Foraging strategies in social animals are often shaped by change in an organism's natural surrounding. Foraging behavior can hence be highly plastic, time, and condition dependent. The motivation of my research is to explore the effects of dispersal behavior in predators or parasites on population dynamics in heterogeneous environments by developing varied models in different contexts through closely working with ecologists. My models include Ordinary Differential Equation (ODE)-type meta population models and Delay Differential Equation (DDE) models with validation through data. I applied dynamical theory and bifurcation theory with carefully designed numerical simulations to have a better understanding on the profitability and cost of an adaptive dispersal in organisms. My work on the prey-predator models provide important insights on how different dispersal strategies may have different impacts on the spatial patterns and also shows that the change of dispersal strategy in organisms may have stabilizing or destabilizing effects leading to extinction or coexistence of species. I also develop models for honeybee population dynamics and its interaction with the parasitic Varroa mite. At first, I investigate the effect of dispersal on honeybee colonies under infestation by the Varroa mites. I then provide another single patch model by considering a stage structure time delay system from brood to adult honeybee. Through a close collaboration with a biologist, a honeybee and mite population data was first used to validate my model and I estimated certain unknown parameters by utilizing least square Monte Carlo method. My analytical, bifurcations, sensitivity analysis, and numerical studies first reveal the dynamical outcomes of migration. In addition, the results point us in the direction of the most sensitive life history parameters affecting the population size of a colony. These results provide novel insights on the effects of foraging and Varroa mites on colony survival.

Head and neck squamous cell carcinoma (HNSCC), the sixth most common cancer

type worldwide, accounts for more than 630,000 new cases and 350,000 deaths

annually. Drug-resistance and tumor recurrence are the most challenging problems

in head and neck cancer treatment. It is hypothesized that a very small fraction

of stem-like cells within HNSCC tumor, called cancer stem cells (CSCs), is

responsible for tumor initiation, progression, resistance and recurrence. It has also

been shown that IL-6 secreted by head and neck tumor-associated endothelial cells

(ECs) enhances the survival, self-renewal and tumorigenic potential of head and

neck CSCs. In this study we will use a mathematical multi-scale model which operates

at the intracellular, molecular, and tissue level to investigate the impacts of

EC-secreted IL-6 signaling on the crosstalk between tumor cells and ECs during

tumor growth. This model will be calibrated by using the experimental in vivo

data.

Eventually the model will be modified to explore the responses of head and neck

cancer cells to combination therapy involving Tocilizumab (an anti-IL-6R antibody)

and Cisplatin (the most frequently used chemotherapy for head and neck

cancer). The model will be able to predict the final proportion of CSCs in response

to endothelial cell-secreted IL-6 and drug therapies. The model will be validated

by directly comparing the experimental treatment data and the model predictions.

This could potentially provide a condition under which we could control enlargement

of the head and neck CSC pool and tumor recurrence. It may also suggest

the best bounds for Cisplatin and/or Tocilizumab dose and frequency to be tested

in the clinical trial.

This dissertation will look at large scale collaboration through the lens of online communities to answer questions about what makes a collaboration persist. Results address how collaborations attract contributions, behaviors that could give rise to patterns seen in the data, and the properties of collaborations that drive those behaviors.

It is understood that collaborations, online and otherwise, must retain users to remain productive. However, before users can be retained they must be recruited. In the first project, a few necessary properties of the ``attraction'' function are identified by constraining the dynamics of an ODE (Ordinary Differential Equation) model. Additionally, more than 100 communities of the Stack Exchange networks are parameterized and their distributions reported.

Collaborations do not exist in a vacuum, they compete with and share users with other collaborations. To address this, the second project focuses on an agent-based model (ABM) of a community of online collaborations using a mechanistic approach. The ABM is compared to data obtained from the Stack Exchange network and produces similar distributional patterns.

The third project is a thorough sensitivity analysis of the model created in the second project. A variance based sensitivity analysis is performed to evaluate the relative importance of 21 parameters of the model. Results indicate that population parameters impact many outcome metrics, though even those parameters that tend towards a low impact can be crucial for some outcomes.

A two-patch mathematical model of Dengue virus type 2 (DENV-2) that accounts for vectors’ vertical transmission and between patches human dispersal is introduced. Dispersal is modelled via a Lagrangian approach. A host-patch residence-times basic reproduction number is derived and conditions under which the disease dies out or persists are established. Analytical and numerical results highlight the role of hosts’ dispersal in mitigating or exacerbating disease dynamics. The framework is used to explore dengue dynamics using, as a starting point, the 2002 outbreak in the state of Colima, Mexico.

