# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

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## Mathematical Biosciences & Engineering

2013 , Volume 10 , Issue 5&6

Special issue dedicated to Carlos Castillo-Chavez on his 60th birthday

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2013, 10(5&6): xxix-xxxvii doi: 10.3934/mbe.2013.10.5xxix +[Abstract](1477) +[PDF](224.3KB)
Abstract:
Carlos Castillo-Chavez's tenure at Cornell University with Simon Levine, also marks the beginning of my life as his daughter. I was nine months old when I arrived to Ithaca, and my recollections of my father in elementary school and middle school were of him furiously writing equations at his desk, or outside on the chalk board in our shed, or on napkins, notepads or anything he could get his hands on at restaurants; but more likely than not, away. When I was young, my father was becoming the researcher that today makes him a three-time Presidential honoree, a member of Barak Obama's Presidential Committee on the National Medal of Science, and of course, the purpose of this volume. Even in those early days, he was away a lot -- either traveling to conferences or increasingly as an invited lecturer, or at the office. Of course, I was still (and am) a daddy's little girl, bonded forever by a shared obsession with the same movies (The Godfather, My Name is Nobody, The Man from Snowy River); the same TV shows (Law & Order); and all things sports related, but I also knew that my father was a very busy man and his time was limited. So I would watch him work, often with my own little extra homework he would give me to keep me entertained, peck him on the check and let him know that I would take over his job when I was old enough.

2013, 10(5&6): i-xxiv doi: 10.3934/mbe.2013.10.5i +[Abstract](2218) +[PDF](598.5KB)
Abstract:
Carlos Castilo-Chavez is a Regents Professor, a Joaquin Bustoz Jr. Professor of Mathematical Biology, and a Distinguished Sustainability Scientist at Arizona State University. His research program is at the interface of the mathematical and natural and social sciences with emphasis on (i) the role of dynamic social landscapes on disease dispersal; (ii) the role of environmental and social structures on the dynamics of addiction and disease evolution, and (iii) Dynamics of complex systems at the interphase of ecology, epidemiology and the social sciences. Castillo-Chavez has co-authored over two hundred publications (see goggle scholar citations) that include journal articles and edited research volumes. Specifically, he co-authored a textbook in Mathematical Biology in 2001 (second edition in 2012); a volume (with Harvey Thomas Banks) on the use of mathematical models in homeland security published in SIAM's Frontiers in Applied Mathematics Series (2003); and co-edited volumes in the Series Contemporary Mathematics entitled Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges'' (American Mathematical Society, 2006) and Mathematical and Statistical Estimation Approaches in Epidemiology (Springer-Verlag, 2009) highlighting his interests in the applications of mathematics in emerging and re-emerging diseases. Castillo-Chavez is a member of the Santa Fe Institute's external faculty, adjunct professor at Cornell University, and contributor, as a member of the Steering Committee of the Committee for the Review of the Evaluation Data on the Effectiveness of NSF-Supported and Commercially Generated Mathematics Curriculum Materials,'' to a 2004 NRC report. The CBMS workshop Mathematical Epidemiology with Applications'' lectures delivered by C. Castillo-Chavez and F. Brauer in 2011 have been published by SIAM in 2013.

2013, 10(5&6): xxv-xxvii doi: 10.3934/mbe.2013.10.5xxv +[Abstract](1794) +[PDF](148.6KB)
Abstract:
A little more than a quarter-century ago, I received an inquiry from a young Assistant Professor of Applied Mathematics at the University of Tulsa, the honoree of this volume, Carlos Castillo-Chavez. Though he was well situated in a faculty job, he was not satisfied: He was interested in mathematical biology, having written an excellent thesis in population biology with Fred Brauer at Wisconsin entitled Linear and Nonlinear Deterministic Character-Dependent Models with Time Delay in Population Dynamics. But that success had only whetted his appetite to become more deeply embedded in biology, and he was prepared to give up his faculty job to start a postdoctoral fellowship in ecology. It is always difficult to read in such letters what potential exists in the author; but there was something about what Carlos wrote, the obvious sacrifice he was prepared to make, and my regard for Fred Brauer that convinced me that I must meet this fellow. We did meet, for lunch in an LA restaurant, and the qualities that have led to his remarkable career were immediately obvious. I resolved on the spot to make sure he joined our group. Carlos arrived at Cornell shortly thereafter, and did not leave for nearly twenty years.

