Roles of astrocytes and prions in Alzheimer's disease: insights from mathematical modeling.

We present a mathematical model that explores the progression of Alzheimer's disease, with a particular focus on the involvement of disease-related proteins and astrocytes. Our model consists of a coupled system of differential equations that delineates the dynamics of amyloid beta plaques, amyloid beta protein, tau protein, and astrocytes. Amyloid beta plaques can be considered fibrils that depend on both the plaque size and time. We change our mathematical model to a temporal system by applying an integration operation with respect to the plaque size. Theoretical analysis including existence, uniqueness, positivity, and boundedness is performed in our model. We extend our mathematical model by adding two populations, namely prion protein and amyloid beta-prion complex. We characterize the system dynamics by locating biologically feasible steady states and their local stability analysis for both models. The characterization of the proposed model can help inform in advancing our understanding of the development of Alzheimer's disease as well as its complicated dynamics. We investigate the global stability analysis around the interior equilibrium point by constructing a suitable Lyapunov function. We validate our theoretical analysis with the aid of extensive numerical illustrations.

An eco-epidemiological model with the impact of fear.

In this study, we propose and analyze an eco-epidemiological model with disease in prey and incorporated the effect of fear on prey species due to predator population. We assume that the prey population grows logistically in the absence of predator species, and the disease is limited to the prey population only. We divide the total prey population into two distinct classes: susceptible prey and infected prey. Predator populations are not infected by the diseases, though feed both the susceptible and infected prey. Due to the fear of predators, the prey population becomes more vigilant and moves away from suspected predators. Such a foraging activity of prey reduces the chance of infection among susceptible prey by lowering the contact with infected prey. We assume that the fear of predators has no effect on infected prey as they are more vigilant. Positivity, boundedness, and uniform persistence of the proposed model are investigated. The biologically feasible equilibrium points and their stability are analyzed. We establish the conditions for the Hopf bifurcation of the proposed model around the endemic steady state. As the level of fear increases, the system moves toward the steady state from a limit cycle oscillation. The increasing level of fear cannot wipe out the diseases from the system, but the amplitude of the infected prey decreases as the level of fear is increased. The system changes its stability as the rate of infection increases, and the predator becomes extinct when the rate of infection in prey is high enough though predators are not infected by the disease.

Mathematical modeling and optimal intervention strategies of the COVID-19 outbreak.

34,354,966 active cases and 460,787 deaths because of COVID-19 pandemic were recorded on November 06, 2021, in India. To end this ongoing global COVID-19 pandemic, there is an urgent need to implement multiple population-wide policies like social distancing, testing more people and contact tracing. To predict the course of the pandemic and come up with a strategy to control it effectively, a compartmental model has been established. The following six stages of infection are taken into consideration: susceptible (S), asymptomatic infected (A), clinically ill or symptomatic infected (I), quarantine (Q), isolation (J) and recovered (R), collectively termed as SAIQJR. The qualitative behavior of the model and the stability of biologically realistic equilibrium points are investigated in terms of the basic reproduction number. We performed sensitivity analysis with respect to the basic reproduction number and obtained that the disease transmission rate has an impact in mitigating the spread of diseases. Moreover, considering the non-pharmaceutical and pharmaceutical intervention strategies as control functions, an optimal control problem is implemented to mitigate the disease fatality. To reduce the infected individuals and to minimize the cost of the controls, an objective functional has been constructed and solved with the aid of Pontryagin's maximum principle. The implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Extensive numerical simulations show that the implementation of intervention strategy has an impact in controlling the transmission dynamics of COVID-19 epidemic. Further, our numerical solutions exhibit that the combination of three controls are more influential when compared with the combination of two controls as well as single control. Therefore, the implementation of all the three control strategies may help to mitigate novel coronavirus disease transmission at this present epidemic scenario. Supplementary Information: The online version supplementary material available at 10.1007/s11071-022-07235-7.

Impact of social media advertisements on the transmission dynamics of COVID-19 pandemic in India.

