A mathematical model for within-host Toxoplasma gondii invasion dynamics
Adam Sullivan Folashade Agusto Sharon Bewick Chunlei Su Suzanne Lenhart Xiaopeng Zhao
Mathematical Biosciences & Engineering 2012, 9(3): 647-662 doi: 10.3934/mbe.2012.9.647
Toxoplasma gondii (T. gondii) is a protozoan parasite that infects a wide range of intermediate hosts, including all mammals and birds. Up to 20% of the human population in the US and 30% in the world are chronically infected. This paper presents a mathematical model to describe intra-host dynamics of T. gondii infection. The model considers the invasion process, egress kinetics, interconversion between fast-replicating tachyzoite stage and slowly replicating bradyzoite stage, as well as the host's immune response. Analytical and numerical studies of the model can help to understand the influences of various parameters to the transient and steady-state dynamics of the disease infection.
keywords: intra-host dynamics. parasite infection stage conversion within-host invasion Toxoplasma gondii
Optimal control of vaccine distribution in a rabies metapopulation model
Erika Asano Louis J. Gross Suzanne Lenhart Leslie A. Real
Mathematical Biosciences & Engineering 2008, 5(2): 219-238 doi: 10.3934/mbe.2008.5.219
We consider an SIR metapopulation model for the spread of rabies in raccoons. This system of ordinary differential equations considers subpop- ulations connected by movement. Vaccine for raccoons is distributed through food baits. We apply optimal control theory to find the best timing for dis- tribution of vaccine in each of the linked subpopulations across the landscape. This strategy is chosen to limit the disease optimally by making the number of infections as small as possible while accounting for the cost of vaccination.
keywords: optimal control vaccine. epidemiology mathematical model rabies in raccoons
Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations
Peng Zhong Suzanne Lenhart
Evolution Equations & Control Theory 2013, 2(4): 749-769 doi: 10.3934/eect.2013.2.749
Integrodifference equations are discrete in time and continuous in space, and are used to model populations that are growing at discrete times, and dispersing spatially. A harvesting problem modeled by integrodifference equations involves three events: growth, dispersal and harvesting. The order of arranging the three events affects the optimized harvesting behavior. In this paper we investigate all six possible cases of orders of events, study the equivalences among them under certain conditions, and show how the six cases can be reduced to three cases.
keywords: Integrodifference equations population model order of events. harvesting optimal control
Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2014, 19(10): i-ii doi: 10.3934/dcdsb.2014.19.1i
Chris Cosner turned 60 on June 3, 2012 and now, at age 62, continues his stellar career at the interface of mathematics and biology. He received his Ph.D. in 1977 at the University of California, Berkeley under the direction of Murray Protter, winning the Bernard Friedman prize for the best dissertation in applied mathematics. From 1977 until 1982 he was on the faculty of Texas A&M University. In 1982 he left A&M to join the faculty of the Department of Mathematics of the University of Miami as Associate Professor, rising to the rank of Professor in 1988. The academic year 2013-2014 marked his 32nd year of distinguished service to the University of Miami and its research and pedagogical missions.

