Bulletin of Mathematical Biology

Zmurchok, Cole; Small, Tim; Ward, Michael; Edelstein-Keshet, Leah

doi: 10.1007/s11538-017-0314-1pmid: 28707220

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction–advection–diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.

Lee, Junehyuk; Adler, Frederick; Kim, Peter

doi: 10.1007/s11538-017-0315-0pmid: 28741104

Respiratory viral infections are common in the general population and one of the most important causes of asthma aggravation and exacerbation. Despite many studies, it is not well understood how viral infections cause more severe symptoms and exacerbations in asthmatics. We develop a mathematical model of two types of macrophages that play complementary roles in fighting viral infection: classically $$(\hbox {CA}$$ ( CA - $$\hbox {M}\Phi )$$ M Φ ) and alternatively activated macrophages $$(\hbox {AA}$$ ( AA - $$\hbox {M}\Phi )$$ M Φ ) . $$\hbox {CA}$$ CA - $$\hbox {M}\Phi $$ M Φ destroy infected cells and tissues to remove viruses, while $$\hbox {AA}$$ AA - $$\hbox {M}\Phi $$ M Φ repair damaged tissues. We show that a higher viral load or longer duration of infection provokes a stronger immune response from the macrophage system. By adjusting the parameters, we model the differences in response to respiratory viral infection in normal and asthmatic subjects and show how this skews the system toward a response that generates more severe symptoms in asthmatic patients.

Maliyoni, Milliward; Chirove, Faraimunashe; Gaff, Holly; Govinder, Keshlan

doi: 10.1007/s11538-017-0317-ypmid: 28707219

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton–Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.

Gambette, P.; Huber, K.; Scholz, G.

doi: 10.1007/s11538-017-0318-xpmid: 28762018

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system $$\Sigma (N)$$ Σ ( N ) induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems.

Magal, P.; Noussair, A.; Pasquier, J.; Zongo, P.; Foll, F.

doi: 10.1007/s11538-017-0319-9pmid: 28721472

In this paper, we consider a direct protein transfer process between cells in co-culture. Assuming that cells continually encounter each other, and from some hypotheses on cell-to-cell rules of transfer, we derive discrete and continuous Boltzmann-like integro-differential equations. The novelty of this model is to take into account multiple transfer rules. This new transfer model is used to fit the experimental data of cell-to-cell protein transfer in breast cancer.

Gaivão, Maria; Dionisio, Francisco; Gjini, Erida

doi: 10.1007/s11538-017-0320-3pmid: 28741105

Humans are often colonized by polymorphic bacteria such as Streptococcus pneumoniae, Bordetella pertussis, Staphylococcus Aureus, and Haemophilus influenzae. Two co-colonizing pathogen clones may interact with each other upon host entry and during within-host dynamics, ranging from competition to facilitation. Here we examine the significance of these exploitation strategies for bacterial spread and persistence in host populations. We model SIS epidemiological dynamics to capture the global behavior of such multi-strain systems, focusing on different parameters of single and dual colonization. We analyze the impact of heterogeneity in clearance and transmission rates of single and dual colonization and find the criteria under which these asymmetries enhance endemic persistence. We obtain a backward bifurcation near $$R_0 = 1$$ R 0 = 1 if the reproductive value of the parasite in dually infected hosts is sufficiently higher than that in singly infected ones. In such cases, the parasite is able to persist even in sub-threshold conditions, and reducing the basic reproduction number below 1 would be insufficient for elimination. The fitness superiority in co-colonized hosts can be attained by lowering net parasite clearance rate ( $$\gamma _\mathrm{{d}}$$ γ d ), by increasing transmission rate ( $$\beta _\mathrm{{d}}$$ β d ), or both, and coupling between these traits critically constrains opportunities of pathogen survival in the $$R_0<1$$ R 0 < 1 regime. Finally, using an adaptive dynamics approach, we verify that despite their importance for sub-threshold endemicity, traits expressed exclusively in coinfection should generally evolve independently of single infection traits. In particular, for $$\beta _\mathrm{{d}}$$ β d a saturating parabolic or hyperbolic function of $$\gamma _\mathrm{{d}}$$ γ d , co-colonization traits evolve to an intermediate optimum (evolutionarily stable strategy, ESS), determined only by host lifespan and the trade-off parameters linking $$\beta _\mathrm{{d}}$$ β d and $$\gamma _\mathrm{{d}}$$ γ d . Our study invites more empirical attention to the dynamics and evolution of parasite life-history traits expressed exclusively in coinfection.

Nåsell, Ingemar

doi: 10.1007/s11538-017-0321-2pmid: 28721470

Moment closure methods are widely used to analyze mathematical models. They are specifically geared toward derivation of approximations of moments of stochastic models, and of similar quantities in other models. The methods possess several weaknesses: Conditions for validity of the approximations are not known, magnitudes of approximation errors are not easily evaluated, spurious solutions can be generated that require large efforts to eliminate, and expressions for the approximations are in many cases too complex to be useful. We describe an alternative method that provides improvements in these regards. The new method leads to asymptotic approximations of the first few cumulants that are explicit in the model’s parameters. We analyze the univariate stochastic logistic Verhulst model and a bivariate stochastic epidemic SIR model with the new method. Errors that were made in early applications of moment closure to the Verhulst model are explained and corrected.

Yang, Chayu; Wang, Xueying; Gao, Daozhou; Wang, Jin

doi: 10.1007/s11538-017-0322-1pmid: 28748506

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.

Chan, Matthew; Hawkes, Kristen; Kim, Peter

doi: 10.1007/s11538-017-0323-0pmid: 28707221

We present a mathematical simplification for the evolutionary dynamics of a heritable trait within a two-sex population. This trait is assumed to control the timing of sex-specific life-history events, such as the age of sexual maturity and end of female fertility, and each sex has a distinct fitness trade-off associated with the trait. We provide a formula for the fitness landscape of the population and show a natural extension of the result to an arbitrary number of heritable traits. Our method can be viewed as a dynamical systems generalisation of the Price equation to include two sexes, age structure and multiple traits. We use this formula to examine the effect of grandmothering, whereby post-fertile females subsidise their daughter’s fertility by provisioning grandchildren. Grandmothering can drive a shift towards increasingly male-biased mating sex ratios due to a post-fertile life stage in females, while male fertility continues to older ages. Our fitness landscapes show a net increase in fitness for both males and females at longer lifespans, and as a result, we find that grandmothering alone provides an evolutionary trajectory to higher longevities.

Wang, Xiaojing; Shi, Yangyang; Feng, Zhilan; Cui, Jingan

doi: 10.1007/s11538-017-0324-zpmid: 28721471

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511–1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.