In this paper, we propose a mathematical model to assess the impact of social media advertisements in combating the coronavirus pandemic in India. We assume that dissemination of awareness among susceptible individuals modifies public attitudes and behaviours towards this contagious disease which results in reducing the chance of contact with the coronavirus and hence decreasing the disease transmission. Moreover, the individual's behavioral response in the presence of global information campaigns accelerate the rate of hospitalization of symptomatic individuals and also encourage the asymptomatic individuals for conducting health protocols, such as self-isolation, social distancing, etc. We calibrate the proposed model with the cumulative confirmed COVID-19 cases for the Republic of India. We estimate eight epidemiologically important parameters, and also the size of basic reproduction number for India. We find that the basic reproduction number for India is greater than unity, which represents the substantial outbreak of COVID-19 in the country. Sophisticated techniques of sensitivity analysis are employed to determine the impacts of model parameters on basic reproduction number and symptomatic infected population. Our results reveal that to reduce disease burden in India, non-pharmaceutical interventions strategies should be implemented effectively to decrease basic reproduction number below unity. Continuous propagation of awareness through the internet and social media platforms should be regularly circulated by the health authorities/government officials for hospitalization of symptomatic individuals and quarantine of asymptomatic individuals to control the prevalence of disease in India.

Forecasting the daily and cumulative number of cases for the COVID-19 pandemic in India.

The ongoing novel coronavirus epidemic was announced a pandemic by the World Health Organization on March 11, 2020, and the Government of India declared a nationwide lockdown on March 25, 2020 to prevent community transmission of the coronavirus disease (COVID)-19. Due to the absence of specific antivirals or vaccine, mathematical modeling plays an important role in better understanding the disease dynamics and in designing strategies to control the rapidly spreading infectious disease. In our study, we developed a new compartmental model that explains the transmission dynamics of COVID-19. We calibrated our proposed model with daily COVID-19 data for four Indian states, namely, Jharkhand, Gujarat, Andhra Pradesh, and Chandigarh. We study the qualitative properties of the model, including feasible equilibria and their stability with respect to the basic reproduction number R0. The disease-free equilibrium becomes stable and the endemic equilibrium becomes unstable when the recovery rate of infected individuals increases, but if the disease transmission rate remains higher, then the endemic equilibrium always remains stable. For the estimated model parameters, R0>1 for all four states, which suggests the significant outbreak of COVID-19. Short-time prediction shows the increasing trend of daily and cumulative cases of COVID-19 for the four states of India.

Modeling and forecasting the COVID-19 pandemic in India.

In India, 100,340 confirmed cases and 3155 confirmed deaths due to COVID-19 were reported as of May 18, 2020. Due to absence of specific vaccine or therapy, non-pharmacological interventions including social distancing, contact tracing are essential to end the worldwide COVID-19. We propose a mathematical model that predicts the dynamics of COVID-19 in 17 provinces of India and the overall India. A complete scenario is given to demonstrate the estimated pandemic life cycle along with the real data or history to date, which in turn divulges the predicted inflection point and ending phase of SARS-CoV-2. The proposed model monitors the dynamics of six compartments, namely susceptible (S), asymptomatic (A), recovered (R), infected (I), isolated infected (Iq ) and quarantined susceptible (Sq ), collectively expressed SARIIqSq . A sensitivity analysis is conducted to determine the robustness of model predictions to parameter values and the sensitive parameters are estimated from the real data on the COVID-19 pandemic in India. Our results reveal that achieving a reduction in the contact rate between uninfected and infected individuals by quarantined the susceptible individuals, can effectively reduce the basic reproduction number. Our model simulations demonstrate that the elimination of ongoing SARS-CoV-2 pandemic is possible by combining the restrictive social distancing and contact tracing. Our predictions are based on real data with reasonable assumptions, whereas the accurate course of epidemic heavily depends on how and when quarantine, isolation and precautionary measures are enforced.

A mathematical model for COVID-19 transmission dynamics with a case study of India.

