Chapter 1 : Introduction and Literature Review 1.1 General Introduction............................. more Chapter 1 : Introduction and Literature Review 1.1 General Introduction.............................................................................................................1 1.1.1 A brief review of mathematical models in epidemiology.............................................3 1.2 Biological preliminaries.......................................................................................................4 1.3 Tools of analysis.................................................................................................................5 1.4 Problem description and solution strategies.......................................................................6 Chapter 2 : Prevention and control of Ebola Virus: A Mathematical Modeling Analysis 2.1 Formulation of Epidemic Model.........................................................................................8 2.2 Analysis of behavior of Epidemic Model.........................................................................
The Zika virus (ZIKV) epidemic is depicted to have high spatial diversity and slow growth, attrib... more The Zika virus (ZIKV) epidemic is depicted to have high spatial diversity and slow growth, attributable to the dynamics of the mosquito vector and mobility of the human populations. In an effort to understand the transmission dynamics of Zika virus, we formulate a new compartmental epidemic model with a system of seven differential equations and 11 parameters incorporating the decaying transmission rate and study the impact of protection measure on basic public health. We do not fit the model to the observed pattern of spread, rather we use parameter values estimated in the past and examine the extent to which the designed model prediction agrees with the pattern of spread seen in Brazil, via reaction–diffusion modeling. Our work includes estimation of key epidemiological parameters such as basic reproduction number ([Formula: see text], and gives a rough estimate of how many individuals can be typically infected during an outbreak if it occurs in India. We used partial rank correla...
In this paper, we have formulated a compartmental epidemic model with exponentially decaying tran... more In this paper, we have formulated a compartmental epidemic model with exponentially decaying transmission rates to understand the Ebola transmission dynamics and study the impact of control measures to basic public health. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. We have calculated the basic reproduction number through next generation matrix and investigated the spatial spread of the epidemic via reaction–diffusion modeling. Instead of fitting the model to the observed pattern of spread, we have used previously estimated parameter values and examined the efficacy of predictions of the designed model vis-à -vis the pattern of spread observed in Sierra Leone, West Africa. Further, we conducted a sensitivity analysis to determine the extent to which improvement in predictions is achievable through better parameterization. We performed numerical simulations with and without control measure for the designed model system. A signifi...
International Journal of Bifurcation and Chaos, 2016
Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. I... more Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. In this paper, an attempt has been made for modeling the virus dynamics using an SEIR model to better understand and characterize the transmission trajectories of the Ebola outbreak. We compare the simulated results with the most recent reported data of Ebola infected cases in the three most affected countries Guinea, Liberia and Sierra Leone. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. Existence and local stability of these equilibria are explored. Using central manifold theory, it is established that the transcritical bifurcation occurs when basic reproduction number passes through unity. The proposed Ebola epidemic model provides an estimate to the potential number of future cases. The model indicates that the disease will decline after peaking if multisectorial and multinational efforts to control the spread of infection are main...
International Journal of Bifurcation and Chaos, 2016
In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradien... more In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradient has been shown to exhibit temporal chaos at a fixed point in space. The proposed model is a modification of the model recently presented by Upadhyay and Roy [2014]. The spatial interactions among the species have been represented in the form of reaction–diffusion equations. The model incorporates the intrinsic growth rate of fish population which varies linearly with the depth of water. Numerical results show that diffusion can drive otherwise stable system into aperiodic behavior with sensitivity to initial conditions. We show that spatially induced chaos plays an important role in spatial pattern formation in heterogeneous environment. Spatiotemporal distributions of species have been simulated using the diffusivity assumptions realistic for natural eco-epidemic systems. We found that in heterogeneous environment, the temporal dynamics of both the species are drastically different an...
Samples of particulate matter of size 10 micron (PM10) were collected in Talcher, Orissa (India) ... more Samples of particulate matter of size 10 micron (PM10) were collected in Talcher, Orissa (India) from six sites with different land-uses. The sampling was done concurrently twice a week during the months of June 2008, November 2008 and January 2009. The ambient mass concentration and the elemental composition in these PM10 samples were determined. The annual average concentrations of PM10 samples at each site were 144 ± 29 μg/m3, 191 ± 61 μg/m3, 90 ± 28 μg/m3, 60 ± 15 μg/m3, 106 ± 35 μg/m3, and 150 ± 36 μg/m3 respectively, indicating severe air pollution levels in Talcher. Variation of particulate matter with meteorological parameters like wind speed, relative humidity and temperature was observed. The study reveals that the particulate matter concentration drops substantially with the rise of wind speed above 1m/s. Elemental concentrations of PM10 were analyzed using an atomic absorption spectrophotometer. Correlation and multivariate analysis techniques, such as principal componen...