2013, 10(5&6): 1265-1279 doi: 10.3934/mbe.2013.10.1265 +[Abstract](1902) +[PDF](571.0KB)
Abstract:
The woodwasp Sirex noctilio is a major pest of pine plantations worldwide. Economically significant damage is however limited to outbreak populations. To understand what determines outbreaks dynamics in this species, we developed an individual based model for a wasp population developing within a pine plantation. We show that outbreaks may be the result of the insect's life history. Specifically we show that limited dispersal may not only increase population persistence but also create the conditions for eruptive dynamics. When the probability of long distance dispersal is greater than zero, but relatively small ($P_{LDD}$= 0.1) large outbreaks are the norm, with all of the suitable trees dead at the end of the simulation. For $P_{LDD}$= 0 (only local dispersal allowed) outbreaks are smaller in size, and in some cases not well defined and spread over longer periods. For $P_{LDD}$= 1 (only long distance dispersal allowed), the frequency of local population extinction (without outbreaks) increases significantly. Aggregated attacks may induce physiological changes in the trees which could allow other wasps to detect them. These changes may in turn trigger an outbreak. In contrast, healthy, vigorous trees are not suitable for wasp oviposition. In our model the density of suitable trees (healthy trees but yet suitable for oviposition) are a key factor determining population persistence before outbreaks. From an applied perspective, our results emphasize the importance of adequate plantation management in preventing woodwasp infestation.
2013, 10(5&6): 1281-1300 doi: 10.3934/mbe.2013.10.1281 +[Abstract](2252) +[PDF](1735.7KB)
Abstract:
Alcohol abuse is a major problem, especially among students on and around college campuses. We use the mathematical framework of [16] and study the role of environmental factors on the long term dynamics of an alcohol drinking population. Sensitivity and uncertainty analyses are carried out on the relevant functions (for example, on the drinking reproduction number and the extinction time of moderate and heavy drinking because of interventions) to understand the impact of environmental interventions on the distributions of drinkers. The reproduction number helps determine whether or not the high-risk alcohol drinking behavior will spread and become persistent in the population, whereas extinction time of high-risk drinking measures the effectiveness of control programs. We found that the reproduction number is most sensitive to social interactions, while the time to extinction of high-risk drinkers is significantly sensitive to the intervention programs that reduce initiation, and the college drop-out rate. The results also suggest that in a population, higher rates of intervention programs in low-risk environments (more than intervention rates in high-risk environments) are needed to reduce heavy drinking in the population.
2013, 10(5&6): 1301-1333 doi: 10.3934/mbe.2013.10.1301 +[Abstract](2334) +[PDF](842.1KB)
Abstract:
In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.
2013, 10(5&6): 1335-1349 doi: 10.3934/mbe.2013.10.1335 +[Abstract](2564) +[PDF](371.5KB)
Abstract:
A new model for the dynamics of cholera is formulated that incorporates both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is defined and proved to be a sharp threshold determining whether or not cholera dies out. Final size relations for cholera outbreaks are derived for simplified models when input and death are neglected.
2013, 10(5&6): 1351-1363 doi: 10.3934/mbe.2013.10.1351 +[Abstract](1744) +[PDF](274.6KB)
Abstract:
This article details the history, logistical operations, and design philosophy of the Mathematical and Theoretical Biology Institute (MTBI), a nationally recognized research program with an 18-year history of mentoring researchers at every level from high school through university faculty, increasing the number of researchers from historically underrepresented minorities, and motivating them to pursue research careers by allowing them to work on problems of interest to them and supporting them in this endeavor. This mosaic profile highlights how MTBI provides a replicable multi-level model for research mentorship.
2013, 10(5&6): 1365-1379 doi: 10.3934/mbe.2013.10.1365 +[Abstract](1543) +[PDF](515.0KB)
Abstract:
This study compares the effects of two types of metering (periodic resetting and periodic increments) on one variable in a dynamical system, relative to the behavior of the corresponding system with an equivalent level of constant recruitment (influx). While the level of the target population in the constant-influx system generally remains between the local extrema of the same population in the metered model, the same is not always true for other state variables in the system. These effects are illustrated by applications to models for chemotherapy dosing and for eating disorders in a school setting.
