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Mathematical Biosciences & Engineering

2010 , Volume 7 , Issue 4

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Stability of a delay equation arising from a juvenile-adult model
Azmy S. Ackleh and Keng Deng
2010, 7(4): 729-737 doi: 10.3934/mbe.2010.7.729 +[Abstract](267) +[PDF](251.2KB)
Abstract:
We consider a delay equation that has been formulated from a juvenile-adult population model. We give respective conditions on the vital rates to ensure local stability of the positive equilibrium and global stability of the trivial equilibrium. We also show that under certain conditions the equation undergoes a Hopf bifurcation. We then study global asymptotic stability and present bifurcation diagrams for two special cases of the model.
Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation
Judy Day, Jonathan Rubin and Gilles Clermont
2010, 7(4): 739-763 doi: 10.3934/mbe.2010.7.739 +[Abstract](316) +[PDF](612.0KB)
Abstract:
Modulation of the inflammatory response has become a key focal point in the treatment of critically ill patients. Much of the computational work in this emerging field has been carried out with the goal of unraveling the primary drivers, interconnections, and dynamics of systemic inflammation. To translate these theoretical efforts into clinical approaches, the proper biological targets and specific manipulations must be identified. In this work, we pursue this goal by implementing a nonlinear model predictive control (NMPC) algorithm in the context of a reduced computational model of the acute inflammatory response to severe infection. In our simulations, NMPC successfully identifies patient-specific therapeutic strategies, based on simulated observations of clinically accessible inflammatory mediators, which outperform standardized therapies, even when the latter are derived using a general optimization routine. These results imply that a combination of computational modeling and NMPC may be of practical use in suggesting novel immuno-modulatory strategies for the treatment of intensive care patients.
A model of varicella-zoster reactivation
Jonathan E. Forde and Bailey Meeker
2010, 7(4): 765-777 doi: 10.3934/mbe.2010.7.765 +[Abstract](243) +[PDF](515.4KB)
Abstract:
Mathematical models have been used to study the dynamic interaction of many infectious diseases with the host's immune system. In this paper, we study Varicella Zoster Virus, which is responsible for chicken pox (varicella), and after a long period of latency, herpes zoster (shingles). After developing the model and demonstrating that is exhibits the type of periodic behavior necessary for long term latency and reactivation, we examine the implications of the model for vaccine booster programs aimed at preventing herpes zoster.
A model of drug resistance with infection by health care workers
Avner Friedman, Najat Ziyadi and Khalid Boushaba
2010, 7(4): 779-792 doi: 10.3934/mbe.2010.7.779 +[Abstract](281) +[PDF](393.9KB)
Abstract:
Antibiotic resistant organisms (ARO) pose an increasing serious threat in hospitals. One of the most life threatening ARO is methicillin-resistant staphylococcus aureus (MRSA). In this paper, we introduced a new mathematical model which focuses on the evolution of two bacterial strains, drug-resistant and non-drug resistant, residing within the population of patients and health care workers in a hospital. The model predicts that as soon as drug is administered, the average load of the non-resistant bacteria will decrease and eventually (after 6 weeks of the model's simulation) reach a very low level. However, the average load of drug-resistant bacteria will initially decrease, after treatment, but will later bounce back and remain at a high level. This level can be made lower if larger amount of drug is given or if the contact between health care workers and patients is reduced.
Minimal state models for ionic channels involved in glucagon secretion
Virginia González-Vélez, Amparo Gil and Iván Quesada
2010, 7(4): 793-807 doi: 10.3934/mbe.2010.7.793 +[Abstract](241) +[PDF](267.1KB)
Abstract:
Pancreatic alpha cells synthesize and release glucagon. This hormone along with insulin, preserves blood glucose levels within a physiological range. During low glucose levels, alpha cells exhibit electrical activity related to glucagon secretion. In this paper, we introduce minimal state models for those ionic channels involved in this electrical activity in mice alpha cells. For estimation of model parameters, we use Monte Carlo algorithms to fit steady-state channel currents. Then, we simulate dynamic ionic currents following experimental protocols. Our aims are 1) To understand the individual ionic channel functioning and modulation that could affect glucagon secretion, and 2) To simulate ionic currents actually measured in voltage-clamp alpha-cell experiments in mice. Our estimations indicate that alpha cells are highly permeable to sodium and potassium which mainly manage action potentials. We have also found that our estimated N-type calcium channel population and density in alpha cells is in good agreement to those reported for L-type calcium channels in beta cells. This finding is strongly relevant since both, L-type and N-type calcium channels, play a main role in insulin and glucagon secretion, respectively.