International Journal of Bifurcation and Chaos, 2015
In this paper, an attempt has been made to study the spatial and temporal dynamical interactions ... more In this paper, an attempt has been made to study the spatial and temporal dynamical interactions among the species of wetland ecosystem through a mathematical model. The model represents the population dynamics of phytoplankton, zooplankton and fish species found in Chilika lake, Odisha, India. Nonlinear stability analysis of both the temporal and spatial models has been carried out. Maximum sustainable yield and optimal harvesting policy have been studied for a nonspatial model system. Numerical simulation has been performed to figure out the parameters responsible for the complex dynamics of the wetland system. Significant outcomes of our numerical findings and their interpretations from an ecological point of view are provided in this paper. Numerical simulation of spatial model exhibits some interesting and beautiful patterns. We have also pointed out the parameters that are responsible for the good health of wetland ecosystem.
Advances in Intelligent Systems and Computing, 2014
ABSTRACT An attempt has been made to investigate the dynamics of a diffusive epidemic model with ... more ABSTRACT An attempt has been made to investigate the dynamics of a diffusive epidemic model with strong Allee effect in the susceptible population and with an asymptotic transmission rate. We show the asymptotic stability of the endemic equilibria. Turing patterns selected by the reaction-diffusion system under zero flux boundary conditions have been explored. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots and spot–stripe mixture patterns. It was found that diffusion has appreciable influence on spatial spread of epidemics. Wave of chaos appears to be a dominant mode of disease dispersal.
ABSTRACT The world's most endangered feline species; the Iberian Lynx has suffered severe... more ABSTRACT The world's most endangered feline species; the Iberian Lynx has suffered severe population decline and is now on the verge of extinction despite recovery plans. In this paper, an attempt has been made to understand the extinction dynamics of this endangered cat species. The paper focuses on the spread of rabbit haemorrhagic disease in the European rabbit population and its effect on the survival of the Iberian Lynx. A qualitative analysis of an eco-epidemiological model with simple law of mass action and Holling type II functional response is carried out. Existence and uniqueness of solutions are established and shown to be uniformly bounded. The basic reproduction number R0 is obtained and the occurrence of a backward bifurcation at R0=1 is shown to be possible using central manifold theory. The global stability of endemic equilibrium is established using a geometric approach. Criteria for diffusion-driven instability caused by local random movements of European rabbits and Iberian Lynx are obtained. Detailed analysis of Turing patterns formation selected by the reaction-diffusion system under zero flux boundary conditions is presented. We found that diffusion coefficients and transmission rate have appreciable influence on spatial spread of the epidemic. Numerical simulation results confirm the analytical finding and generate beautiful patterns that are consistent with the field observations and suggest that Iberian Lynx might have become extinct from Portugal and neighbouring countries. Suggestions for disease eradication and its control which in turn may increase the population of Iberian Lynx are discussed.
ABSTRACT In this paper, we study the complex dynamics of a spatial nonlinear predator-prey system... more ABSTRACT In this paper, we study the complex dynamics of a spatial nonlinear predator-prey system under harvesting. A modified Leslie–Gower model with Holling type IV functional response and nonlinear harvesting of prey is considered. We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. Numerical simulations were performed to figure out how Turing patterns evolve under nonlinear harvesting. Simulation study leads to a few interesting sequences of pattern formation, which may be relevant in real world situations.
International Journal of Bifurcation and Chaos, 2014
In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptoti... more In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptotic transmission rate. Existence and uniqueness of solutions are established and shown to be uniformly bounded for all non-negative initial values. We have also found a sufficient condition which ensures the persistence of the model system. This implies that both susceptible and infected will always coexist at any location of the inhabited domain. This coexistence is independent of values of the diffusivity constants for two subpopulations. The global stability of the endemic equilibrium is established by constructing a Lyapunov function. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation, conditions for Hopf and Turing bifurcations are obtained. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Turing patterns select...