2013, 10(5&6): 1381-1398 doi: 10.3934/mbe.2013.10.1381 +[Abstract](2111) +[PDF](1631.5KB)
Abstract:
Prisoner's Dilemma is a game theory model used to describe altruistic behavior seen in various populations. This theoretical game is important in understanding why a seemingly selfish strategy does persist and spread throughout a population that is mixing homogeneously at random. For a population with structure determined by social interactions, Prisoner's Dilemma brings to light certain requirements for the altruistic strategy to become established. Monte Carlo simulations of Prisoner's Dilemma are carried out using both simulated social networks and a dataset of a real social network. In both scenarios we confirm the requirements for the persistence of altruism in a population.
2013, 10(5&6): 1399-1417 doi: 10.3934/mbe.2013.10.1399 +[Abstract](1775) +[PDF](999.7KB)
Abstract:
A discrete SIS epidemic model with the bilinear incidence depending on the new infection is formulated and studied. The condition for the global stability of the disease free equilibrium is obtained. The existence of the endemic equilibrium and its stability are investigated. More attention is paid to the existence of the saddle-node bifurcation, the flip bifurcation, and the Hopf bifurcation. Sufficient conditions for those bifurcations have been obtained. Numerical simulations are conducted to demonstrate our theoretical results and the complexity of the model.
2013, 10(5&6): 1419-1436 doi: 10.3934/mbe.2013.10.1419 +[Abstract](1886) +[PDF](469.6KB)
Abstract:
The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called linear determinacy'' conjecture equates full nonlinear model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading edge of the invasion. A survey that accounts for recent developments in the identification of conditions under which linear determinacy gives the right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions covered in this survey are also discussed.
2013, 10(5&6): 1437-1453 doi: 10.3934/mbe.2013.10.1437 +[Abstract](1813) +[PDF](3385.4KB)
Abstract:
The exponential is among the most important family functions in mathematics; the foundation for the solution of linear differential equations, linear difference equations, and stochastic processes. However there is little research and superficial agreement on how the concepts of exponential growth are learned and/or should be taught initially. In order to investigate these issues, I preformed a teaching experiment with two high school students, which focused on building understandings of exponential growth leading up to the (nonlinear) logistic differential equation model. In this paper, I highlight some of the ways of thinking used by participants in this teaching experiment. From these results I discuss how mathematicians using exponential growth routinely make use of multiple --- sometimes contradictory --- ways of thinking, as well as the danger that these multiple ways of thinking are not being made distinct to students.
2013, 10(5&6): 1455-1474 doi: 10.3934/mbe.2013.10.1455 +[Abstract](2956) +[PDF](693.5KB)
Abstract:
We use a stochastic simulation model to explore the effect of reactive intervention strategies during the 2002 dengue outbreak in the small population of Easter Island, Chile. We quantified the effect of interventions on the transmission dynamics and epidemic size as a function of the simulated control intensity levels and the timing of initiation of control interventions. Because no dengue outbreaks had been reported prior to 2002 in Easter Island, the 2002 epidemic provided a unique opportunity to estimate the basic reproduction number $R_0$ during the initial epidemic phase, prior to the start of control interventions. We estimated $R_0$ at $27.2$ ($95 \%$CI: $14.8$, $49.3$). We found that the final epidemic size is highly sensitive to the timing of start of interventions. However, even when the control interventions start several weeks after the epidemic onset, reactive intervention efforts can have a significant impact on the final epidemic size. Our results indicate that the rapid implementation of control interventions can have a significant effect in reducing the epidemic size of dengue epidemics.