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Optimal control and stability analysis of an epidemic model with education campaign and treatment
Sanjukta Hota Folashade Agusto Hem Raj Joshi Suzanne Lenhart
Conference Publications 2015, 2015(special): 621-634 doi: 10.3934/proc.2015.0621
In this paper we investigated a SIR epidemic model in which education campaign and treatment are both important for the disease management. Optimal control theory was used on the system of differential equations to achieve the goal of minimizing the infected population and slow down the epidemic outbreak. Stability analysis of the disease free equilibrium of the system was completed. Numerical results with education campaign levels and treatment rates as controls are illustrated.
keywords: treatment and education. epidemic model stability analysis Optimal Control
Modeling the effect of information campaigns on the HIV epidemic in Uganda
Hem Joshi Suzanne Lenhart Kendra Albright Kevin Gipson
Mathematical Biosciences & Engineering 2008, 5(4): 757-770 doi: 10.3934/mbe.2008.5.757
The increasing prevalence of HIV/AIDS in Africa over the past twenty-five years continues to erode the continent's health care and overall welfare. There have been various responses to the pandemic, led by Uganda, which has had the greatest success in combating the disease. Part of Uganda's success has been attributed to a formalized information, education, and communication (IEC) strategy, lowering estimated HIV/AIDS infection rates from 18.5% in 1995 to 4.1% in 2003. We formulate a model to investigate the effects of information and education campaigns on the HIV epidemic in Uganda. These campaigns affect people's behavior and can divide the susceptibles class into subclasses with different infectivity rates. Our model is a system of ordinary differential equations and we use data about the epidemics and the number of organizations involved in the campaigns to estimate the model parameters. We compare our model with three types of susceptibles to a standard SIR model.
keywords: differential equations epidemic model HIV education campaign
Optimal control of integrodifference equations with growth-harvesting-dispersal order
Peng Zhong Suzanne Lenhart
Discrete & Continuous Dynamical Systems - B 2012, 17(6): 2281-2298 doi: 10.3934/dcdsb.2012.17.2281
Integrodifference equations are discrete in time and continuous in space, and are used to model the spread of populations that are growing in discrete generations, or at discrete times, and dispersing spatially. We investigate optimal harvesting strategies, in order to maximize the profit and minimize the cost of harvesting. Theoretical results on the existence, uniqueness and characterization, as well as numerical results of optimized harvesting rates are obtained. The order of how the three events, growth, dispersal and harvesting, are arranged affects the harvesting behavior.
keywords: order of events. optimal control Integrodifference equations population model harvesting
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Discrete & Continuous Dynamical Systems - B 2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i
The movement and dispersal of organisms have long been recognized as key components of ecological interactions and as such, they have figured prominently in mathematical models in ecology. More recently, dispersal has been recognized as an equally important consideration in epidemiology and in environmental science. Recognizing the increasing utility of employing mathematics to understand the role of movement and dispersal in ecology, epidemiology and environmental science, The University of Miami in December 2012 held a workshop entitled ``Everything Disperses to Miami: The Role of Movement and Dispersal in Ecology, Epidemiology and Environmental Science" (EDM).

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Management strategies in a malaria model combining human and transmission-blocking vaccines
Jemal Mohammed-Awel Ruijun Zhao Eric Numfor Suzanne Lenhart
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 977-1000 doi: 10.3934/dcdsb.2017049

We propose a new mathematical model studying control strategies of malaria transmission. The control is a combination of human and transmission-blocking vaccines and vector control (larvacide). When the disease induced death rate is large enough, we show the existence of a backward bifurcation analytically if vaccination control is not used, and numerically if vaccination is used. The basic reproduction number is a decreasing function of the vaccination controls as well as the vector control parameters, which means that any effort on these controls will reduce the burden of the disease. Numerical simulation suggests that the combination of the vaccinations and vector control may help to eradicate the disease. We investigate optimal strategies using the vaccinations and vector controls to gain qualitative understanding on how the combinations of these controls should be used to reduce disease prevalence in malaria endemic setting. Our results show that the combination of the two vaccination controls integrated with vector control has the highest impact on reducing the number of infected humans and mosquitoes.

keywords: Stability malaria transmission-blocking vaccine optimal control differential equations
Optimal control of advective direction in reaction-diffusion population models
Heather Finotti Suzanne Lenhart Tuoc Van Phan
Evolution Equations & Control Theory 2012, 1(1): 81-107 doi: 10.3934/eect.2012.1.81
We investigate optimal control of the advective coefficient in a class of parabolic partial differential equations, modeling a population with nonlinear growth. This work is motivated by the question: Does movement toward a better resource environment benefit a population? Our objective functional is formulated with interpreting "benefit" as the total population size integrated over our finite time interval. Results on existence, uniqueness, and characterization of the optimal control are established. Our numerical illustrations for several growth functions and resource functions indicate that movement along the resource spatial gradient benefits the population, meaning that the optimal control is close to the spatial gradient of the resource function.
keywords: advective direction population model. optimal control reaction-diffusion maximum principle Parabolic partial differential equations

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