The ongoing COVID-19 has precipitated a major global crisis, with 968,117 total confirmed cases, 612,782 total recovered cases and 24,915 deaths in India as of July 15, 2020. In absence of any effective therapeutics or drugs and with an unknown epidemiological life cycle, predictive mathematical models can aid in understanding of both coronavirus disease control and management. In this study, we propose a compartmental mathematical model to predict and control the transmission dynamics of COVID-19 pandemic in India with epidemic data up to April 30, 2020. We compute the basic reproduction number R 0, which will be used further to study the model simulations and predictions. We perform local and global stability analysis for the infection free equilibrium point E 0 as well as an endemic equilibrium point E* with respect to the basic reproduction number R 0. Moreover, we showed the criteria of disease persistence for R 0 > 1. We conduct a sensitivity analysis in our coronavirus model to determine the relative importance of model parameters to disease transmission. We compute the sensitivity indices of the reproduction number R 0 (which quantifies initial disease transmission) to the estimated parameter values. For the estimated model parameters, we obtained R 0 = 1.6632 , which shows the substantial outbreak of COVID-19 in India. Our model simulation demonstrates that the disease transmission rate ßs is more effective to mitigate the basic reproduction number R 0. Based on estimated data, our model predict that about 60 days the peak will be higher for COVID-19 in India and after that the curve will plateau but the coronavirus diseases will persist for a long time.

The influence of time delay in a chaotic cancer model.

The tumor-immune interactive dynamics is an evergreen subject that continues to draw attention from applied mathematicians and oncologists, especially so due to the unpredictable growth of tumor cells. In this respect, mathematical modeling promises insights that might help us to better understand this harmful aspect of our biology. With this goal, we here present and study a mathematical model that describes how tumor cells evolve and survive the brief encounter with the immune system, mediated by effector cells and host cells. We focus on the distribution of eigenvalues of the resulting ordinary differential equations, the local stability of the biologically feasible singular points, and the existence of Hopf bifurcations, whereby the time lag is used as the bifurcation parameter. We estimate analytically the length of the time delay to preserve the stability of the period-1 limit cycle, which arises at the Hopf bifurcation point. We also perform numerical simulations, which reveal the rich dynamics of the studied system. We show that the delayed model exhibits periodic oscillations as well as chaotic behavior, which are often indicators of long-term tumor relapse.

Modeling the dynamics of COVID-19 pandemic with implementation of intervention strategies.

The ongoing COVID-19 epidemic spread rapidly throughout India, with 34,587,822 confirmed cases and 468,980 deaths as of November 30, 2021. Major behavioral, clinical, and state interventions have implemented to mitigate the outbreak and prevent the persistence of the COVID-19 in human-to-human transmission in India and worldwide. Hence, the mathematical study of the disease transmission becomes essential to illuminate the real nature of the transmission behavior and control of the diseases. We proposed a compartmental model that stratify into nine stages of infection. The incidence data of the SRAS-CoV-2 outbreak in India was analyzed for the best fit to the epidemic curve and we estimated the parameters from the best fitted curve. Based on the estimated model parameters, we performed a short-term prediction of our model. We performed sensitivity analysis with respect to R 0 and obtained that the disease transmission rate has an impact in reducing the spread of diseases. Furthermore, considering the non-pharmaceutical and pharmaceutical intervention policies as control functions, an optimal control problem is implemented to reduce the disease fatality. To mitigate the infected individuals and to minimize the cost of the controls, an objective functional has been formulated and solved with the aid of Pontryagin's maximum principle. This study suggest that the implementation of optimal control strategy at the start of a pandemic tends to decrease the intensity of epidemic peaks, spreading the maximal impact of an epidemic over an extended time period. Our numerical simulations exhibit that the combination of two controls is more effective when compared with the combination of single control as well as no control.

How do the contaminated environment influence the transmission dynamics of COVID-19 pandemic?