2013, 10(5&6): 1475-1497 doi: 10.3934/mbe.2013.10.1475 +[Abstract](2716) +[PDF](638.2KB)
Abstract:
The spread of an infectious disease is sensitive to the contact patterns in the population and to precautions people take to reduce the transmission of the disease. We investigate the impact that different mixing assumptions have on the spread an infectious disease in an age-structured ordinary differential equation model. We consider the impact of heterogeneity in susceptibility and infectivity within the population on the disease transmission. We apply the analysis to the spread of a smallpox-like disease, derive the formula for the reproduction number, $\Re_{0}$, and based on this threshold parameter, show the level of human behavioral change required to control the epidemic. We analyze how different mixing patterns can affect the disease prevalence, the cumulative number of new infections, and the final epidemic size. Our analysis indicates that the combination of residual immunity and behavioral changes during a smallpox-like disease outbreak can play a key role in halting infectious disease spread; and that realistic mixing patterns must be included in the epidemic model for the predictions to accurately reflect reality.
2013, 10(5&6): 1499-1499 doi: 10.3934/mbe.2013.10.1499 +[Abstract](1346) +[PDF](117.0KB)
Abstract:
This volume, a tribute to the life and work of Carlos Castillo-Chavez on the occasion of his sixtieth birthday, stands for something greater than what can be presented in its pages. It is the man himself, who remains far above the words. The works collected here by people who know and admire Carlos are an attempt to convey some of his remarkable contributions and achievements.
I first met Carlos in 2005 as part of a visit he made to the National Science Foundation. Carlos had established a tremendous reputation by that time, and as a young program officer I was extremely pleased to have the opportunity to make small talk with him. He spoke about his students with an unmistakable passion and earnestness, and I wondered if this might be the key ingredient to his success in mentoring young scientists. In the years that followed I was able to see that Carlos channels his vitality into a remarkable capacity for work. The emails he sent to me over the years typically arrived at some odd hour of the night, perhaps suggesting a follow-up phone call for early the next morning. I'd call Carlos upon my arrival in the office and find him fully energized with the work of the new day.
The breadth of Carlos' impact is evident in the number of research and education communities that have honored and supported his work. At the National Science Foundation this includes programs spanning three Directorates: Mathematical and Physical Sciences, Biological Sciences, and Education and Human Resources. My interactions with Carlos came primarily in connection with his exceptional research programs for undergraduate and graduate students. His success in recruiting and mentoring underrepresented minorities in the mathematical sciences is almost without peer.
The founding of the Mathematical and Theoretical Biology Institute in 1996 and its subsequent development are rightly celebrated as landmarks in STEM education. What is remarkable is that this lifetime of work is just a fraction of what Carlos has accomplished!
His greatness lies in taking up the challenges that matter most. His most outstanding research contributions are grounded in stopping the spread of infectious diseases. His unparalleled mentoring of minority students is a reflection of his deep belief in the abundance of scientific talent.
Carlos, we in the mathematical sciences community offer you our deepest gratitude. Your legacy as a researcher and educator is already secure and it continues to grow. The immortality of your impact is assured by the timelessness of your research, and through the many alumni of your programs who are now making important contributions of their own. Best wishes for many more years of success!
2013, 10(5&6): 1501-1518 doi: 10.3934/mbe.2013.10.1501 +[Abstract](1595) +[PDF](491.2KB)
Abstract:
Chronic myeloid leukemia, a disorder of hematopoietic stem cells, is currently treated using targeted molecular therapy with imatinib. We compare two models that describe the treatment of CML, a multi-scale model (Model 1) and a simple cell competition model (Model 2). Both models describe the competition of leukemic and normal cells, however Model 1 also describes the dynamics of BCR-ABL, the oncogene targeted by imatinib, at the sub-cellular level. Using clinical data, we analyze the differences in estimated parameters between the models and the capacity for each model to predict drug resistance. We found that while both models fit the data well, Model 1 is more biologically relevant. The estimated parameter ranges for Model 2 are unrealistic, whereas the parameter ranges for Model 1 are close to values found in literature. We also found that Model 1 predicts long-term drug resistance from patient data, which is exhibited by both an increase in the proportion of leukemic cells as well as an increase in BCR-ABL/ABL%. Model 2, however, is not able to predict resistance and accurately model the clinical data. These results suggest that including sub-cellular mechanisms in a mathematical model of CML can increase the accuracy of parameter estimation and may help to predict long-term drug resistance.
