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1551-0018
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Mathematical Biosciences & Engineering
2009 , Volume 6 , Issue 4
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2009, 6(4): 683-700
doi: 10.3934/mbe.2009.6.683
+[Abstract](776)
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Abstract:
In this paper we study a Lotka-Volterra predator-prey system with prey logistic growth under the telegraph noise. The telegraph noise switches at random two prey-predator models. The aim of this work is to determine the subset of omega-limit set of the system and show out the existence of a stationary distribution. We also focus on persistence of the predator and thus we look for conditions that allow persistence of the predator and prey community. We show that the asymptotic behaviour highly depends on the value of some constant $\lambda$ which is useful to make suitable predictions about the persistence of the system.
In this paper we study a Lotka-Volterra predator-prey system with prey logistic growth under the telegraph noise. The telegraph noise switches at random two prey-predator models. The aim of this work is to determine the subset of omega-limit set of the system and show out the existence of a stationary distribution. We also focus on persistence of the predator and thus we look for conditions that allow persistence of the predator and prey community. We show that the asymptotic behaviour highly depends on the value of some constant $\lambda$ which is useful to make suitable predictions about the persistence of the system.
2009, 6(4): 701-718
doi: 10.3934/mbe.2009.6.701
+[Abstract](842)
+[PDF](238.6KB)
Abstract:
We consider a selection mutation predator-prey model for the distribution of individuals with respect to an evolutionary trait. Local stability of the equilibria of this model is studied using the linearized stability principle and taking advantage of the (assumed) asymptotic stability of the equilibria of the resident population adopting an evolutionarily stable strategy.
We consider a selection mutation predator-prey model for the distribution of individuals with respect to an evolutionary trait. Local stability of the equilibria of this model is studied using the linearized stability principle and taking advantage of the (assumed) asymptotic stability of the equilibria of the resident population adopting an evolutionarily stable strategy.
2009, 6(4): 719-742
doi: 10.3934/mbe.2009.6.719
+[Abstract](791)
+[PDF](407.0KB)
Abstract:
Harvesting and predation occur through contact processes in which the rate at which the managed (prey) population can be found depends on the population size, usually saturating at high densities. Many models incorporate saturation in this process without considering the effects of the particular function used to describe it. We show that the sharpness with which this saturation occurs has an important effect upon the resulting population dynamics, with bistability (sometimes involving a stable equilibrium and a stable limit cycle) occurring for saturation that is any sharper than the commonly used Michaelis-Menten (Holling type II) functional response. This sharpness threshold occurs across a wide range of model types, from simple harvesting to density-dependent and ratio-dependent predation.
Harvesting and predation occur through contact processes in which the rate at which the managed (prey) population can be found depends on the population size, usually saturating at high densities. Many models incorporate saturation in this process without considering the effects of the particular function used to describe it. We show that the sharpness with which this saturation occurs has an important effect upon the resulting population dynamics, with bistability (sometimes involving a stable equilibrium and a stable limit cycle) occurring for saturation that is any sharper than the commonly used Michaelis-Menten (Holling type II) functional response. This sharpness threshold occurs across a wide range of model types, from simple harvesting to density-dependent and ratio-dependent predation.
2009, 6(4): 743-752
doi: 10.3934/mbe.2009.6.743
+[Abstract](973)
+[PDF](154.8KB)
Abstract:
For a time-delayed reaction-diffusion equation of age-structured single species population, the linear and nonlinear stability of the traveling wavefronts were proved by Gourley [4] and Li-Mei-Wong [8] respectively. The stability results, however, assume the delay-time is sufficiently small. We now prove that the wavefronts remain stable even when the delay-time is arbitrarily large. This essentially improves the previous stability results obtained in [4, 8]. Finally, we point out that this novel stability can be applied to the age-structured reaction-diffusion equation with a more general maturation rate.
For a time-delayed reaction-diffusion equation of age-structured single species population, the linear and nonlinear stability of the traveling wavefronts were proved by Gourley [4] and Li-Mei-Wong [8] respectively. The stability results, however, assume the delay-time is sufficiently small. We now prove that the wavefronts remain stable even when the delay-time is arbitrarily large. This essentially improves the previous stability results obtained in [4, 8]. Finally, we point out that this novel stability can be applied to the age-structured reaction-diffusion equation with a more general maturation rate.