An application of queuing theory to SIS and SEIS epidemic models
Carlos M. Hernández-Suárez, Carlos Castillo-Chavez, Osval Montesinos López and Karla Hernández-Cuevas
2010, 7(4): 809-823 doi: 10.3934/mbe.2010.7.809 +[Abstract](393) +[PDF](405.0KB)
Abstract:
In this work we consider every individual of a population to be a server whose state can be either busy (infected) or idle (susceptible). This server approach allows to consider a general distribution for the duration of the infectious state, instead of being restricted to exponential distributions. In order to achieve this we first derive new approximations to quasistationary distribution (QSD) of SIS (Susceptible- Infected- Susceptible) and SEIS (Susceptible- Latent- Infected- Susceptible) stochastic epidemic models. We give an expression that relates the basic reproductive number, $R_0$ and the server utilization, $\rho$.
A stoichiometrically derived algal growth model and its global analysis
Xiong Li and Hao Wang
2010, 7(4): 825-836 doi: 10.3934/mbe.2010.7.825 +[Abstract](263) +[PDF](245.9KB)
Abstract:
Organisms are composed of multiple chemical elements such as carbon, nitrogen, and phosphorus. The scarcity of any of these elements can severely restrict organismal and population growth. However, many trophic interaction models only consider carbon limitation via energy flow. In this paper, we construct an algal growth model with the explicit incorporation of light and nutrient availability to characterize both carbon and phosphorus limitations. We provide a global analysis of this model to illustrate how light and nutrient availability regulate algal dynamics.
Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence
C. Connell McCluskey
2010, 7(4): 837-850 doi: 10.3934/mbe.2010.7.837 +[Abstract](401) +[PDF](191.8KB)
Abstract:
An SIR model with distributed delay and a general incidence function is studied. Conditions are given under which the system exhibits threshold behaviour: the disease-free equilibrium is globally asymptotically stable if R0<1 and globally attracting if R0=1; if R0>1, then the unique endemic equilibrium is globally asymptotically stable. The global stability proofs use a Lyapunov functional and do not require uniform persistence to be shown a priori. It is shown that the given conditions are satisfied by several common forms of the incidence function.
Time variations in the generation time of an infectious disease: Implications for sampling to appropriately quantify transmission potential
Hiroshi Nishiura
2010, 7(4): 851-869 doi: 10.3934/mbe.2010.7.851 +[Abstract](417) +[PDF](900.9KB)
Abstract:
Although the generation time of an infectious disease plays a key role in estimating its transmission potential, the impact of the sampling time of generation times on the estimation procedure has yet to be clarified. The present study defines the period and cohort generation times, both of which are time-inhomogeneous, as a function of the infection time of secondary and primary cases, respectively. By means of analytical and numerical approaches, it is shown that the period generation time increases with calendar time, whereas the cohort generation time decreases as the incidence increases. The initial growth phase of an epidemic of Asian influenza A (H2N2) in the Netherlands in 1957 was reanalyzed, and estimates of the basic reproduction number, $R_0$, from the Lotka-Euler equation were examined. It was found that the sampling time of generation time during the course of the epidemic introduced a time-effect to the estimate of $R_0$. Other historical data of a primary pneumonic plague in Manchuria in 1911 were also examined to help illustrate the empirical evidence of the period generation time. If the serial intervals, which eventually determine the generation times, are sampled during the course of an epidemic, direct application of the sampled generation-time distribution to the Lotka-Euler equation leads to a biased estimate of $R_0$. An appropriate quantification of the transmission potential requires the estimation of the cohort generation time during the initial growth phase of an epidemic or adjustment of the time-effect (e.g., adjustment of the growth rate of the epidemic during the sampling time) on the period generation time. A similar issue also applies to the estimation of the effective reproduction number as a function of calendar time. Mathematical properties of the generation time distribution in a heterogeneously mixing population need to be clarified further.