2013, 10(5&6): 1519-1538 doi: 10.3934/mbe.2013.10.1519 +[Abstract](1836) +[PDF](645.3KB)
Abstract:
Spatially homogeneous (ODE) and reaction-diffusion models for plant-herbivore interactions with toxin-determined functional response are analyzed. The models include two plant species that have different levels of toxicity. The plant species with a higher level of toxicity is assumed to be less preferred by the herbivore and to have a relatively lower intrinsic growth rate than the less toxic plant species. Two of the equilibrium points of the system representing significant ecological interests are $E_1$, in which only the less toxic plant is present, and $E_2$, in which the more toxic plant and herbivore coexist while the less toxic plant has gone to extinction. Under certain conditions it is shown that, for the spatially homogeneous system all solutions will converge to the equilibrium $E_2$, whereas for the reaction-diffusion model there exist traveling wave solutions connecting $E_1$ and $E_2$.
2013, 10(5&6): 1539-1540 doi: 10.3934/mbe.2013.10.1539 +[Abstract](1359) +[PDF](122.6KB)
Abstract:
In AY 1994-95 the Alfred P. Sloan Foundation launched a program, later known as the Minority Ph.D. Program, to increase the number of underrepresented minority students earning Ph.D.s in natural sciences, engineering and mathematics (SEM). This program emerged from a recognition that African Americans, Hispanic Americans and American Indians were very underrepresented at all levels and in all aspects of SEM disciplines and that, although undergraduate education had received and continued to received much attention by universities, private funders and government agencies, there was still relatively little attention being paid to the graduate and especially the Ph.D. level. Because earning the Ph.D. is a necessary milestone along the pathway to a faculty position and the effort to diversify the graduates of SEM disciplines depends, in large part, on diversifying the faculty at American universities, this relative lack of attention to Ph.D. education was, in the opinion of the Sloan Foundation, a significant deficiency of national efforts. As a Program Director at the Sloan Foundation, I initiated this Minority Ph.D. Program and ran it until my retirement in June 2011.

2013, 10(5&6): 1541-1560 doi: 10.3934/mbe.2013.10.1541 +[Abstract](1882) +[PDF](400.9KB)
Abstract:
The Michaelis-Menten (MM) function is a fractional linear function depending on two positive parameters. These can be estimated by nonlinear or linear least squares methods. The non-linear methods, based directly on the defect of the MM function, can fail and not produce any minimizer. The linear methods always produce a unique minimizer which, however, may not be positive. Here we give sufficient conditions on the data such that the nonlinear problem has at least one positive minimizer and also conditions for the minimizer of the linear problem to be positive.
We discuss in detail the models and equilibrium relations of a classical operator-repressor system, and we extend our approach to the MM problem with leakage and to reversible MM kinetics. The arrangement of the sufficient conditions exhibits the important role of data that have a concavity property (chemically feasible data).
2013, 10(5&6): 1561-1586 doi: 10.3934/mbe.2013.10.1561 +[Abstract](2116) +[PDF](7579.0KB)
Abstract:
An appropriate choice of strategy for resource allocation may frequently determine whether a population will be able to survive under the conditions of severe resource limitations. Here we focus on two classes of strategies allocation of resources towards rapid proliferation, or towards slower proliferation but increased physiological and environmental maintenance. We propose a generalized framework, where individuals within a population can use either strategy in different proportion for utilization of a common dynamical resource in order to maximize their fitness. We use the model to address two major questions, namely, whether either strategy is more likely to be selected for as a result of natural selection, and, if one allows for the possibility of resource over-consumption, whether either strategy is preferable for avoiding population collapse due to resource exhaustion. Analytical and numerical results suggest that the ultimate choice of strategy is determined primarily by the initial distribution of individuals in the population, and that while investment in physiological and environmental maintenance is a preferable strategy in a homogeneous population, no generalized prediction can be made about heterogeneous populations.
2013, 10(5&6): 1587-1607 doi: 10.3934/mbe.2013.10.1587 +[Abstract](1614) +[PDF](273.3KB)
Abstract:
Mathematical models are well-established as metaphors for biological and epidemiological systems. The framework of epidemic modeling has also been applied to sociological phenomena driven by peer pressure, notably in two dozen dynamical systems research projects developed through the Mathematical and Theoretical Biology Institute, and popularized by authors such as Gladwell (2000). This article reviews these studies and their common structures, and identifies a new mathematical metaphor which uses multiple nonlinearities to describe the multiple thresholds governing the persistence of hierarchical phenomena, including the situation termed a backward bifurcation'' in mathematical epidemiology, where established phenomena can persist in circumstances under which the phenomena could not initially emerge.
