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Aracena, J., Demongeot, J., Fanchon, E., & Montalva, M. (2013). On the number of different dynamics in Boolean networks with deterministic update schedules. Math. Biosci., 242(2), 188–194.
Abstract: Deterministic Boolean networks are a type of discrete dynamical systems widely used in the modeling of genetic networks. The dynamics of such systems is characterized by the local activation functions and the update schedule, i.e., the order in which the nodes are updated. In this paper, we address the problem of knowing the different dynamics of a Boolean network when the update schedule is changed. We begin by proving that the problem of the existence of a pair of update schedules with different dynamics is NPcomplete. However, we show that certain structural properties of the interaction digraph are sufficient for guaranteeing distinct dynamics of a network. In [1] the authors define equivalence classes which have the property that all the update schedules of a given class yield the same dynamics. In order to determine the dynamics associated to a network, we develop an algorithm to efficiently enumerate the above equivalence classes by selecting a representative update schedule for each class with a minimum number of blocks. Finally, we run this algorithm on the well known Arabidopsis thaliana network to determine the full spectrum of its different dynamics. (C) 2013 Elsevier Inc. All rights reserved.
Keywords: Boolean network; Update schedule; Update digraph; Dynamics

Kapitanov, G., Alvey, C., VogtGeisse, K., & Feng, Z. L. (2015). An AgeStructured Model For The Coupled Dynamics Of Hiv And Hsv2. Math. Biosci. Eng., 12(4), 803–840.
Abstract: Evidence suggests a strong correlation between the prevalence of HSV2 (genital herpes) and the perseverance of the HIV epidemic. HSV2 is an incurable viral infection, characterized by periodic reactivation. We construct a model of the coinfection dynamics between the two diseases by incorporating a timesinceinfection variable to track the alternating periods of infectiousness of HSV2. The model considers only heterosexual relationships and distinguishes three population groups: males, general population females, and female sex workers. We calculate the basic reproduction numbers for each disease that provide threshold conditions, which determine whether a disease dies out or becomes endemic in the absence of the other disease. We also derive the invasion reproduction numbers that determine whether or not a disease can invade into a population in which the other disease is endemic. The calculations of the invasion reproduction numbers suggest a new aspect in their interpretation – the class from which the initial disease carrier arises is important for understanding the invasion dynamics and biological interpretation of the expressions of the reproduction numbers. Sensitivity analysis is conducted to examine the role of model parameters in influencing the model outcomes. The results are discussed in the last section.

Towers, S., Vogt Geisse, K., ChiaChun, T., Han, Q., & Feng, Z. L. (2012). The Impact Of School Closures On Pandemic Influenza: Assessing Potential Repercussions Using A Seasonal Sir Model. Math. Biosci. Eng., 9(2), 413–430.
Abstract: When a new pandemic influenza strain has been identified, massproduction of vaccines can take several months, and antiviral drugs are expensive and usually in short supply. Social distancing measures, such as school closures, thus seem an attractive means to mitigate disease spread. However, the transmission of influenza is seasonal in nature, and as has been noted in previous studies, a decrease in the average transmission rate in a seasonal disease model may result in a larger final size. In the studies presented here, we analyze a hypothetical pandemic using a SIR epidemic model with time and agedependent transmission rates; using this model we assess and quantify, for the first time, the the effect of the timing and length of widespread school closures on influenza pandemic final size and average peak time. We find that the effect on pandemic progression strongly depends on the timing of the start of the school closure. For instance, we determine that school closures during a late spring wave of an epidemic can cause a pandemic to become up to 20% larger, but have the advantage that the average time of the peak is shifted by up to two months, possibly allowing enough time for development of vaccines to mitigate the larger size of the epidemic. Our studies thus suggest that when heterogeneity in transmission is a significant factor, decisions of public health policy will be particularly important as to how control measures such as school closures should be implemented.
Keywords: Pandemic influenza; epidemic model; dynamic systems
