Enhancement of Biodiesel Production via Ultrasound Technology: A Mathematical Study.

Biodiesel is one of the alternative renewable energy sources that has received a lot of attention since it is clean, green energy. Different sources can be used for the production of biodiesel, but the most appropriate and economical method relies on the transesterification of methanol with the nonedible vegetable oil from the fruit of the Jatropha curcas plant. Molar ratio, vessel diameter, catalyst concentration, and ultrasound all have a substantial influence on the synthesis of biodiesel by the transesterification process. Among these factors, the diameter of the vessel and the ultrasonic effect through mass transfer limitations have a significant impact on successful reaction completion. In this research work, we have developed a mathematical model to analyze the three-step transesterification process and side saponification reaction in the presence of a potassium hydroxide catalyst. The model considers the influence of mixing intensity variations, including ultrasound, on the mass transfer in different phases. The mass transfer rate is calculated using the modified Dittus-Boelter correlation. An optimal control approach through the minimum principle by Pontryagin is applied to maximize the production of biodiesel at minimal cost. The novelty of this research, which we have derived from our analytical as well as numerical results, considering industrial processes, is that more than 97% biodiesel yield conversion is to be obtained at 50 kHz ultrasound frequency, a 6:1 methanol-to-Jatropha-oil molar ratio, and 1 m of vessel diameter within 50 min using optimal control theory.

Critical observation of WHO recommended multidrug therapy on the disease leprosy through mathematical study.

Leprosy is a skin disease and it is characterized by a disorder of the peripheral nervous system which occurs due to the infection of Schwann cells. In this research article, we have formulated a four-dimensional ODE-based mathematical model which consists of the densities of healthy Schwann cells, infected Schwann cells, M. leprae bacteria, and the concentration of multidrug therapy (MDT). This work primarily aims on exploring the dynamical changes and interrelations of the system cell populations during the disease progression. Also, evaluating a critical value of the drug efficacy rate of MDT remains our key focus in this article so that a safe drug dose regimen for leprosy can be framed more effectively and realistically. We have examined the stability scenario of different equilibria and the occurrence of Hopf-bifurcation for the densities of our system cell populations with respect to the drug efficacy rate of MDT to gain insight on the precise impact of the efficiency rate on both the infected Schwann cell and the bacterial populations. Also, a necessary transversality condition for the occurrence of the bifurcation has been established. Our analytical and numerical investigations in this research work precisely explores that the process of demyelination, nerve regeneration, and infection of the healthy Schwann cells are the three most crucial factors in the leprosy pathogenesis and to control the M. leprae-induced infection of Schwann cells successfully, a more flexible version of MDT regime with efficacy rate varying in the range η∈(0.025,0.059) for 100-120 days in PB cases and 300 days in MB cases obtained in this research article should be applied. All of our analytical outcomes have been verified through numerical simulations and compared with some existing clinical findings.

A vivid cytokines interaction model on psoriasis with the effect of impulse biologic (TNF-α inhibitor) therapy.

Psoriasis is a chronic skin condition that produces plaques of condensed, scaling skin due to excessively rapid proliferation of keratinocytes. During the disease progression, keratinocyte proliferation is influenced by many immune cells and cytokines. This article deals with a five dimensional deterministic model, which has been derived using quasi-steady-state approximation for describing the dynamics of psoriasis in various cytokines environment. Equilibrium analysis of the system shows that either the system converges to a stable steady state or exhibits a periodic oscillation depending upon system parameters. Finally, introducing a one dimensional impulsive system, we have determined the perfect dose and perfect dosing interval for biologic (TNF-α inhibitor) therapy to control the hyper-proliferation of keratinocytes. We have studied the effect of TNF-α inhibitor by considering both perfect and imperfect dosing during the inductive phase. The maximum possible number of drug holidays and the minimal number of doses that must subsequently be taken while avoiding drug resistance have been calculated for imperfect dosing. Since, psoriasis is non-curable but treatable disease, so the aim is to investigate the minimum dose with highest efficacy and proper dosing interval of TNF-α inhibitor for a psoriatic patient. Through numerical simulations, we have given a detailed prediction about the maximum drug holidays, tolerable for a patient, without loss of previous drug effects. Our theoretical predictions and numerical outcomes may be useful in guiding the design of future clinical trials.

Biological pest control using a model-based robust feedback.

Biological control is the artificial manipulation of natural enemies of a pest for its regulation to densities below a threshold for economic damage. The authors address the biological control of a class of pest population models using a model-based robust feedback approach. The proposed control framework is based on a recursive cascade control scheme exploiting the chained form of pest population models and the use of virtual inputs. The robust feedback is formulated considering the non-linear model uncertainties via a simple and intuitive control design. Numerical results on three pest biological control problems show that the proposed model-based robust feedback can regulate the pest population at the desired reference via the manipulation of a biological control action despite model uncertainties.

Mathematical insights on psoriasis regulation: Role of Th1 and Th2 cells.

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th1), Type 2 (Th2) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th1-Cells viz. Interferon gamma (IFN-γ), Transforming growth factor beta (TGF-ß) and Tumor necrosis factor alpha (TNF-α) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 (IL-10), Interleukin 4 (IL-4) etc., released by Th2-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / IL-10) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

Human immunodeficiency virus/acquired immune deficiency syndrome: Using drug from mathematical perceptive.

Entry of acquired immune deficiency syndrome virus into the host immune cell involves the participation of various components of host and viral cell unit. These components may be categorized as attachment of the viral surface envelope protein subunit, gp120, to the CD4(+) receptor and chemokine coreceptors, CCR5 and CXCR4, present on T cell surface. The viral fusion protein, gp41, the second cleaved subunit of Env undergoes reconfiguration and the membrane fusion reaction itself. Since the CD4(+) T cell population is actively involved; the ultimate outcome of human immunodeficiency virus infection is total collapse of the host immune system. Mathematical modeling of the stages in viral membrane protein-host cell receptor-coreceptor interaction and the effect of antibody vaccine on the viral entry into the susceptible host cell has been carried out using as impulsive differential equations. We have studied the effect of antibody vaccination and determined analytically the threshold value of drug dosage and dosing interval for optimum levels of infection. We have also investigated the effect of perfect adherence of drug dose on the immune cell count in extreme cases and observed that systematic drug dosage of the immune cells leads to longer and improved lives.