Time-Optimal Control Studies for Additional Food provided Prey-Predator Systems involving Holling Type-III and Holling Type-IV Functional Responses

In recent years, time-optimal control studies on additional food provided prey-predator systems have gained significant attention from researchers in the field of mathematical biology. In this study, we initially consider an additional food provided prey-predator model exhibiting Holling type-III functional response and the intra-specific competition among predators. We prove the existence and uniqueness of global positive solutions for the proposed model. We do the time optimal control studies with respect quality and quantity of additional food as control variables by transforming the independent variable in the control system. Making use of the Pontraygin maximum principle, we characterize the optimal quality of additional food and optimal quantity of additional food. We show that the findings of these time-optimal control studies on additional food provided prey-predator systems involving Holling type III functional response have the potential to be applied to a variety of problems in pest management. In the later half of this study, we consider an additional food provided prey-predator model exhibiting Holling type-IV functional response and study the above aspects for this system

Stochastic Optimal and Time-Optimal Control Studies for Additional Food provided prey-predator Systems involving Holling Type-III Functional Response

This paper consists of a detailed and novel stochastic optimal control analysis of a coupled non-linear dynamical system. The state equations are modeled as additional food provided prey-predator system with Holling Type-III functional response for predator and intra-specific competition among predators. We firstly discuss the optimal control problem as a Lagrangian problem with a linear quadratic control. Secondly we consider an optimal control problem in the time-optimal control setting. Stochastic maximum principle is used for establishing the existence of optimal controls for both these problems. Numerical simulations are performed based on stochastic forward-backward sweep methods for realizing the theoretical findings. The results obtained in these optimal control problems are discussed in the context of biological conservation and pest management

Stochastic Time-Optimal Control Studies for Additional Food provided Prey-Predator Systems involving Holling Type-IV Functional Response

We consider an additional food provided prey-predator model exhibiting Holling type IV functional response with combined continuous white noise and discontinuous L\'evy noise. We prove the existence and uniqueness of global positive solutions for the considered model. By considering the quality and quantity of additional food as control parameters, we formulate a time-optimal control problem. We obtain the condition for the existence of an optimal control. Furthermore, making use of the arrow condition of the sufficient stochastic maximum principle, we characterize the optimal quality of additional food and optimal quantity of additional food. Numerical results are given to illustrate the theoretical findings with applications in biological conservation and pest management

A Study of Qualitative Correlations Between Crucial Bio-markers and the Optimal Drug Regimen of Type-I Lepra Reaction: A Deterministic Approach

Mycobacterium leprae is a bacteria that causes the disease Leprosy (Hansen's disease), which is a neglected tropical disease. More than 200000 cases are being reported per year world wide. This disease leads to a chronic stage known as Lepra reaction that majorly causes nerve damage of peripheral nervous system leading to loss of organs. The early detection of this Lepra reaction through the level of bio-markers can prevent this reaction occurring and the further disabilities. Motivated by this, we frame a mathematical model considering the pathogenesis of leprosy and the chemical pathways involved in Lepra reactions. The model incorporates the dynamics of the susceptible schwann cells, infected schwann cells and the bacterial load and the concentration levels of the bio markers $IFN-\gamma$, $TNF-\alpha$, $IL-10$, $IL-12$, $IL-15$ and $IL-17$. We consider a nine compartment optimal control problem considering the drugs used in Multi Drug Therapy (MDT) as controls. We validate the model using 2D - heat plots. We study the correlation between the bio-markers levels and drugs in MDT and propose an optimal drug regimen through these optimal control studies. We use the Newton's Gradient Method for the optimal control studies

Achieving Minimum-Time Biological Conservation and Pest Management for Additional Food provided Predator-Prey Systems involving Inhibitory Effect : A Qualitative Investigation

Theoretical and experimental studies on prey-predator systems where predator is supplied with alternate sources of food have received significant attention over the years due to their relevance in achieving biological conservation and biological control. Some of the outcomes of these studies suggest that with appropriate quality and quantity of additional food, the system can be steered towards any desired state eventually with time. One of the limitations of previous studies is that the desired state is reached asymptotically, which makes the outcomes not easily applicable in practical scenarios. To overcome this limitation, in this work, we formulate and study optimal control problems to achieve the desired outcomes in minimum (finite) time. We consider two different models of additional food provided prey-predator systems involving Holling type IV functional response (with inhibitory effect of prey). In the first scenario, additional food is incorporated implicitly into the predator's functional response with a possibility of achieving biological conservation through co-existence of species and biological control by maintaining prey at a level that is least harmful to the system. In the second, the effect of additional food is incorporated explicitly into the predator's compartment with the goal of pest management by maintaining prey density at a very minimal damaging level. For both cases, appropriate optimal control strategies are derived and the theoretical findings are illustrated by numerical simulations. We also discuss the ecological significance of the theoretical findings for both models

A Study of Within-Host Dynamics of Dengue Infection incorporating both Humoral and Cellular Response with a Time Delay for Production of Antibodies

Abstract a. Background: Dengue is an acute illness caused by a virus. The complex behaviour of the virus in human body can be captured using mathematical models. These models helps us to enhance our understanding on the dynamics of the virus. b. Objectives: We propose to study the dynamics of within-host epidemic model of dengue infection which incorporates both innate immune response and adaptive immune response (Cellular and Humoral). The proposed model also incorporates the time delay for production of antibodies from B cells. We propose to understand the dynamics of the this model using the dynamical systems approach by performing the stability and sensitivity analysis. c. Methods used: The basic reproduction number (R0) has been computed using the next generation matrix method. The standard stability analysis and sensitivity analysis were performed on the proposed model. d. Results: The critical level of the antibody recruitment rate(q) was found to be responsible for the existence and stability of various steady states. The stability of endemic state was found to be dependent on time delay(τ). The sensitivity analysis identified the production rate of antibodies (q) to be highly sensitive parameter. e. Conclusions: The existence and stability conditions for the equilibrium points have been obtained. The threshold value of time delay (τ0) has been computed which is critical for change in stability of the endemic state. Sensitivity analysis was performed to identify the crucial and sensitive parameters of the model

A Comprehensive and Detailed Within-Host Modeling Study involving crucial Bio markers and Optimal Drug regimen for Lepra Type-I Reaction : A Deterministic Approach

Leprosy (Hansen's disease) is an infectious, neglected tropical disease caused by the Mycobacterium Leprae (M. Leprae). Each year there are approximately 2,02,189 new cases are detected globally. In the year 2017 more than half million people were disabled due to leprosy and almost 50000 new cases are added every year world wide. In leprosy, lepra reactions are the major cause for nerve damage leading to disability. Early detection of lepra reactions through study of biomarkers have important role in prevention of subsequent disabilities. To our knowledge there seems to be very limited literature available on within-host modeling at cellular level involving the crucial biomarkers and the possible optimal drug regimen for leprosy disease and lepra reactions. Motivated by these observations, in this study, we have proposed and analyzed a three dimensional mathematical model to capture the dynamics of susceptible schwann cells, infected schwann cells and the bacterial load based on the pathogenesis of leprosy. We estimated the parameters from various clinical papers to make the model more practical. The sensitivity of couple of parameters was evaluated through PRCC method to find out the single most influential parameter and also combination of two most influential parameters was studied using SRCC method. The sensitivity of other remaining parameters was evaluated using Sobol's index. We then have framed and studied an optimal control problem considering the different medication involved in the Multi Drug Therapy (MDT) as control variables. We further studied this optimal control problem along with both MDT and steroid interventions. The finding from this novel and comprehensive study will help the clinicians and public health researchers involved in the process of elimination and eradication of leprosy