2013, 10(5&6): 1609-1610 doi: 10.3934/mbe.2013.10.1609 +[Abstract](1521) +[PDF](125.0KB)
Abstract:
It is hard, now, to imagine a time when our math department's graduate program had no minority students. The diversity of our program has become so familiar to us that when, this spring and summer, seven of our minority students earned their doctoral degree it was hardly commented on. Indeed it was only when we began to miss these students -- students who were like family to us -- that the reality of this singular achievement manifested itself to us. Yet there was indeed a time when there were no minority graduate students in mathematics at the University of Iowa. In fact, only two minority students earned their doctoral degrees from our department from1974, when I joined the department, to 1998. And it is no exaggeration at all to state that, without the trust and support of Carlos Castillo-Chavez, it is unlikely that the transformation of our graduate program that took place over the past fifteen years would have occurred.

2013, 10(5&6): 1611-1613 doi: 10.3934/mbe.2013.10.1611 +[Abstract](1474) +[PDF](153.1KB)
Abstract:
When the opportunity to contribute a short essay about Dr. Carlos Castillo-Chavez presented itself in the context of this wonderful birthday celebration my immediate reaction was por supuesto que sí! Sixteen years ago, I travelled to Cornell University with my colleague at the National Security Agency (NSA) Barbara Deuink to meet Carlos and hear about his vision to expand the talent pool of mathematicians in our country. Our motivation was very simple. First of all, the Agency relies heavily on mathematicians to carry out its mission. If the U.S. mathematics community is not healthy, NSA is not healthy. Keeping our country safe requires a team of the sharpest minds in the nation to tackle amazing intellectual challenges on a daily basis. Second, the Agency cares deeply about diversity. Within the mathematical sciences, students with advanced degrees from the Chicano, Latino, Native American, and African-American communities are underrepresented. It was clear that addressing this issue would require visionary leadership and a long-term commitment. Carlos had the vision for a program that would provide promising undergraduates from minority communities with an opportunity to gain confidence and expertise through meaningful research experiences while sharing in the excitement of mathematical and scientific discovery. His commitment to the venture was unquestionable and that commitment has not waivered since the inception of the Mathematics and Theoretical Biology Institute (MTBI) in 1996.

2013, 10(5&6): 1615-1634 doi: 10.3934/mbe.2013.10.1615 +[Abstract](3331) +[PDF](592.1KB)
Abstract:
Optimal control strategies for controlling seasonal influenza transmission in the US are of high interest, because of the significant epidemiological and economic burden of influenza. To evaluate optimal strategies of vaccination and social distancing, we used an age-structured dynamic model of seasonal influenza. We applied optimal control theory to identify the best way of reducing morbidity and mortality at a minimal cost. In combination with the Pontryagins maximum principle, we calculated time-dependent optimal policies of vaccination and social distancing to minimize the epidemiological and economic burden associated with seasonal influenza. We computed optimal age-specific intervention strategies and analyze them under various costs of interventions and disease transmissibility. Our results show that combined strategies have a stronger impact on the reduction of the final epidemic size. Our results also suggest that the optimal vaccination can be achieved by allocating most vaccines to preschool-age children (age under five) followed by young adults (age 20-39) and school age children (age 6-19). We find that the optimal vaccination rates for all age groups are highest at the beginning of the outbreak, requiring intense effort at the early phase of an epidemic. On the other hand, optimal social distancing of clinical cases tends to last the entire duration of an outbreak, and its intensity is relatively equal for all age groups. Furthermore, with higher transmissibility of the influenza virus (i.e. higher R0), the optimal control strategy needs to include more efforts to increase vaccination rates rather than efforts to encourage social distancing. Taken together, public health agencies need to consider both the transmissibility of the virus and ways to encourage early vaccination as well as voluntary social distancing of symptomatic cases in order to determine optimal intervention strategies against seasonal influenza.