Chapter 1 : Introduction and Literature Review 1.1 General Introduction............................. more Chapter 1 : Introduction and Literature Review 1.1 General Introduction.............................................................................................................1 1.1.1 A brief review of mathematical models in epidemiology.............................................3 1.2 Biological preliminaries.......................................................................................................4 1.3 Tools of analysis.................................................................................................................5 1.4 Problem description and solution strategies.......................................................................6 Chapter 2 : Prevention and control of Ebola Virus: A Mathematical Modeling Analysis 2.1 Formulation of Epidemic Model.........................................................................................8 2.2 Analysis of behavior of Epidemic Model.........................................................................
The Zika virus (ZIKV) epidemic is depicted to have high spatial diversity and slow growth, attrib... more The Zika virus (ZIKV) epidemic is depicted to have high spatial diversity and slow growth, attributable to the dynamics of the mosquito vector and mobility of the human populations. In an effort to understand the transmission dynamics of Zika virus, we formulate a new compartmental epidemic model with a system of seven differential equations and 11 parameters incorporating the decaying transmission rate and study the impact of protection measure on basic public health. We do not fit the model to the observed pattern of spread, rather we use parameter values estimated in the past and examine the extent to which the designed model prediction agrees with the pattern of spread seen in Brazil, via reaction–diffusion modeling. Our work includes estimation of key epidemiological parameters such as basic reproduction number ([Formula: see text], and gives a rough estimate of how many individuals can be typically infected during an outbreak if it occurs in India. We used partial rank correla...
In this paper, we have formulated a compartmental epidemic model with exponentially decaying tran... more In this paper, we have formulated a compartmental epidemic model with exponentially decaying transmission rates to understand the Ebola transmission dynamics and study the impact of control measures to basic public health. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. We have calculated the basic reproduction number through next generation matrix and investigated the spatial spread of the epidemic via reaction–diffusion modeling. Instead of fitting the model to the observed pattern of spread, we have used previously estimated parameter values and examined the efficacy of predictions of the designed model vis-à -vis the pattern of spread observed in Sierra Leone, West Africa. Further, we conducted a sensitivity analysis to determine the extent to which improvement in predictions is achievable through better parameterization. We performed numerical simulations with and without control measure for the designed model system. A signifi...
International Journal of Bifurcation and Chaos, 2016
Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. I... more Recently, the 2014 Ebola virus (EBOV) outbreak in West Africa was the largest outbreak to date. In this paper, an attempt has been made for modeling the virus dynamics using an SEIR model to better understand and characterize the transmission trajectories of the Ebola outbreak. We compare the simulated results with the most recent reported data of Ebola infected cases in the three most affected countries Guinea, Liberia and Sierra Leone. The epidemic model exhibits two equilibria, namely, the disease-free and unique endemic equilibria. Existence and local stability of these equilibria are explored. Using central manifold theory, it is established that the transcritical bifurcation occurs when basic reproduction number passes through unity. The proposed Ebola epidemic model provides an estimate to the potential number of future cases. The model indicates that the disease will decline after peaking if multisectorial and multinational efforts to control the spread of infection are main...
International Journal of Bifurcation and Chaos, 2016
In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradien... more In this paper, an eco-epidemiological model in which both species diffuse along a spatial gradient has been shown to exhibit temporal chaos at a fixed point in space. The proposed model is a modification of the model recently presented by Upadhyay and Roy [2014]. The spatial interactions among the species have been represented in the form of reaction–diffusion equations. The model incorporates the intrinsic growth rate of fish population which varies linearly with the depth of water. Numerical results show that diffusion can drive otherwise stable system into aperiodic behavior with sensitivity to initial conditions. We show that spatially induced chaos plays an important role in spatial pattern formation in heterogeneous environment. Spatiotemporal distributions of species have been simulated using the diffusivity assumptions realistic for natural eco-epidemic systems. We found that in heterogeneous environment, the temporal dynamics of both the species are drastically different an...
Samples of particulate matter of size 10 micron (PM10) were collected in Talcher, Orissa (India) ... more Samples of particulate matter of size 10 micron (PM10) were collected in Talcher, Orissa (India) from six sites with different land-uses. The sampling was done concurrently twice a week during the months of June 2008, November 2008 and January 2009. The ambient mass concentration and the elemental composition in these PM10 samples were determined. The annual average concentrations of PM10 samples at each site were 144 ± 29 μg/m3, 191 ± 61 μg/m3, 90 ± 28 μg/m3, 60 ± 15 μg/m3, 106 ± 35 μg/m3, and 150 ± 36 μg/m3 respectively, indicating severe air pollution levels in Talcher. Variation of particulate matter with meteorological parameters like wind speed, relative humidity and temperature was observed. The study reveals that the particulate matter concentration drops substantially with the rise of wind speed above 1m/s. Elemental concentrations of PM10 were analyzed using an atomic absorption spectrophotometer. Correlation and multivariate analysis techniques, such as principal componen...