COVID-19 is an infectious disease caused by the SARS-CoV-2 virus that first appeared in Wuhan city and then globally. The COVID-19 pandemic exudes public health and socio-economic burden globally. Mathematical modeling plays a significant role to comprehend the transmission dynamics and controlling factors of rapid spread of the disease. Researchers focus on the human-to-human transmission of the virus but the SARS-CoV-2 virus also contaminates the environment. In this study we proposed a nonlinear mathematical model for the COVID-19 pandemic to analyze the transmission dynamics of the disease in India. We have also incorporated the environment contamination by the infected individuals as the population density is very high in India. The model is fitted and parameterized using daily new infection data from India. Analytical study of the proposed COVID-19 model, including feasibility of critical points and their stability reveals that the infection-free steady state is stable if the basic reproduction number is less than unity otherwise the system shows significant outbreak. Numerical illustrations demonstrates that if the rate of environment contamination increased then the number of infected persons also increased. But if the environment is disinfected by sanitization then the number of infected persons cannot drastically increase.

Spatiotemporal dynamics of a glioma immune interaction model.

We report a mathematical model which depicts the spatiotemporal dynamics of glioma cells, macrophages, cytotoxic-T-lymphocytes, immuno-suppressive cytokine TGF-ß and immuno-stimulatory cytokine IFN-γ through a system of five coupled reaction-diffusion equations. We performed local stability analysis of the biologically based mathematical model for the growth of glioma cell population and their environment. The presented stability analysis of the model system demonstrates that the temporally stable positive interior steady state remains stable under the small inhomogeneous spatiotemporal perturbations. The irregular spatiotemporal dynamics of gliomas, macrophages and cytotoxic T-lymphocytes are discussed extensively and some numerical simulations are presented. Performed some numerical simulations in both one and two dimensional spaces. The occurrence of heterogeneous pattern formation of the system has both biological and mathematical implications and the concepts of glioma cell progression and invasion are considered. Simulation of the model shows that by increasing the value of time, the glioma cell population, macrophages and cytotoxic-T-lymphocytes spread throughout the domain.

Dynamics of coronavirus pandemic: effects of community awareness and global information campaigns.

The effects of social media advertisements together with local awareness in controlling COVID-19 are explored in the present investigation by means of a mathematical model. The expression for the basic reproduction number is derived. Sufficient conditions for the global stability of endemic equilibrium are obtained. We perform sensitivity analysis to identify the key parameters of the model having great impacts on the prevalence and control of COVID-19. We calibrate the proposed model to fit the data set of COVID-19 cases for India. Our simulation results show that dissemination rate of awareness among susceptible individuals at community level and individual level plays pivotal role in curtailing the COVID-19 disease. Moreover, we observe that the global information distributing from social media and local awareness coming from mouth-to-mouth communication between unaware susceptible and aware people, together with hospitalization of symptomatic individuals and quarantine of asymptomatic individuals, are much beneficial in reducing COVID-19 cases in India. Our study suggests that both global and local awareness must be implemented effectively to manage the burden of COVID-19 pandemic.

Mathematical modeling of the COVID-19 pandemic with intervention strategies.

Mathematical modeling plays an important role to better understand the disease dynamics and designing strategies to manage quickly spreading infectious diseases in lack of an effective vaccine or specific antivirals. During this period, forecasting is of utmost priority for health care planning and to combat COVID-19 pandemic. In this study, we proposed and extended classical SEIR compartment model refined by contact tracing and hospitalization strategies to explain the COVID-19 outbreak. We calibrated our model with daily COVID-19 data for the five provinces of India namely, Kerala, Karnataka, Andhra Pradesh, Maharashtra, West Bengal and the overall India. To identify the most effective parameters we conduct a sensitivity analysis by using the partial rank correlation coefficients techniques. The value of those sensitive parameters were estimated from the observed data by least square method. We performed sensitivity analysis for R 0 to investigate the relative importance of the system parameters. Also, we computed the sensitivity indices for R 0 to determine the robustness of the model predictions to parameter values. Our study demonstrates that a critically important strategy can be achieved by reducing the disease transmission coefficient ß s and clinical outbreak rate q a to control the COVID-19 outbreaks. Performed short-term predictions for the daily and cumulative confirmed cases of COVID-19 outbreak for all the five provinces of India and the overall India exhibited the steady exponential growth of some states and other states showing decays of daily new cases. Long-term predictions for the Republic of India reveals that the COVID-19 cases will exhibit oscillatory dynamics. Our research thus leaves the option open that COVID-19 might become a seasonal disease. Our model simulation demonstrates that the COVID-19 cases across India at the end of September 2020 obey a power law.