2013, 10(5&6): 1635-1650 doi: 10.3934/mbe.2013.10.1635 +[Abstract](1953) +[PDF](414.6KB)
Abstract:
In a chemostat, several species compete for the same nutrient, while in an epidemic, several strains of the same pathogen may compete for the same susceptible hosts. As winner, chemostat models predict the species with the lowest break-even concentration, while epidemic models predict the strain with the largest basic reproduction number. We show that these predictions amount to the same if the per capita functional responses of consumer species to the nutrient concentration or of infective individuals to the density of susceptibles are proportional to each other but that they are different if the functional responses are nonproportional. In the second case, the correct prediction is given by the break-even concentrations. In the case of nonproportional functional responses, we add a class for which the prediction does not only rely on local stability and instability of one-species (strain) equilibria but on the global outcome of the competition. We also review some results for nonautonomous models.
2013, 10(5&6): 1651-1668 doi: 10.3934/mbe.2013.10.1651 +[Abstract](1610) +[PDF](458.3KB)
Abstract:
A set of deterministic SIS models with density-dependent treatments are studied to understand the disease dynamics when different treatment strategies are applied. Qualitative analyses are carried out in terms of general treatment functions. It has become customary that a backward bifurcation leads to bistable dynamics. However, this study finds that finds that bistability may not be an option at all; the disease-free equilibrium could be globally stable when there is a backward bifurcation. Furthermore, when a backward bifurcation occurs, the fashion of bistability could be the coexistence of either dual stable equilibria or the disease-free equilibrium and a stable limit cycle. We also extend the formula for mean infection period from density-independent treatments to density-dependent ones. Finally, the modeling results are applied to the transmission of gonorrhea in China, suggesting that these gonorrhea patients may not seek medical treatments in a timely manner.
2013, 10(5&6): 1669-1672 doi: 10.3934/mbe.2013.10.1669 +[Abstract](1438) +[PDF](160.5KB)
Abstract:
The first two sections of this paper entitled The Conference and The History are taken directly from the webpage of the NSF sponsored Mathematical Sciences Institutes [1]. In the third section I share personal views.
2013, 10(5&6): 1673-1686 doi: 10.3934/mbe.2013.10.1673 +[Abstract](1834) +[PDF](707.8KB)
Abstract:
Treating HIV-infected individuals reduces their viral load, consequently increasing their survival time and decreasing their infectivity. It has been proposed that universal testing and treatment (i.e., universal test & treat'') could lead to HIV elimination and would be extremely cost-effective. It is now being debated whether to use a universal test and treat'' approach in the real-world'' as a prevention strategy to control HIV epidemics. However current modeling predictions of the impact, and cost-effectiveness, of universal test & treat'' strategies are based on an unrealistically short survival time for treated individuals. Here we use mathematical modeling and a longer, more realistic, survival time. We model the potential impact of a universal test & treat'' strategy in South Africa. Our results show that increasing the length of the survival time on treatment, although beneficial to individuals, reduces the probability of eliminating HIV and decreases the cost-effectiveness of using universal test & treat'' strategies. Therefore our results show that individual-level benefits and public health benefits will conflict when using test & treat'' strategies to reduce HIV transmission.
2013, 10(5&6): 1687-1689 doi: 10.3934/mbe.2013.10.1687 +[Abstract](1591) +[PDF](167.1KB)
Abstract:
We prove that Carlos is a Canadian, and document some important contribution of Carlos to the growth of mathematical epidemiology in Canada.
2013, 10(5&6): 1691-1701 doi: 10.3934/mbe.2013.10.1691 +[Abstract](2064) +[PDF](369.9KB)
Abstract:
A classical deterministic SIR model is modified to take into account of limited resources for diagnostic confirmation/medical isolation. We show that this modification leads to four different scenarios (instead of three scenarios in comparison with the SIR model) for optimal isolation strategies, and obtain analytic solutions for the optimal control problem that minimize the outbreak size under the assumption of limited resources for isolation. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for isolation (such as the number of intensive care units). With sufficient resources, the optimal control strategy is the normal Bang-Bang control. However, with limited resources the optimal control strategy requires to switch to time-variant isolation at an optimal rate proportional to the ratio of isolated cases over the entire infected population once the maximum capacity is reached.

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