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Leprosy (Hansen’s disease) is an infectious, neglected tropical disease caused by the Mycobacterium Leprae (M. Leprae). About 2,02,189 new cases are diagnosed worldwide each year. Lepra reactions are an off shoot of leprosy infection causing major nerve damage leading to disability. Early detection of lepra reactions through the study of biomarkers can prevent subsequent disabilities. 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 did the stability analysis, numerical simulations, and also performed the sensitivity analysis using Spearman’s rank correlation coefficient, partial rank correlation coefficient, and Sobol’s index methods. We later performed the optimal control studies with both multi-drug therapy and steroid interventions as control variables. Finally, we did the comparative and effectiveness study of these different control interventions.

The COVID-19 pandemic recently caused a huge impact on India, not only in terms of health but also in terms of economy. Understanding the spatio-temporal patterns of the disease spread is crucial for controlling the outbreak. In this study, we apply the compatible window-wise dynamic mode decomposition (CwDMD) and dynamic mode decomposition (DMD) techniques to the COVID-19 data of India to model the spatial-temporal patterns of the epidemic. We preprocess the COVID-19 data into weekly time-series at the state-level and apply both the CwDMD and DMD methods to decompose the data into a set of spatial-temporal modes. We identify the key modes that capture the dominant features of the COVID-19 spread in India and analyze their phase, magnitude, and frequency relationships to extract the temporal and spatial patterns. By incorporating rank truncation in each window, we have achieved greater control over the system’s output, leading to better results. Our results reveal that the COVID-19 outbreak in India is driven by a complex interplay of regional, demographic, and environmental factors. We identify several key modes that capture the patterns of disease spread in different regions and over time, including seasonal fluctuations, demographic trends, and localized outbreaks. Overall, our study provides valuable insights into the patterns of the COVID-19 outbreak in India using both CwDMD and DMD methods. These findings can help public health organizations to develop more effective strategies for controlling the spread of the pandemic. The CwDMD and DMD methods can be applied to other countries to identify the unique drivers of the outbreak and develop effective control strategies.

Ecotourism is a form of tourism involving responsible travel to natural areas, conserving the environment, and improving the well-being of the local people. Its purpose may be to educate the traveler, to provide funds for ecological censervation, to directly benifit the economic development, and political empowerment of local communities. Ecotourism has come up as an important conservation strategy in the tropical areas where diversity of species and habitats are threatened because of the traditional forms of development. This study deals with a non-linear mathematical model with a novel idea for sustainable development of biomass with ecotourism which is imperative in the present scenario. Stability and bifurcation analysis of the model is done and it is observed from our study that the system predicts unstability and exhibits bifurcation if ecotourism goes beyond a threshold value.

The declaration of a nationwide lockdown in India led to millions of migrant workers, particularly from Uttar Pradesh (UP) and Bihar, returning to their home states without proper transportation and social distancing from cities such as Delhi, Mumbai, and Hyderabad. This unforeseen migration and social mixing accelerated the transmission of diseases across the country. To analyze the impact of reverse migration on disease progression, we have developed a disease transmission model for the neighboring Indian states of Delhi and UP. The model’s essential mathematical properties, including positivity, boundedness, equilibrium points (EPs), and their linear stability, as well as computation of the basic reproduction number ( R 0 ) \left({R}_{0}) , are studied. The mathematical analysis reveals that the model with active reverse migration cannot reach a disease-free equilibrium, indicating that the failure of restrictive mobility intervention caused by reverse migration kept the disease propagation alive. Further, PRCC analysis highlights the need for effective home isolation, better disease detection techniques, and medical interventions to curb the spread. The study estimates a significantly shorter doubling time for exponential growth of the disease in both regions. In addition, the occurrence of synchronous patterns between epidemic trajectories of the Delhi and UP regions accentuates the severe implications of migrant plight on UP’s already fragile rural health infrastructure. By using COVID-19 incidence data, we quantify key epidemiological parameters, and our scenario analyses demonstrate how different lockdown plans might have impacted disease prevalence. Based on our observations, the transmission rate has the most significant impact on COVID-19 cases. This case study exemplifies the importance of carefully considering these issues before implementing lockdowns and social isolation throughout the country to combat future outbreaks.

This article presents a cost-effective optimal control analysis of interventions applied to a S2EI2RS type deterministic compartmental model of COVID-19, considering community awareness and immunity loss. We introduce two time-dependent controls, namely, home quarantine and treatment, to the model for defining an optimal control problem (OCP). In addition to some basic qualitative properties, we obtain the reproductive threshold R 0 {R}_{0} by using the next-generation method and see the impact of controls on it. We also investigate the effect of community awareness and waning immunity, when no controls are applied. The existence and characterization of optimal controls is proved to establish the optimality system, and the OCP is solved using the forward–backward sweep method. The results are simulated using MATLAB. Our comparative cost-effective analysis indicates that implementing both control strategies simultaneously, along with community awareness, is the most optimal and sustainable way to flatten COVID-19 curves in a short period of time than that of implementing single controls. This article offers valuable insights that can assist policymakers and public health experts in designing targeted and effective control measures for COVID-19 and future epidemics in the post-COVID era. Therefore, this piece of work could be a valuable contribution to the existing literature.