This dissertation discusses the Cournot competition and competitions in the exploitation of common pool resources and its extension to the tragedy of the commons. I address these models by using potential games and inquire how these models reflect the real competitions for provisions of environmental resources. The Cournot models are dependent upon how many firms there are so that the resultant Cournot-Nash equilibrium is dependent upon the number of firms in oligopoly. But many studies do not take into account how the resultant Cournot-Nash equilibrium is sensitive to the change of the number of firms. Potential games can find out the outcome when the number of firms changes in addition to providing the "traditional" Cournot-Nash equilibrium when the number of firms is fixed. Hence, I use potential games to fill the gaps that exist in the studies of competitions in oligopoly and common pool resources and extend our knowledge in these topics. In specific, one of the rational conclusions from the Cournot model is that a firm's best policy is to split into separate firms. In real life, we usually witness the other way around; i.e., several firms attempt to merge and enjoy the monopoly profit by restricting the amount of output and raising the price. I aim to solve this conundrum by using potential games. I also clarify, within the Cournot competition model, how regulatory intervention in the management of environmental pollution externalities affects the equilibrium number of polluters. In addition, the tragedy of the commons is the term widely used to describe the overexploitation of open-access common-pool resources. Open-access encourages potential resource users to continue to enter the resource up to the point where rents are exhausted. The resulting level of resource use is higher than is socially optimal, and in extreme cases can lead to the collapse of the resource and the communities that may depend on it. In this paper I use the concept of potential games to evaluate the relation between the cost of resource use and the equilibrium number of resource users in open access regimes. I find that costs of access and costs of production are sufficient to determine the equilibrium number of resource users, and that there is in fact a continuum between Cournot competition and the tragedy of the commons. I note that the various common pool resource management regimes identified in the empirical literature are associated with particular cost structures, and hence that this may be the mechanism that determines the number of resource users accessing the resource.

Background
In the weeks following the first imported case of Ebola in the U. S. on September 29, 2014, coverage of the very limited outbreak dominated the news media, in a manner quite disproportionate to the actual threat to national public health; by the end of October, 2014, there were only four laboratory confirmed cases of Ebola in the entire nation. Public interest in these events was high, as reflected in the millions of Ebola-related Internet searches and tweets performed in the month following the first confirmed case. Use of trending Internet searches and tweets has been proposed in the past for real-time prediction of outbreaks (a field referred to as “digital epidemiology”), but accounting for the biases of public panic has been problematic. In the case of the limited U. S. Ebola outbreak, we know that the Ebola-related searches and tweets originating the U. S. during the outbreak were due only to public interest or panic, providing an unprecedented means to determine how these dynamics affect such data, and how news media may be driving these trends.
Methodology
We examine daily Ebola-related Internet search and Twitter data in the U. S. during the six week period ending Oct 31, 2014. TV news coverage data were obtained from the daily number of Ebola-related news videos appearing on two major news networks. We fit the parameters of a mathematical contagion model to the data to determine if the news coverage was a significant factor in the temporal patterns in Ebola-related Internet and Twitter data.
Conclusions
We find significant evidence of contagion, with each Ebola-related news video inspiring tens of thousands of Ebola-related tweets and Internet searches. Between 65% to 76% of the variance in all samples is described by the news media contagion model.

Background
Seroepidemiological studies before and after the epidemic wave of H1N1-2009 are useful for estimating population attack rates with a potential to validate early estimates of the reproduction number, R, in modeling studies.
Methodology/Principal Findings
Since the final epidemic size, the proportion of individuals in a population who become infected during an epidemic, is not the result of a binomial sampling process because infection events are not independent of each other, we propose the use of an asymptotic distribution of the final size to compute approximate 95% confidence intervals of the observed final size. This allows the comparison of the observed final sizes against predictions based on the modeling study (R = 1.15, 1.40 and 1.90), which also yields simple formulae for determining sample sizes for future seroepidemiological studies. We examine a total of eleven published seroepidemiological studies of H1N1-2009 that took place after observing the peak incidence in a number of countries. Observed seropositive proportions in six studies appear to be smaller than that predicted from R = 1.40; four of the six studies sampled serum less than one month after the reported peak incidence. The comparison of the observed final sizes against R = 1.15 and 1.90 reveals that all eleven studies appear not to be significantly deviating from the prediction with R = 1.15, but final sizes in nine studies indicate overestimation if the value R = 1.90 is used.
Conclusions
Sample sizes of published seroepidemiological studies were too small to assess the validity of model predictions except when R = 1.90 was used. We recommend the use of the proposed approach in determining the sample size of post-epidemic seroepidemiological studies, calculating the 95% confidence interval of observed final size, and conducting relevant hypothesis testing instead of the use of methods that rely on a binomial proportion.

