadrature. Dual solutions for the upper branch and lower branch separated by a critical point are identified. Visualization of velocity, temperature, and induced magnetic field function are presented graphically including validation of solutions with previous studies. Furthermore, skin-friction coefficient and the local Nusselt number are also computed. The impact of the controlling parameters, i.e. Prandtl number nanoparticle volume fraction parameter reciprocal of magnetic Prandtl number magnetic parameter and stretching rate ratio parameter have been illustrated through graphs and evaluated when a desire heat transfer can occur. Furthermore, resistance between fluid and the plate can be increased with the growing magnetic Prandtl number values. Increment in magnetic parameter () produces an elevation in the induced magnetic field magnitudes. Skin friction and Nusselt number are found to be greater for cobalt nanoparticles when compared to magnetite and Mn-Zn ferromagnetic nanoparticles when there is an increase in reciprocal magnetic Prandtl number. The simulations provide a deeper insight into the manufacturing flows of functional nano-ferromagnetic materials of relevance to deposition and coating systems.

Abstract B009: Integrating mathematical modelling and wet-lab experiments to examine the scope for adaptive treatment scheduling of PARP inhibitors in ovarian cancer

Abstract PARP inhibitors (PARPis) represent a great advancement in the treatment of ovarian cancer, yet these drugs often fail after a few months due to emerging drug resistance. A recent clinical trial in prostate cancer showed that evolutionary-inspired, adaptive drug scheduling significantly delayed time to progression. This approach adaptively skipped treatment to maintain a pool of drug-sensitive cells that suppressed resistant cells through competition. Here, we present results from a combined modelling and experimental study in which we investigated whether adaptive therapy could delay resistance to the PARPi olaparib in ovarian cancer.We performed a series of in vitro experiments in which we used Incucyte Zoom time-lapse microscopy to characterize the cell population dynamics under different PARPi schedules. Leveraging these data we developed an ordinary differential equation mathematical model of treatment response, and used this model to test different plausible adaptive treatment schedules. Our model can accurately predict the in vitro treatment dynamics, even to new schedules, and suggests that treatment modifications need to be carefully timed, or one risks losing control over tumor growth, even in the absence of any resistance. This is because multiple rounds of cell division are required for cells to acquire sufficient DNA damage to induce apoptosis. As a result, adaptive therapy algorithms that modulate treatment but never completely withdraw it are predicted to perform better in this setting than strategies based on treatment interruptions. Subsequent experiments confirm this prediction in vivo. To conclude, we present preliminary work aiming at investigating the scope for clinical translation of our results by confronting our model with longitudinal CA-125 data from 53 ovarian cancer patients receiving olaparib at Moffitt. Overall, this study contributes to a better understanding of the impact of scheduling on treatment outcome for PARPis, and showcases some of challenges involved in developing adaptive therapies for new treatment settings. Citation Format: Maximilian Strobl, Mehdi Damaghi, Alexandra Martin, Samantha Byrne, Mark Robertson-Tessi, Robert Gatenby, Robert Wenham, Philip Maini, Alexander R.A. Anderson. Integrating mathematical modelling and wet-lab experiments to examine the scope for adaptive treatment scheduling of PARP inhibitors in ovarian cancer [abstract]. In: Proceedings of the AACR Special Conference on the Evolutionary Dynamics in Carcinogenesis and Response to Therapy; 2022 Mar 14-17. Philadelphia (PA): AACR; Cancer Res 2022;82(10 Suppl):Abstract nr B009.

Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis.

Retinitis pigmentosa (RP) is the most common inherited retinal dystrophy with a prevalence of about 1 in 4,000, affecting approximately 1.5 million people worldwide. Patients with RP experience progressive visual field loss as the retina degenerates, destroying light-sensitive photoreceptor cells (rods and cones), with rods affected earlier and more severely than cones. Spatio-temporal patterns of retinal degeneration in human RP have been well characterised; however, the mechanism(s) giving rise to these patterns have not been conclusively determined. One such mechanism, which has received a wealth of experimental support, is described by the trophic factor hypothesis. This hypothesis suggests that rods produce a trophic factor necessary for cone survival; the loss of rods depletes this factor, leading to cone degeneration. In this article, we formulate a partial differential equation mathematical model of RP in one spatial dimension, spanning the region between the retinal centre (fovea) and the retinal edge (ora serrata). Using this model we derive and solve an inverse problem, revealing for the first time experimentally testable conditions under which the trophic factor mechanism will qualitatively recapitulate the spatio-temporal patterns of retinal regeneration observed in human RP.

Modeling the COVID-19 Epidemic With Multi-Population and Control Strategies in the United States.