A two-sex model for the influence of heavy alcohol consumption on the spread of HIV/AIDS
Gigi Thomas and Edward M. Lungu
2010, 7(4): 871-904 doi: 10.3934/mbe.2010.7.871 +[Abstract](294) +[PDF](2375.6KB)
Abstract:
The HIV/AIDS epidemic, one of the leading public health problems to have affected sub-Sahara Africa, is a multifaceted problem with social, behavioral and biological aspects. In the absence of a cure, behavioral change has been advocated as an intervention strategy for reversing the epidemic. Empirical studies have found heavy alcohol consumption to be a fueling factor for HIV/AIDS infection and progression. Previously [20], we formulated and analyzed a one-sex deterministic model to capture the dynamics of this deadly interaction. But, since alcohol drinking habits, consequent risky sexual practices, alcohol-induced immune suppression, etc., can be different for men and women, the primary objective of our present paper is to construct a two-sex model aimed at shedding light on how both sexes, with varying heavy alcohol consumption trends, contribute differently to the HIV/AIDS spread. Based on numerical simulations, supported by the UNAIDS epidemiological software SPECTRUM and using the available data, our study identifies heavy drinking among men and women to be a major driving force for HIV/AIDS in Botswana and sub-Sahara Africa and quantifies its hazardous outcomes in terms of increased number of active TB cases and economic burden caused by increased need for AntiRetroviral Therapy (ART). Our simulations point to the heavy-drinking habits of men as a major reason for the continuing disproportionate impact of HIV/AIDS on women in sub-Sahara Africa. Our analysis has revealed the possibility of the phenomenon of backward bifurcation. In contrast to the result in some HIV vaccination models [52], backward bifurcation in our model is not removed by replacing the corresponding standard incidence function with a mass action incidence, but is removed by merging the two susceptible classes of the same sex into one, i.e., by ignoring acquisition of, and ongoing recovery from, heavy-drinking habits among the susceptible population.
An elementary approach to modeling drug resistance in cancer
Cristian Tomasetti and Doron Levy
2010, 7(4): 905-918 doi: 10.3934/mbe.2010.7.905 +[Abstract](299) +[PDF](188.1KB)
Abstract:
Resistance to drugs has been an ongoing obstacle to a successful treatment of many diseases. In this work we consider the problem of drug resistance in cancer, focusing on random genetic point mutations. Most previous works on mathematical models of such drug resistance have been based on stochastic methods. In contrast, our approach is based on an elementary, compartmental system of ordinary differential equations. We use our very simple approach to derive results on drug resistance that are comparable to those that were previously obtained using much more complex mathematical techniques. The simplicity of our model allows us to obtain analytic results for resistance to any number of drugs. In particular, we show that the amount of resistance generated before the start of the treatment, and present at some given time afterward, always depends on the turnover rate, no matter how many drugs are simultaneously used in the treatment.
Rotating antibiotics does not minimize selection for resistance
Sebastian Bonhoeffer, Pia Abel zur Wiesch and Roger D. Kouyos
2010, 7(4): 919-922 doi: 10.3934/mbe.2010.7.919 +[Abstract](299) +[PDF](200.0KB)
Abstract:
N/A
Antibiotic cycling versus mixing: The difficulty of using mathematical models to definitively quantify their relative merits
Robert E. Beardmore and Rafael Peña-Miller
2010, 7(4): 923-933 doi: 10.3934/mbe.2010.7.923 +[Abstract](381) +[PDF](534.8KB)
Abstract:
We ask the question Which antibiotic deployment protocols select best against drug-resistant microbes: mixing or periodic cycling? and demonstrate that the statistical distribution of the performances of both sets of protocols, mixing and periodic cycling, must have overlapping supports. In other words, it is a general, mathematical result that there must be mixing policies that outperform cycling policies and vice versa.
   As a result, we agree with the tenet of Bonhoefer et al. [1] that one should not apply the results of [2] to conclude that an antibiotic cycling policy that implements cycles of drug restriction and prioritisation on an ad-hoc basis can select against drug-resistant microbial pathogens in a clinical setting any better than random drug use. However, nor should we conclude that a random, per-patient drug-assignment protocol is the de facto optimal method for allocating antibiotics to patients in any general sense.

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