Agent-Based Modeling and Simulation in Mathematics and Biology Education.

With advances in computing, agent-based models (ABMs) have become a feasible and appealing tool to study biological systems. ABMs are seeing increased incorporation into both the biology and mathematics classrooms as powerful modeling tools to study processes involving substantial amounts of stochasticity, nonlinear interactions, and/or heterogeneous spatial structures. Here we present a brief synopsis of the agent-based modeling approach with an emphasis on its use to simulate biological systems, and provide a discussion of its role and limitations in both the biology and mathematics classrooms.

The potential impact of a prophylactic vaccine for Ebola in Sierra Leone.

The 2014 outbreak of Ebola virus disease (EVD) in West Africa was multinational and of an unprecedented scale primarily affecting the countries of Guinea, Liberia, and Sierra Leone. One of the qualities that makes EVD of high public concern is its potential for extremely high mortality rates (up to 90%). A prophylactic vaccine for ebolavirus (rVSV-ZEBOV) has been developed, and clinical trials show near-perfect efficacy. We have developed an ordinary differential equations model that simulates an EVD epidemic and takes into account (1) transmission through contact with infectious EVD individuals and deceased EVD bodies, (2) the heterogeneity of the risk of becoming infected with EVD, and (3) the increased survival rate of infected EVD patients due to greater access to trained healthcare providers. Using fitted parameter values that closely simulate the dynamics of the 2014 outbreak in Sierra Leone, we utilize our model to predict the potential impact of a prophylactic vaccine for the ebolavirus using various vaccination strategies including ring vaccination. Our results show that an rVSV-ZEBOV vaccination coverage as low as 40% in the general population and 95% in healthcare workers will prevent another catastrophic outbreak like the 2014 outbreak from occurring.

A proton therapy model using discrete difference equations with an example of treating hepatocellular carcinoma.

Proton therapy is a type of radiation therapy used to treat cancer. It provides more localized particle exposure than other types of radiotherapy (e.g., x-ray and electron) thus reducing damage to tissue surrounding a tumor and reducing unwanted side effects. We have developed a novel discrete difference equation model of the spatial and temporal dynamics of cancer and healthy cells before, during, and after the application of a proton therapy treatment course. Specifically, the model simulates the growth and diffusion of the cancer and healthy cells in and surrounding a tumor over one spatial dimension (tissue depth) and the treatment of the tumor with discrete bursts of proton radiation. We demonstrate how to use data from in vitro and clinical studies to parameterize the model. Specifically, we use data from studies of Hepatocellular carcinoma, a common form of liver cancer. Using the parameterized model we compare the ability of different clinically used treatment courses to control the tumor. Our results show that treatment courses which use conformal proton therapy (targeting the tumor from multiple angles) provides better control of the tumor while using lower treatment doses than a non-conformal treatment course, and thus should be recommend for use when feasible.

Evaluating the potential impact of vaginal microbicides to reduce the risk of acquiring HIV in female sex workers.

OBJECTIVES: The following questions were addressed: would the introduction of vaginal microbicides substantially reduce the risk of female sex workers (FSWs) acquiring HIV? Which factor would it be most important to maximize, microbicide efficacy or microbicide use? What level of microbicide efficacy and use would be necessary to counterbalance a possible reduction in condom use? DESIGN: Mathematical modeling, with parameter estimations from available literature. METHODS: Risk equations were developed and Monte Carlo simulations were performed to model a FSW's daily risk of HIV acquisition currently, and after, microbicide introduction. Uncertainty and sensitivity analyses were used as well as tornado plots for two ranges of microbicide efficacy (30-50%) and (50-80%). Risk was estimated for FSWs whose clients sometimes (10-50%) use condoms, and those whose clients never use condoms. An analytical threshold for which reducing condom use increases risk was estimated. RESULTS: For both groups of FSWs, daily risk would decrease by approximately 17% or approximately 28% using 30-50% or 50-80% effective microbicides, respectively. Increasing microbicide use would have greater impact on reducing risk than increasing microbicide efficacy. The microbicide efficacy and usage required to ensure that 'condom replacement' does not increase a FSW's risk of acquiring HIV was calculated. CONCLUSIONS: Microbicides could substantially reduce FSWs' risk of acquiring HIV; absolute decrease in risk would be greatest in high-prevalence regions. The public health impact of microbicides will depend upon usage and efficacy. Even if the microbicides that become available are only low-to-moderately effective, the probability that risk in FSWs will increase (due to replacing condoms with microbicides) is low.

Predicting the potential public health impact of disease-modifying HIV vaccines in South Africa: the problem of subtypes.