As of January 19, 2021, the cumulative number of people infected with coronavirus disease-2019 (COVID-19) in the United States has reached 24,433,486, and the number is still rising. The outbreak of the COVID-19 epidemic has not only affected the development of the global economy but also seriously threatened the lives and health of human beings around the world. According to the transmission characteristics of COVID-19 in the population, this study established a theoretical differential equation mathematical model, estimated model parameters through epidemiological data, obtained accurate mathematical models, and adopted global sensitivity analysis methods to screen sensitive parameters that significantly affect the development of the epidemic. Based on the established precise mathematical model, we calculate the basic reproductive number of the epidemic, evaluate the transmission capacity of the COVID-19 epidemic, and predict the development trend of the epidemic. By analyzing the sensitivity of parameters and finding sensitive parameters, we can provide effective control strategies for epidemic prevention and control. After appropriate modifications, the model can also be used for mathematical modeling of epidemics in other countries or other infectious diseases.

Mathematical assessment of the impact of human-antibodies on sporogony during the within-mosquito dynamics of Plasmodium falciparum parasites

We develop and analyze a deterministic ordinary differential equation mathematical model for the within-mosquito dynamics of the Plasmodium falciparum malaria parasite. Our model takes into account the action and effect of blood resident human-antibodies, ingested by the mosquito during a blood meal from humans, in inhibiting gamete fertilization. The model also captures subsequent developmental processes that lead to the different forms of the parasite within the mosquito. Continuous functions are used to model the switching transition from oocyst to sporozoites as well as human antibody density variations within the mosquito gut are proposed and used. In sum, our model integrates the developmental stages of the parasite within the mosquito such as gametogenesis, fertilization and sporogenesis culminating in the formation of sporozoites. Quantitative and qualitative analyses including a sensitivity analysis for influential parameters are performed. We quantify the average sporozoite load produced at the end of the within-mosquito malaria parasite’s developmental stages. Our analysis shows that an increase in the efficiency of the ingested human antibodies in inhibiting fertilization within the mosquito’s gut results in lowering the density of oocysts and hence sporozoites that are eventually produced by each mosquito vector. So, it is possible to control and limit oocysts development and hence sporozoites development within a mosquito by boosting the efficiency of antibodies as a pathway to the development of transmission-blocking vaccines which could potentially reduce oocysts prevalence among mosquitoes and hence reduce the transmission potential from mosquitoes to human.

A Nonlinear differential equation model of Asthma effect of environmental pollution using LHAM

In this paper, we investigated a nonlinear differential equation mathematical model to study the spread of asthma in the environmental pollutants from industry and mainly from tobacco smoke from smokers in different type of population. Smoking is the main cause to spread Asthma in the environment. Numerical simulation is also discussed. Finally by using Liao’s Homotopy analysis Method (LHAM), we found that the approximate analytical solution of Asthmatic disease in the environmental.

A genetic algorithm optimized Morlet wavelet artificial neural network to study the dynamics of nonlinear Troesch’s system

In this work, a new stochastic computing technique is developed to study the nonlinear dynamics of Troesch’s problem by designing the mathematical models of Morlet Wavelets Artificial Neural Networks (MW-ANNs) optimized with Genetic Algorithm (GA) integrated with Sequential Quadratic Programming (SQP). The differential equation mathematical model for MW-ANNs are designed for Troesch’s system by incorporating a windowing kernel based on Morlet Wavelets as an activation function and these networks are constructed to define a fitness function of Troesch’s system in the mean squared sense. The unknown adjustable parameters of MW-ANNs are trained initially by an effective global search using GAs hybridized with SQP for rapid local refinement of the results. The proposed scheme is evaluated to solve the Troesch’s problems for small and large values of the critical parameter in the system. Comparison of the proposed results with standard reference solutions of Adams method shows good agreement. Validation of accuracy and convergence of the proposed scheme is made using statistical analysis based on a sufficiently large number of independent runs, this is done in terms of performance measures of mean absolute deviation and root mean squared error.

Assessment of Ice Volume Changes in the Cryosphere via Simplified Heat Transport Model

In order to describe changes in ice volume in the cryosphere, a differential equation mathematical model is used in this paper. The dominating effects of freezing and thawing in the cryosphere enable simplification of the heat transport equations. This results in a mathematical model that can be solved exactly and is useful in investigating other climatic components, which in turn may be similarly analyzed for possible Global Circulation Model (GCM) validation. Data forms representing temperature and ice volume during the Pleistocene are available and can be directly compared with the exact solution of the simplified differential equation used in this paper. The model parameters may then be adjusted to approximate the effects of climate change; the adjusted model then run in reverse time, to develop an alternative history of ice volume of the cryosphere to be compared with the actual data interpretations previously published in the literature. In this fashion, an assessment may be made as to possible climate impacts in the cryosphere.

Analysis of a malaria model with mosquito-dependent transmission coefficient for humans

In this paper, we discuss an ordinary differential equation mathematical model for the spread of malaria in human and mosquito population. We suppose the human population to act as a reservoir. Both the species follow a logistic population model. The transmission coefficient or the interaction coefficient of humans is considered to be dependent on the mosquito population. It is seen that as the factors governing the transmission coefficient of humans increase, so does the number of infected humans. Further, it is observed that as the immigration constant increases, it leads to a rise in infected humans, giving an endemic shape to the disease.