2009, 6(4): 753-778
doi: 10.3934/mbe.2009.6.753
+[Abstract](645)
+[PDF](598.9KB)
Abstract:
Type 1 diabetes (T1DM) is a chronic autoimmune disease with a long prodrome, which is characterized by dysfunction and ultimately destruction of pancreatic $\beta$-cells. Because of the limited access to pancreatic tissue and pancreatic lymph nodes during the normoglycemic phase of the disease, little is known about the dynamics involved in the chain of events leading to the clinical onset of the disease in humans. In particular, during T1DM progression there is limited information about temporal fluctuations of immunologic abnormalities and their effect on pancreatic $\beta$-cell function and mass. Therefore, our understanding of the pathoetiology of T1DM relies almost entirely on studies in animal models of this disease. In an effort to elucidate important mechanisms that may play a critical role in the progression to overt disease, we propose a mathematical model that takes into account the dynamics of functional and dysfunctional $\beta$-cells, regulatory T cells, and pathogenic T cells. The model assumes that all individuals carrying susceptible HLA haplotypes will develop variable degrees of T1DM-related immunologic abnormalities. The results provide information about the concentrations and ratios of pathogenic T cells and regulatory T cells, the timing in which $\beta$-cells become dysfunctional, and how certain kinetic parameters affect the progression to T1DM. Our model is able to describe changes in the ratio of pathogenic T cells and regulatory T cells after the appearance of islet antibodies in the pancreas. Finally, we discuss the robustness of the model and its ability to assist experimentalists in designing studies to test complicated theories about the disease.
Type 1 diabetes (T1DM) is a chronic autoimmune disease with a long prodrome, which is characterized by dysfunction and ultimately destruction of pancreatic $\beta$-cells. Because of the limited access to pancreatic tissue and pancreatic lymph nodes during the normoglycemic phase of the disease, little is known about the dynamics involved in the chain of events leading to the clinical onset of the disease in humans. In particular, during T1DM progression there is limited information about temporal fluctuations of immunologic abnormalities and their effect on pancreatic $\beta$-cell function and mass. Therefore, our understanding of the pathoetiology of T1DM relies almost entirely on studies in animal models of this disease. In an effort to elucidate important mechanisms that may play a critical role in the progression to overt disease, we propose a mathematical model that takes into account the dynamics of functional and dysfunctional $\beta$-cells, regulatory T cells, and pathogenic T cells. The model assumes that all individuals carrying susceptible HLA haplotypes will develop variable degrees of T1DM-related immunologic abnormalities. The results provide information about the concentrations and ratios of pathogenic T cells and regulatory T cells, the timing in which $\beta$-cells become dysfunctional, and how certain kinetic parameters affect the progression to T1DM. Our model is able to describe changes in the ratio of pathogenic T cells and regulatory T cells after the appearance of islet antibodies in the pancreas. Finally, we discuss the robustness of the model and its ability to assist experimentalists in designing studies to test complicated theories about the disease.
2009, 6(4): 779-813
doi: 10.3934/mbe.2009.6.779
+[Abstract](1088)
+[PDF](574.6KB)
Abstract:
After two phases of AIDS control activities in India, the third phase of the National AIDS Control Programme (NACP III) was launched in July 2007. Our focus here is to predict the number of people living with HIV/AIDS (PLHA) in India so that the results can assist the NACP III planning team to determine appropriate targets to be activated during the project period (2007-2012). We have constructed a dynamical model that captures the mixing patterns between susceptibles and infectives in both low-risk and high-risk groups in the population. Our aim is to project the HIV estimates by taking into account general interventions for susceptibles and additional interventions, such as targeted interventions among high risk groups, provision of anti-retroviral therapy, and behavior change among HIV-positive individuals. Continuing the current level of interventions in NACP II, the model estimates there will be 5.06 million PLHA by the end of 2011. If 50 percent of the targets in NACP III are achieved by the end of the above period then about 0.8 million new infections will be averted in that year. The current status of the epidemic appears to be less severe compared to the trend observed in the late 1990s. The projections based on the second phase and the third phase of the NACP indicate prevention programmes which are directed towards the general and high-risk populations, and HIV-positive individuals will determine the decline or stabilization of the epidemic. Model based results are derived separately for the revised HIV estimates released in 2007. According to revised projections there will be 2.08 million PLHA by 2012 if 50 percent of the targets in NACP III are reached. We perform a Monte Carlo procedure for sensitivity analysis of parameters and model validation. We also predict a positive role of implementation of anti-retroviral therapy treatment of 90 percent of the eligible people in the country. We present methods for obtaining disease progression parameters using convolution approaches. We also extend our models to age-structured populations.