Influence of multiple delays in brain tumor and immune system interaction with T11 target structure as a potent stimulator.

We present a mathematical model which describes the growth of malignant gliomas in presence of immune responses by considering the role of immunotherapeutic agent T11 target structure (T11TS). The model consider five populations, namely, glioma cells, macrophages, cytotoxic T-lymphocytes, TGF - ß and IFN - γ. The model system has highly nonlinear terms with four discrete time lags, but remains tractable. The goal of this work is to better understand the effect of multiple delays on the interaction between gliomas and immune components in conjunction with an administration of T11 target structure. Analytically, we investigate the conditions for the asymptotic stability of equilibrium points, the existence of Hopf bifurcations and the maximum value of the delay to preserve the stability of limit cycle. For the set of parameter values estimated from experimental data, time delays have hardly any influence on the system behavior. Numerical simulations are carried out to investigate the dynamics of the model with different values for delays with and without administration of T11 target structure.

Quantifying the role of immunotherapeutic drug T11 target structure in progression of malignant gliomas: Mathematical modeling and dynamical perspective.

The paper describes a mathematical model with synergistic interaction between the malignant glioma cells and the immune system, namely, macrophages, activated Cytotoxic T-Lymphocytes (CTLs), the immunosuppressive cytokine Transforming Growth Factor - ß (TGF-ß) and the immuno-stimulatory cytokine Interferon - γ (IFN-γ), using a system of coupled non-linear ordinary differential equations (ODEs). We have introduced a new immunotherapeutic drug T11 Target structure (T11TS) into the model, which boosts the macrophages and CTLs to kill the glioma cells. In our analysis, we have established a criteria for the threshold level of immunotherapeutic drug T11TS for which the system will be gliomas free or tumor free. The analytical findings are supported by numerical simulations using parameters estimated from experimental data.

How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment?

Although recent advances in oncology emphasized the role of microenvironment in tumor growth, the role of delays for modeling tumor growth is still uncertain. In this paper, we considered a model, describing the interactions of tumor cells with their microenvironment made of immune cells and host cells, in which we inserted, as suggested by the clinicians, two time delays, one in the interactions between tumor cells and immune cells and, one in the action of immune cells on tumor cells. We showed analytically that the singular point associated with the co-existence of the three cell populations loses its stability via a Hopf bifurcation. We analytically calculated a range of the delays over which tumor cells are inhibited by immune cells and over which a period-1 limit cycle induced by this Hopf bifurcation is observed. By using a global modeling technique, we investigated how the dynamics observed with two delays can be reproduced by a similar model without delays. The effects of these two delays were thus interpreted in terms of interactions between the cell populations.

A mathematical model to elucidate brain tumor abrogation by immunotherapy with T11 target structure.

T11 Target structure (T11TS), a membrane glycoprotein isolated from sheep erythrocytes, reverses the immune suppressed state of brain tumor induced animals by boosting the functional status of the immune cells. This study aims at aiding in the design of more efficacious brain tumor therapies with T11 target structure. We propose a mathematical model for brain tumor (glioma) and the immune system interactions, which aims in designing efficacious brain tumor therapy. The model encompasses considerations of the interactive dynamics of glioma cells, macrophages, cytotoxic T-lymphocytes (CD8(+) T-cells), TGF-ß, IFN-γ and the T11TS. The system undergoes sensitivity analysis, that determines which state variables are sensitive to the given parameters and the parameters are estimated from the published data. Computer simulations were used for model verification and validation, which highlight the importance of T11 target structure in brain tumor therapy.