It is now known that early government interventions in pandemic management helps in slowing down the pandemic in the initial phase, during which a conservative basic reproduction number can be maintained. There have been several ways to evaluate these early response strategies for COVID-19 during its outbreak globally in 2020. As a novelty, we evaluate them through the lens of patient recovery logistics. Here, we use a data-driven approach of recovery analysis in a case study of Singapore during January 22–April 01, 2020, which is effectively the analysis of length-of-stay in the government healthcare facility, National Center for Infectious Diseases. We propose the use of a data-driven method involving periodization, statistical analysis, regression models, and epidemiological models. We demonstrate that the estimates of reproduction number in Singapore shows variation in different age groups and periods, indicating the success of early intervention strategy in the initial transmission stages of the pandemic.

Switching mechanism is adopted by predator populations when they are provided with two types of prey: susceptible and infected. In this study, we propose a modification of an eco-epidemiological model with the predator switching mechanism. In the presence of switching behavior, the existence of steady states and their stability have been discussed. The qualitative changes in the proposed model have been observed by the existence of transcritical and Hopf bifurcation. Numerical simulations are performed to support our numerical findings. In the context of species’ survival when disease is present in the system, it gives some theoretical views for eco-managers to understand the dynamics.

The magnetohydrodynamic (MHD) Stokes equations have several applications in the field of biofluid dynamics. In the present study, we propose the staggered finite volume method (S-FVM) for MHD Stokes equations and establish its equivalence to a nonconforming finite element approximation. We also theoretically establish the convergence of the proposed S-FVM. The error estimation is carried out in an unstructured grid framework which is known for its flexibility and robustness in dealing with complex domains. The apriori estimate shows that the L 2 {L}_{2} -norm of the error for the pressure and velocity components is of order h h , the spacial grid size. After validating the numerical performance of the scheme against benchmark test cases, we do numerical simulations for the blood flow through an injured arteriole and analyze the influence of the magnetic force on hemodynamics in the arteriole under an injured condition.

In this work, the dynamics of a food chain model with disease in the predator and the Allee effect in the prey have been investigated. The model also incorporates a Holling type-III functional response, accounting for both disease transmission and predation. The existence of equilibria and their stability in the model have also been investigated. The primary objective of this research is to examine the effects of the Allee parameter. Hopf bifurcations are explored about the interior and disease-free equilibrium point, where the Allee is taken as a bifurcation point. In numerical simulation, phase portraits have been used to look into the existence of equilibrium points and their stability. The bifurcation diagrams that have been drawn clearly demonstrate the presence of significant local bifurcations, including Hopf, transcritical, and saddle-node bifurcations. Through the phase portrait, limit cycle, and time series, the stability and oscillatory behaviour of the equilibrium point of the model are investigated. The numerical simulation has been done using MATLAB and Matcont.

The aim of this article is to determine the optimal intensity of lock-down measures and vaccination rates to control the spread of coronavirus disease 2019. The study uses a stochastic susceptible-infected-recovered (SIR) model with infection dynamics. A Feynman-type path integral control approach is used to derive a forward Fokker-Plank-type equation for the system, which helps in performing a stochastic control analysis. The simulation study concludes that increasing the diffusion coefficients leads to a downward trend in the susceptible and recovery curves, while the infection curve becomes ergodic. Additionally, the study shows that the optimal lock-down intensity is stable around zero, and the vaccination rate increases over time.

A mathematical model is proposed and discussed to study the effect of cell-to-cell transmission, the non-cytolytic process, and the effect of logistic growth on the dynamics of HIV in vivo . The model system consists of one disease-free steady state and another endemic steady state. The disease-free steady state is globally asymptotically stable and the disease eradicated if the basic reproduction number is smaller than one. However, the endemic steady state is globally stable under specific parametric conditions, when it exists. At R 0 = 1 {R}_{0}=1 , the forward transcritical bifurcation is obtained. Also, by considering proliferation rate as bifurcation parameter, we get Hopf and Hopf–Hopf bifurcations. We have performed numerical simulations using MATLAB to support our analytical results and show the effects of cell-to-cell infection, proliferation rate, and non-cytolytic cure on all three populations. In the end, we have performed data fitting and note the same behaviour of observed data with predicted data.