Current HIV vaccines in development appear unlikely to prevent infection, but could provide benefits by increasing survival; such vaccines are described as disease-modifying vaccines. We review the current status of vaccines and modeling vaccines. We also predict the impact that disease-modifying vaccines could have in South Africa, where multiple subtypes are co-circulating. We model transmissibility/fitness differences among subtypes. We used uncertainty analyses to model vaccines with four characteristics: (i) take, (ii) duration of immunity, (iii) reduction in transmissibility/fitness, and (iv) increase in survival. We reconstructed, and forecasted, the South African epidemic from 1940 to 2140 (assuming no vaccination). We predict that: (i) incidence will peak in 2014, decline, and stabilize, (ii) prevalence will continue to rise, and (iii) the AIDS death rate curve will peak in 2022. Our predictions show that (over the next 135 years) the epidemic in South Africa will switch from a predominantly Subtype C epidemic to an epidemic driven by other subtypes. We predict that the epidemic could remain unchanged, even with mass vaccination with a vaccine that is equally effective against all co-circulating subtypes. However, if the non-C subtypes are less (or equally) transmissible as Subtype C then disease-modifying vaccines could result in eradication. Thus, in countries where multiple-subtypes are co-circulating it is critical to realize that small biological differences among subtypes will have dramatic consequences for the effectiveness of HIV vaccination campaigns. A slight difference in fitness will determine whether a disease-modifying vaccine has almost no impact on the epidemic or can achieve eradication.

Order of events matter: comparing discrete models for optimal control of species augmentation.

We investigate optimal timing of augmentation of an endangered/threatened species population in a target region by moving individuals from a reserve or captive population. This is formulated as a discrete-time optimal control problem in which augmentation occurs once per time period over a fixed number of time periods. The population model assumes the Allee effect growth functions in both target and reserve populations and the control objective is to maximize the target and reserve population sizes over the time horizon while accounting for costs of augmentation. Two possible orders of events are considered for different life histories of the species relative to augmentation time: move individuals either before or after population growth occurs. The control variable is the proportion of the reserve population to be moved to the target population. We develop solutions and illustrate numerical results which indicate circumstances for which optimal augmentation strategies depend upon the order of events.

Evolutionary dynamics of complex networks of HIV drug-resistant strains: the case of San Francisco.

Over the past two decades, HIV resistance to antiretroviral drugs (ARVs) has risen to high levels in the wealthier countries of the world, which are able to afford widespread treatment. We have gained insights into the evolution and transmission dynamics of ARV resistance by designing a biologically complex multistrain network model. With this model, we traced the evolutionary history of ARV resistance in San Francisco and predict its future dynamics. By using classification and regression trees, we identified the key immunologic, virologic, and treatment factors that increase ARV resistance. Our modeling shows that 60% of the currently circulating ARV-resistant strains in San Francisco are capable of causing self-sustaining epidemics, because each individual infected with one of these strains can cause, on average, more than one new resistant infection. It is possible that a new wave of ARV-resistant strains that pose a substantial threat to global public health is emerging.

Optimal control applied to a model for species augmentation.

Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. In this paper, we develop a differential equations model and optimal control formulation for a continuous time augmentation of a general declining population. We find a characterization for the optimal control and show numerical results for scenarios of different illustrative parameter sets. The numerical results provide considerably more detail about the exact dynamics of optimal augmentation than can be readily intuited. The work and results presented in this paper are a first step toward building a general theory of population augmentation, which accounts for the complexities inherent in many conservation biology applications.

Effectiveness and efficiency of imperfect therapeutic HSV-2 vaccines.

BACKGROUND: Efforts are currently underway to develop therapeutic vaccines for Herpes Simplex Virus type 2 (HSV-2). METHODS: We use a mathematical model to predict the potential public health impact of imperfect, therapeutic HSV-2 vaccines. We evaluate vaccine effectiveness and efficiency for the general population in the United States where HSV-2 prevalence is currently 22%. We assume that therapeutic vaccines will produce two therapeutic benefits in vaccinated infected-individuals: (i) the rate of viral reactivation will decrease (hence infected-individuals will experience fewer viral shedding episodes), and (ii) the average length of the viral shedding episodes will be shortened. In addition, we assume that therapeutic vaccines will benefit uninfected individuals by reducing viral shedding in (and hence transmission from) vaccinated infected-individuals. RESULTS: Our predictions show that therapeutic vaccines could substantially reduce HSV-2 epidemics by reducing new infections by 77% and preventing 0.84 new infections for each vaccinated individual. These vaccines could prevent 212,600 (median; IQR, 156,064-288,558) new infections after only one year. We show that increased effectiveness and efficiency are more strongly correlated with a vaccine-induced reduction in transmission probability than with either of the two therapeutic benefits that accrue directly to the infected individuals (specifically, the reduction in episode length and number of episodes). CONCLUSIONS: We suggest that current vaccine development efforts target mechanisms that reduce viral shedding (thereby reducing transmission) thus providing both a beneficial therapeutic and a beneficial epidemic-level impact. Our results also demonstrate that therapeutic vaccines would be substantially more useful than prophylactic vaccines for epidemic control.

Modelling an outbreak of an emerging pathogen.

To illustrate the usefulness of mathematical models to the microbiology and medical communities, we explain how to construct and apply a simple transmission model of an emerging pathogen. We chose to model, as a case study, a large (>8,000 reported cases) on-going outbreak of community-acquired meticillin-resistant Staphylococcus aureus (CA-MRSA) in the Los Angeles County Jail. A major risk factor for CA-MRSA infection is incarceration. Here, we show how to design a within-jail transmission model of CA-MRSA, parameterize the model and reconstruct the outbreak. The model is then used to assess the severity of the outbreak, predict the epidemiological consequences of a catastrophic outbreak and design effective interventions for outbreak control.