International Journal of Bifurcation and Chaos, 2015
In this paper, an attempt has been made to study the spatial and temporal dynamical interactions ... more In this paper, an attempt has been made to study the spatial and temporal dynamical interactions among the species of wetland ecosystem through a mathematical model. The model represents the population dynamics of phytoplankton, zooplankton and fish species found in Chilika lake, Odisha, India. Nonlinear stability analysis of both the temporal and spatial models has been carried out. Maximum sustainable yield and optimal harvesting policy have been studied for a nonspatial model system. Numerical simulation has been performed to figure out the parameters responsible for the complex dynamics of the wetland system. Significant outcomes of our numerical findings and their interpretations from an ecological point of view are provided in this paper. Numerical simulation of spatial model exhibits some interesting and beautiful patterns. We have also pointed out the parameters that are responsible for the good health of wetland ecosystem.
Advances in Intelligent Systems and Computing, 2014
ABSTRACT An attempt has been made to investigate the dynamics of a diffusive epidemic model with ... more ABSTRACT An attempt has been made to investigate the dynamics of a diffusive epidemic model with strong Allee effect in the susceptible population and with an asymptotic transmission rate. We show the asymptotic stability of the endemic equilibria. Turing patterns selected by the reaction-diffusion system under zero flux boundary conditions have been explored. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots and spot–stripe mixture patterns. It was found that diffusion has appreciable influence on spatial spread of epidemics. Wave of chaos appears to be a dominant mode of disease dispersal.
ABSTRACT The world's most endangered feline species; the Iberian Lynx has suffered severe... more ABSTRACT The world's most endangered feline species; the Iberian Lynx has suffered severe population decline and is now on the verge of extinction despite recovery plans. In this paper, an attempt has been made to understand the extinction dynamics of this endangered cat species. The paper focuses on the spread of rabbit haemorrhagic disease in the European rabbit population and its effect on the survival of the Iberian Lynx. A qualitative analysis of an eco-epidemiological model with simple law of mass action and Holling type II functional response is carried out. Existence and uniqueness of solutions are established and shown to be uniformly bounded. The basic reproduction number R0 is obtained and the occurrence of a backward bifurcation at R0=1 is shown to be possible using central manifold theory. The global stability of endemic equilibrium is established using a geometric approach. Criteria for diffusion-driven instability caused by local random movements of European rabbits and Iberian Lynx are obtained. Detailed analysis of Turing patterns formation selected by the reaction-diffusion system under zero flux boundary conditions is presented. We found that diffusion coefficients and transmission rate have appreciable influence on spatial spread of the epidemic. Numerical simulation results confirm the analytical finding and generate beautiful patterns that are consistent with the field observations and suggest that Iberian Lynx might have become extinct from Portugal and neighbouring countries. Suggestions for disease eradication and its control which in turn may increase the population of Iberian Lynx are discussed.
ABSTRACT In this paper, we study the complex dynamics of a spatial nonlinear predator-prey system... more ABSTRACT In this paper, we study the complex dynamics of a spatial nonlinear predator-prey system under harvesting. A modified Leslie–Gower model with Holling type IV functional response and nonlinear harvesting of prey is considered. We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. Numerical simulations were performed to figure out how Turing patterns evolve under nonlinear harvesting. Simulation study leads to a few interesting sequences of pattern formation, which may be relevant in real world situations.
International Journal of Bifurcation and Chaos, 2014
In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptoti... more In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptotic transmission rate. Existence and uniqueness of solutions are established and shown to be uniformly bounded for all non-negative initial values. We have also found a sufficient condition which ensures the persistence of the model system. This implies that both susceptible and infected will always coexist at any location of the inhabited domain. This coexistence is independent of values of the diffusivity constants for two subpopulations. The global stability of the endemic equilibrium is established by constructing a Lyapunov function. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation, conditions for Hopf and Turing bifurcations are obtained. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Turing patterns select...
Uploads