This article discusses a novel idea from cell therapy in which nanoparticles (NPs) are adsorbed on red blood cells (RBCs). RBCs serve as a drug carrier for NPs or nanodrugs adsorbed on the cell membrane of RBC. For the purpose of examination, Fe 3 O 4 {{\rm{Fe}}}_{3}{{\rm{O}}}_{4} NPs are adsorbed on RBCs, collectively called NP-RBC complex. RBCs being a natural vascular carrier, have high transfusion rates and biocompatibility. This mathematical study provides a basis to attempt nanodrug delivery via RBCs, as carriers for nanodrugs, to the stenosed sites in an artery. The mathematical model is developed for an artery with stenosis and a catheter that regards the temperature and velocity of the NP-RBC complex. Catheter coated with the NP-RBC complex is inserted into the lumen of the stenosed artery. The mathematical problem is solved numerically using Bernstein polynomials. The physical features were discussed through graphs plotted using MATLAB. The influence of parameters such as volume fraction, radius of the NP-RBC complex in blood, and the thickness of the nanolayer on RBCs was studied. A noticeable outcome states that the nanolayer of optimum thickness about 50–40 nm is suitable for this purpose. Thus, this is an attempt to study the delivery of NPs adsorbed on the surface of RBCs to develop newfangled strategies in nanomedicine bearing high precision and efficiency.

The study of dynamics of diabetic population infected by COVID-19 is of pressing concern as people with diabetes are considered to be at higher risk of severe illness from COVID-19. A three-compartment mathematical model to describe the interactions of diabetic population and non-diabetic population both infected by COVID-19 with a susceptible population is considered. Time delays in incubation periods of COVID-19 in diabetic and non-diabetic populations are introduced. Besides the basic properties of such a dynamical system, both local and global stability of endemic equilibrium, are studied. The lengths of time delays are estimated for which the stability of the system is preserved locally, while sufficient conditions on system parameters are obtained for global stability. Numerical examples are provided to establish the theory, and simulations are provided to visualize the examples. It is noted that an increase in length of time delay in either of infected populations leads to oscillations in susceptible population but has no impact on infected populations.

The four-dimensional food-web system consisting of two prey species for a generalist middle predator and a top predator is proposed and investigated. The middle predator is predating over both the prey species with a modified Holling type-II functional response. The food-web model is effectively formulated, exhibits bounded behavior, and displays dissipative dynamics. The proposed model’s essential dynamical features are studied regarding local stability. We investigated the four species’ survival and established their persistence criteria. In the proposed model, a transcritical bifurcation occurs at the axial equilibrium point. The numerical simulations reveal the persistence of a chaotic attractor or stable focus. The conclusion is that increasing the food available to the middle predator may make it possible to manage and mitigate the chaos within the food chain.

Mycobacterium leprae is a bacterium that causes the disease leprosy (Hansen’s disease), which is a neglected tropical disease. More than 2,00,000 cases are being reported per year worldwide. This disease leads to a chronic stage known as lepra reaction that majorly causes nerve damage of the 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 interferon- γ \hspace{0.1em}\text{interferon-}\hspace{0.1em}\gamma , tumor necrosis factor- α \hspace{0.1em}\text{tumor necrosis factor-}\hspace{0.1em}\alpha , IL (interleukin)- 10 \hspace{0.1em}\text{IL (interleukin)-}\hspace{0.1em}10 , IL- 12 \hspace{0.1em}\text{IL-}\hspace{0.1em}12 , IL- 15 \hspace{0.1em}\text{IL-}\hspace{0.1em}15 , and IL- 17 \hspace{0.1em}\text{IL-}\hspace{0.1em}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.

This article consists of a detailed and novel stochastic optimal control analysis of a coupled non-linear dynamical system. The state equations are modelled as an additional food-provided prey–predator system with Holling type III functional response for predator and intra-specific competition among predators. We first discuss the optimal control problem as a Lagrangian problem with a linear quadratic control. Second, we consider an optimal control problem in the time-optimal control setting. We initially establish the existence of optimal controls for both these problems and later characterize these optimal controls using the Stochastic maximum principle. Further 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.

This study discusses an SIR epidemic model with modified saturated incidence rates and Holling functional type-II therapy. In this study, we take the new alert compartment (A) in the SIR compartment model. Consider the modified non-linear incidence rate from the susceptible to the infected class and the second non-linear incidence rate from the alert to the infected class. Further, we investigate the elementary reproduction number, the equilibrium points of the model, and their stability. We apply manifold theory to discuss bifurcations of the disease-free equilibrium point. This study shows that the infected population decreases with the Holling functional type II treatment rate. It also shows that the number of infected people decreases when the psychological rate increases and the contact rate decreases.