After two phases of AIDS control activities in India, the third phase of the National AIDS Control Programme (NACP III) was launched in July 2007. Our focus here is to predict the number of people living with HIV/AIDS (PLHA) in India so that the results can assist the NACP III planning team to determine appropriate targets to be activated during the project period (2007-2012). We have constructed a dynamical model that captures the mixing patterns between susceptibles and infectives in both low-risk and high-risk groups in the population. Our aim is to project the HIV estimates by taking into account general interventions for susceptibles and additional interventions, such as targeted interventions among high risk groups, provision of anti-retroviral therapy, and behavior change among HIV-positive individuals. Continuing the current level of interventions in NACP II, the model estimates there will be 5.06 million PLHA by the end of 2011. If 50 percent of the targets in NACP III are achieved by the end of the above period then about 0.8 million new infections will be averted in that year. The current status of the epidemic appears to be less severe compared to the trend observed in the late 1990s. The projections based on the second phase and the third phase of the NACP indicate prevention programmes which are directed towards the general and high-risk populations, and HIV-positive individuals will determine the decline or stabilization of the epidemic. Model based results are derived separately for the revised HIV estimates released in 2007. According to revised projections there will be 2.08 million PLHA by 2012 if 50 percent of the targets in NACP III are reached. We perform a Monte Carlo procedure for sensitivity analysis of parameters and model validation. We also predict a positive role of implementation of anti-retroviral therapy treatment of 90 percent of the eligible people in the country. We present methods for obtaining disease progression parameters using convolution approaches. We also extend our models to age-structured populations.
2009, 6(4): 815-837
doi: 10.3934/mbe.2009.6.815
+[Abstract](1226)
+[PDF](856.5KB)
Abstract:
Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proportion of individuals at risk for both diseases is not known. Here, we consider a highly simplified deterministic model that incorporates the joint dynamics of TB and HIV, a model that is quite hard to analyze. We compute independent reproductive numbers for TB ($\R_1$) and HIV ($\R_2$) and the overall reproductive number for the system, $\R =\max \{\R_1, \R_2\}$. The focus is naturally (given the highly simplified nature of the framework) on the qualitative analysis of this model. We find that if $\R <1$ then the disease-free equilibrium is locally asymptotically stable. The TB-only equilibrium $E_T$ is locally asymptotically stable if $\R_1>1$ and $\R_2<1$. However, the symmetric condition, $\R_1<1$ and $\R_2>1$, does not necessarily guarantee the stability of the HIV-only equilibrium $E_H$, and it is possible that TB can coexist with HIV when $\R_2>1$. In other words, in the case when $\R_1<1$ and $\R_2>1$ (or when $\R_1>1$ and $\R_2>1$), we are able to find a stable HIV/TB coexistence equilibrium. Moreover, we show that the prevalence level of TB increases with $\R_2>1$ under certain conditions. Through simulations, we find that i) the increased progression rate from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and ii) the increased progression rates from HIV to AIDS have not only increased the prevalence level of HIV while decreasing TB prevalence, but also generated damped oscillations in the system.
Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proportion of individuals at risk for both diseases is not known. Here, we consider a highly simplified deterministic model that incorporates the joint dynamics of TB and HIV, a model that is quite hard to analyze. We compute independent reproductive numbers for TB ($\R_1$) and HIV ($\R_2$) and the overall reproductive number for the system, $\R =\max \{\R_1, \R_2\}$. The focus is naturally (given the highly simplified nature of the framework) on the qualitative analysis of this model. We find that if $\R <1$ then the disease-free equilibrium is locally asymptotically stable. The TB-only equilibrium $E_T$ is locally asymptotically stable if $\R_1>1$ and $\R_2<1$. However, the symmetric condition, $\R_1<1$ and $\R_2>1$, does not necessarily guarantee the stability of the HIV-only equilibrium $E_H$, and it is possible that TB can coexist with HIV when $\R_2>1$. In other words, in the case when $\R_1<1$ and $\R_2>1$ (or when $\R_1>1$ and $\R_2>1$), we are able to find a stable HIV/TB coexistence equilibrium. Moreover, we show that the prevalence level of TB increases with $\R_2>1$ under certain conditions. Through simulations, we find that i) the increased progression rate from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and ii) the increased progression rates from HIV to AIDS have not only increased the prevalence level of HIV while decreasing TB prevalence, but also generated damped oscillations in the system.