Background
Several past studies have found that media reports of suicides and homicides appear to subsequently increase the incidence of similar events in the community, apparently due to the coverage planting the seeds of ideation in at-risk individuals to commit similar acts.
Methods
Here we explore whether or not contagion is evident in more high-profile incidents, such as school shootings and mass killings (incidents with four or more people killed). We fit a contagion model to recent data sets related to such incidents in the US, with terms that take into account the fact that a school shooting or mass murder may temporarily increase the probability of a similar event in the immediate future, by assuming an exponential decay in contagiousness after an event.
Conclusions
We find significant evidence that mass killings involving firearms are incented by similar events in the immediate past. On average, this temporary increase in probability lasts 13 days, and each incident incites at least 0.30 new incidents (p = 0.0015). We also find significant evidence of contagion in school shootings, for which an incident is contagious for an average of 13 days, and incites an average of at least 0.22 new incidents (p = 0.0001). All p-values are assessed based on a likelihood ratio test comparing the likelihood of a contagion model to that of a null model with no contagion. On average, mass killings involving firearms occur approximately every two weeks in the US, while school shootings occur on average monthly. We find that state prevalence of firearm ownership is significantly associated with the state incidence of mass killings with firearms, school shootings, and mass shootings.

Background
The transmission dynamics of Tuberculosis (TB) involve complex epidemiological and socio-economical interactions between individuals living in highly distinct regional conditions. The level of exogenous reinfection and first time infection rates within high-incidence settings may influence the impact of control programs on TB prevalence. The impact that effective population size and the distribution of individuals’ residence times in different patches have on TB transmission and control are studied using selected scenarios where risk is defined by the estimated or perceive first time infection and/or exogenous re-infection rates.
Methods
This study aims at enhancing the understanding of TB dynamics, within simplified, two patch, risk-defined environments, in the presence of short term mobility and variations in reinfection and infection rates via a mathematical model. The modeling framework captures the role of individuals’ ‘daily’ dynamics within and between places of residency, work or business via the average proportion of time spent in residence and as visitors to TB-risk environments (patches). As a result, the effective population size of Patch i (home of i-residents) at time t must account for visitors and residents of Patch i, at time t.
Results
The study identifies critical social behaviors mechanisms that can facilitate or eliminate TB infection in vulnerable populations. The results suggest that short-term mobility between heterogeneous patches contributes to significant overall increases in TB prevalence when risk is considered only in terms of direct new infection transmission, compared to the effect of exogenous reinfection. Although, the role of exogenous reinfection increases the risk that come from large movement of individuals, due to catastrophes or conflict, to TB-free areas.
Conclusions
The study highlights that allowing infected individuals to move from high to low TB prevalence areas (for example via the sharing of treatment and isolation facilities) may lead to a reduction in the total TB prevalence in the overall population. The higher the population size heterogeneity between distinct risk patches, the larger the benefit (low overall prevalence) under the same “traveling” patterns. Policies need to account for population specific factors (such as risks that are inherent with high levels of migration, local and regional mobility patterns, and first time infection rates) in order to be long lasting, effective and results in low number of drug resistant cases.

A key factor in the success of social animals is their organization of work. Mathematical models have been instrumental in unraveling how simple, individual-based rules can generate collective patterns via self-organization. However, existing models offer limited insights into how these patterns are shaped by behavioral differences within groups, in part because they focus on analyzing specific rules rather than general mechanisms that can explain behavior at the individual-level. My work argues for a more principled approach that focuses on the question of how individuals make decisions in costly environments.

In Chapters 2 and 3, I demonstrate how this approach provides novel insights into factors that shape the flexibility and robustness of task organization in harvester ant colonies (Pogonomyrmex barbatus). My results show that the degree to which colonies can respond to work in fluctuating environments depends on how individuals weigh the costs of activity and update their behavior in response to social information. In Chapter 4, I introduce a mathematical framework to study the emergence of collective organization in heterogenous groups. My approach, which is based on the theory of multi-agent systems, focuses on myopic agents whose behavior emerges out of an independent valuation of alternative choices in a given work environment. The product of this dynamic is an equilibrium organization in which agents perform different tasks (or abstain from work) with an analytically defined set of threshold probabilities. The framework is minimally developed, but can be extended to include other factors known to affect task decisions including individual experience and social facilitation. This research contributes a novel approach to developing (and analyzing) models of task organization that can be applied in a broader range of contexts where animals cooperate.