In this article, the behavior of an susceptible exposed infected recovered (SEIR) epidemic model with nonlinear incidence rate and Holling type II treatment function is presented and analyzed. Reproduction number of the model is calculated. Equilibrium points are determined. Disease-free equilibrium exists when R 0 is below 1. Behavior of disease-free equilibrium is examined at R 0 = 1. Endemic equilibrium exists when R 0 crosses 1. Stability of both equilibrium points is investigated locally and globally. Simulation is provided to support the result.

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.

In this article, a nonlinear mathematical model is proposed and analyzed to study the spread of coronavirus disease (COVID-19) and its control. Due to sudden emergence of a peculiar kind of infection, no vaccines were available, and therefore, the nonpharmaceutical interventions such as lockdown, isolation, and hospitalization were imposed to stop spreading of the infectious disease. The proposed model consists of six dependent variables, namely, susceptible population, infective population, isolated susceptible population who are aware of the undesirable consequences of the COVID-19, quarantined population of known infectives (symptomatic), recovered class, and the coronavirus population. The model exhibits two equilibria namely, the COVID-19-free equilibrium and the COVID-19-endemic equilibrium. It is observed that if basic reproduction number R 0 < 1 {R}_{0}\lt 1 , then the COVID-19-free equilibrium is locally asymptotically stable. However, the endemic equilibrium is locally as well as nonlinearly asymptotically stable under certain conditions if R 0 > 1 {R}_{0}\gt 1 . Model analysis shows that if safety measures are adopted by way of isolation of susceptibles and quarantine of infectives, the spread of COVID-19 disease can be kept under control.

Dengue fever is an infectious viral fever. The complex behavior of the virus within the body can be explained through mathematical models to understand the virus’s dynamics. We propose two different with-in host models of dengue virus transmission with humoral immune response. The proposed models differ from one another because one of the models assumes that newly formed viruses infect healthy cells again. To understand the dynamics of the proposed models, we perform a comparative study of stability analysis, numerical simulation, and sensitivity analysis. The basic reproduction number (BRN) of the two models is computed using next-generation matrix method. The local stability (l.s) analysis is discussed using the linearization method. The Lyapunov’s direct method is used to check the global stability (g.s) of the models. It has been found that both the equilibrium states for both the models, namely, virus-free equilibrium state and endemic equilibrium state, are globally stable, based on the value of BRN. Results show the influence of immune response on the cell dynamics and virus particles. The virus neutralization rate by antibodies and rate that affects the antibody growth are highly sensitive for the two models. Optimal control is applied to explore the possible control strategies to prevent virus spread in the host system. It is evident from the results that the strategy to administrate antibiotic drugs and home remedies slow down the virus spread in the host.

Measles is a highly communicable viral infection that mostly affects children aged 5 years and below. Maternal antibodies in neonates help protect them from infectious diseases, including measles. However, maternal antibodies disappear a few months after birth, necessitating vaccination against measles. A mathematical model of measles, incorporating maternal antibodies and a double-dose vaccination, was proposed. Whenever ℛ 0 < 1 {{\mathcal{ {\mathcal R} }}}_{0}\lt 1 , the model is shown to be locally asymptotically stable. This means that the measles disease can be eliminated under such conditions in a finite time. It was established that ℛ 0 {{\mathcal{ {\mathcal R} }}}_{0} is highly sensitive to β \beta (the transmission rate). A numerical simulation of the model using the Runge-Kutta fourth-order scheme was carried out, showing that varying the parameters to reduce ℛ 0 {{\mathcal{ {\mathcal R} }}}_{0} will help control the measles disease and ultimately lead to eradication. The measles-mumps-rubella (MMR) vaccine dosage should be adjusted for babies from recovered mothers, as maternal antibodies are usually high in such babies and can interfere with the effectiveness of the MMR vaccine.

More than half of the coronavirus disease 19 (COVID-19) related mortality rates in the United States and Europe are associated with long-term-care facilities (LTCFs) such as old-age organizations, nursing homes, and disability centers. These facilities are considered most vulnerable to spread of an pandemic like COVID-19 because of multiple reasons including high density of elderly population with a diverse range of medical requirements, limited resources, nursing activities/medications, and the role of external visitors. In this study, we aim to understand the role of visitor’s family members and specific interventions (such as use of face masks and restriction of visiting hours) on the dynamics of infection in a community using a mathematical model. The model considers two types of social contexts (community and LTCFs) with three different groups of interacting populations (non-mobile community individuals, mobile community individuals, and long-term facility residents). The goal of this work is to compare the outbreak burden between different centre of disease control (CDC) planning scenarios, which capture distinct types of intensity of diseases spread in LTCF observed during COVID-19 outbreak. The movement of community mobile members is captured via their average relative times in and out of the long-term facilities to understand the strategies that would work well in these facilities the CDC planning scenarios. Our results suggest that heterogeneous mixing worsens epidemic scenario as compared to homogeneous mixing and the epidemic burden is hundreds times greater for community spread than within the facility population.