2009, 6(4): 839-854
doi: 10.3934/mbe.2009.6.839
+[Abstract](1117)
+[PDF](545.1KB)
Abstract:
Rubella is a highly contagious childhood disease that causes relatively mild symptoms. However, rubella can result in severe congenital defects, known as congenital rubella syndrome (CRS), if transmitted from a mother to a fetus. Consequently, women have higher incentive to vaccinate against rubella than men do. Within the population vaccination reduces transmission but also increases the average age of infection and possibly the risk of CRS among unvaccinated females. To evaluate how the balance among these factors results in optimal coverage of vaccination, we developed a game theoretic age-structured epidemiological model of rubella transmission and vaccination. We found that high levels of vaccination for both genders are most effective in maximizing average utility across the population by decreasing the risk of CRS and reducing transmission of rubella. By contrast, the demands for vaccines driven by self-interest among males and females are $0\%$ and $100\%$ acceptance, respectively, if the cost of vaccination is relatively low. Our results suggest that the rubella vaccination by males that is likely to be achieved on voluntary basis without additional incentives would have been far lower than the population optimum, if rubella vaccine were offered separately instead of combined with measles and mumps vaccination as the MMR vaccine.
Rubella is a highly contagious childhood disease that causes relatively mild symptoms. However, rubella can result in severe congenital defects, known as congenital rubella syndrome (CRS), if transmitted from a mother to a fetus. Consequently, women have higher incentive to vaccinate against rubella than men do. Within the population vaccination reduces transmission but also increases the average age of infection and possibly the risk of CRS among unvaccinated females. To evaluate how the balance among these factors results in optimal coverage of vaccination, we developed a game theoretic age-structured epidemiological model of rubella transmission and vaccination. We found that high levels of vaccination for both genders are most effective in maximizing average utility across the population by decreasing the risk of CRS and reducing transmission of rubella. By contrast, the demands for vaccines driven by self-interest among males and females are $0\%$ and $100\%$ acceptance, respectively, if the cost of vaccination is relatively low. Our results suggest that the rubella vaccination by males that is likely to be achieved on voluntary basis without additional incentives would have been far lower than the population optimum, if rubella vaccine were offered separately instead of combined with measles and mumps vaccination as the MMR vaccine.
2009, 6(4): 855-871
doi: 10.3934/mbe.2009.6.855
+[Abstract](980)
+[PDF](483.9KB)
Abstract:
We study an eco-epidemic model with two trophic levels in which the dynamics are determined by predator-prey interactions as well as the vulnerability of the predator to a disease. Using the concept of generalized models we show that for certain classes of eco-epidemic models quasiperiodic and chaotic dynamics are generic and likely to occur. This result is based on the existence of bifurcations of higher codimension such as double Hopf bifurcations. We illustrate the emergence of chaotic behavior with one example system.
We study an eco-epidemic model with two trophic levels in which the dynamics are determined by predator-prey interactions as well as the vulnerability of the predator to a disease. Using the concept of generalized models we show that for certain classes of eco-epidemic models quasiperiodic and chaotic dynamics are generic and likely to occur. This result is based on the existence of bifurcations of higher codimension such as double Hopf bifurcations. We illustrate the emergence of chaotic behavior with one example system.
2009, 6(4): 873-887
doi: 10.3934/mbe.2009.6.873
+[Abstract](896)
+[PDF](337.1KB)
Abstract:
As obesity and its related health problems grow around the world, efforts to control and manage weight is increasing in importance. It is well known that altering and maintaining weight is problematic and this has led to specific studies trying to determine the cause of the difficulty. Recent research has identified that the body reacts to forced weight change by adapting individual total energy expenditure. Key factors are an adaptation of resting metabolic rate, non-exercise activity thermogenesis and dietary induced thermogenesis. We develop a differential equation model based on the first law of thermodynamics that incorporates all three adjustments along with natural age related reduction of the resting metabolic rate. Forward time simulations of the model compare well with mean data in both overfeeding and calorie restriction studies.
As obesity and its related health problems grow around the world, efforts to control and manage weight is increasing in importance. It is well known that altering and maintaining weight is problematic and this has led to specific studies trying to determine the cause of the difficulty. Recent research has identified that the body reacts to forced weight change by adapting individual total energy expenditure. Key factors are an adaptation of resting metabolic rate, non-exercise activity thermogenesis and dietary induced thermogenesis. We develop a differential equation model based on the first law of thermodynamics that incorporates all three adjustments along with natural age related reduction of the resting metabolic rate. Forward time simulations of the model compare well with mean data in both overfeeding and calorie restriction studies.
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