A simple mathematical model for genetic effects in pneumococcal carriage and transmission

Streptococcus pneumoniae ( S. pneumoniae) is a bacterium commonly found in the throat of young children. Pneumococcal serotypes can cause a variety of invasive and non-invasive diseases such as meningitis and pneumonia. In 2000 a vaccine was introduced in the USA that not only prevents vaccine type disease but has also been shown to eliminate carriage of the vaccine serotypes. One key problem with the vaccine is that it has been observed that the same sequence types (genetic material found in the serotypes) are able to manifest in more than one serotype. This is a potential problem if sequence types associated with invasive disease may express themselves in multiple serotypes. We present a basic differential equation mathematical model for exploring the relationship between sequence types and serotypes where a sequence type is able to manifest itself in one vaccine serotype and one non-vaccine serotype. An expression for the effective reproduction number is found and an equilibrium and then a global stability analysis carried out. We illustrate our analytical results by using simulations with realistic parameter values.

A delay differential equation mathematical model for the control of the hormonal system of the hypothalamus, the pituitary and the testis in man

In this paper we develop previously studied mathematical models of the regulation of testosterone by luteinizing hormone and luteinizing hormone release hormone in the human body. We propose a delay differential equation mathematical model which improves on earlier simpler models by taking into account observed experimental facts. We show that our model has four possible equilibria, but only one unique equilibrium where all three hormones are present. We perform stability and Hopf bifurcation analyses on the equilibrium where all three hormones are present. With no time delay, this equilibrium is unstable, but as the time delay increases through an infinite sequence of positive values, Hopf bifurcation occurs repeatedly. This is of practical interest as biological evidence shows that the levels of these hormones in the body oscillate periodically. We next discuss stability of the other equilibria heuristically using analytical methods. Then we describe simulations with realistic parameter values and show that our model can mimic the regular fluctuations of the three hormones in the body and explore, numerically, some of our heuristic conjectures. A brief discussion concludes the paper.

A kinetic model of bone marrow neutrophil production that characterizes late phenotypic maturation

Acute inflammatory stimuli rapidly mobilize neutrophils from the bone marrow by shortening postmitotic maturation time and releasing younger neutrophils; however, the kinetics of this change in maturation time remains unknown. We propose a kinetic model that examines the rate of change in neutrophil average age at exit from the bone marrow during active mobilization to quantify this response and use this model to examine the temporal profile of late neutrophil phenotypic maturation. Total and CD10(-)/CD16(low) circulating neutrophils were quantified in cardiac surgery patients during extracorporeal circulation (ECC). Net growth in the circulating neutrophil pool occurred during the procedural (0.04 +/- 0.02 x 10(9) x l(-1) x min(-1)), warming (0.14 +/- 0.02 x 10(9) x l(-1) x min(-1)), and weaning (0.12 +/- 0.06 x 10(9) x l(-1) x min(-1)) phases of ECC. When applied to our differential equation mathematical model, these results predict that neutrophil average age at exit from the bone marrow decreased continually during ECC, resulting in average neutrophil release 8.44 +/- 2.20 h earlier during the weaning phase than at the beginning of ECC sampling. Modeling of concurrent changes in CD10(-)/CD16(low) neutrophil numbers indicates that CD10 expression is directly related to neutrophil mean age and predicts that the proportion of mobilizable postmitotic neutrophils that are CD10(+) increases from 64 to 81% during these sampled 8.4 h of maturation.

Bifurcation Analysis of a Mathematical Model for Malaria Transmission

We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, and recovered classes, before reentering the susceptible class. Susceptible mosquitoes can become infected when they bite infectious or recovered humans, and once infected they move through the exposed and infectious classes. Both species follow a logistic population model, with humans having immigration and disease‐induced death. We define a reproductive number, $R_0$, for the number of secondary cases that one infected individual will cause through the duration of the infectious period. We find that the disease‐free equilibrium is locally asymptotically stable when $R_0 1$. We prove the existence of at least one endemic equilibrium point for all $R_0 > 1$. In the absence of disease‐induced death, we prove that the tran...

Modelling and Guidance of Surface Ships Using an Artificial Intelligence Approach

Abstract The conventional approach to the modelling and control of dynamic systems is to firstly construct a differential equation mathematical model of the system and then to design a controller that optimises a given performance criterion. If the system parameters are unknown, the mathematical model can be obtained from a set of input-output measurements made on the physical system. This paper investigates the possibility of employing a neural network approach to both of these tasks, when applied to the modelling and control of a surface ship. A neural network is trained to represent the ship dynamic behaviour using selected data from a range of manoeuvres. A second network is then trained using optimal guidance data. It is demonstrated that the neural network solution has comparable performance to the original system.