Modelling the Impact of NETosis During the Initial Stage of Systemic Lupus Erythematosus.

The development of autoimmune diseases often takes years before clinical symptoms become detectable. We propose a mathematical model for the immune response during the initial stage of Systemic Lupus Erythematosus which models the process of aberrant apoptosis and activation of macrophages and neutrophils. NETosis is a type of cell death characterised by the release of neutrophil extracellular traps, or NETs, containing material from the neutrophil's nucleus, in response to a pathogenic stimulus. This process is hypothesised to contribute to the development of autoimmunogenicity in SLE. The aim of this work is to study how NETosis contributes to the establishment of persistent autoantigen production by analysing the steady states and the asymptotic dynamics of the model by numerical experiment.

Modeling repellent-based interventions for control of vector-borne diseases with constraints on extent and duration.

We study a simple model for a vector-borne disease with control intervention based on clothes and household items treated with mosquito repellents, which has constraints on the extent (population coverage) and on the time duration reflecting technological and physical properties. We compute first, the viability kernel of initial data of the model for which exists an optimal control that maintains the infected host population below a given cap for all future times. Second, we use the viability kernel to compute the set of initial data of the model for which exists an optimal control that brings this population below the cap in a time period not exceeding the intervention's duration. We discuss applications of this framework in predicting and evaluating the performance of control interventions under the given type of constraints.

Complexity of host-vector dynamics in a two-strain dengue model.

We introduce a compartmental host-vector model for dengue with two viral strains, temporary cross-immunity for the hosts, and possible secondary infections. We study the conditions on existence of endemic equilibria where one strain displaces the other or the two virus strains co-exist. Since the host and vector epidemiology follow different time scales, the model is described as a slow-fast system. We use the geometric singular perturbation technique to reduce the model dimension. We compare the behaviour of the full model with that of the model with a quasi-steady approximation for the vector dynamics. We also perform numerical bifurcation analysis with parameter values from the literature and compare the bifurcation structure to that of previous two-strain host-only models.

Modelling the Host Immune Response to Mature and Immature Dengue Viruses.

Immature dengue virions contained in patient blood samples are essentially not infectious because the uncleaved surface protein prM renders them incompetent for membrane fusion. However, the immature virions regain full infectivity when they interact with anti-prM antibodies, and once opsonised virion fusion into Fc receptor-expressing cells is facilitated. We propose a within-host mathematical model for the immune response which takes into account the dichotomy between mature infectious and immature noninfectious dengue virions. The model accounts for experimental observations on the different interactions of plasmacytoid dendritic cells with infected cells producing virions with different infectivity. We compute the basic reproduction number as a function of the proportion of infected cells producing noninfectious virions and use numerical simulations to compare the host's immune response in a primary and a secondary dengue infections. The results can be placed in the immunoregulatory framework with plasmacytoid dendritic cells serving as a bridge between the innate and adaptive immune response, and pose questions for potential experimental work to validate hypothesis about the evolutionary context whereby the virus strives to maximise its chance for transmission from the human host to the mosquito vector.

On the role of vector modeling in a minimalistic epidemic model.

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

Stability of Cross-Feeding Polymorphisms in Microbial Communities.

Cross-feeding, a relationship wherein one organism consumes metabolites excreted by another, is a ubiquitous feature of natural and clinically-relevant microbial communities and could be a key factor promoting diversity in extreme and/or nutrient-poor environments. However, it remains unclear how readily cross-feeding interactions form, and therefore our ability to predict their emergence is limited. In this paper we developed a mathematical model parameterized using data from the biochemistry and ecology of an E. coli cross-feeding laboratory system. The model accurately captures short-term dynamics of the two competitors that have been observed empirically and we use it to systematically explore the stability of cross-feeding interactions for a range of environmental conditions. We find that our simple system can display complex dynamics including multi-stable behavior separated by a critical point. Therefore whether cross-feeding interactions form depends on the complex interplay between density and frequency of the competitors as well as on the concentration of resources in the environment. Moreover, we find that subtly different environmental conditions can lead to dramatically different results regarding the establishment of cross-feeding, which could explain the apparently unpredictable between-population differences in experimental outcomes. We argue that mathematical models are essential tools for disentangling the complexities of cross-feeding interactions.

Kinase Inhibition Leads to Hormesis in a Dual Phosphorylation-Dephosphorylation Cycle.

Many antimicrobial and anti-tumour drugs elicit hormetic responses characterised by low-dose stimulation and high-dose inhibition. While this can have profound consequences for human health, with low drug concentrations actually stimulating pathogen or tumour growth, the mechanistic understanding behind such responses is still lacking. We propose a novel, simple but general mechanism that could give rise to hormesis in systems where an inhibitor acts on an enzyme. At its core is one of the basic building blocks in intracellular signalling, the dual phosphorylation-dephosphorylation motif, found in diverse regulatory processes including control of cell proliferation and programmed cell death. Our analytically-derived conditions for observing hormesis provide clues as to why this mechanism has not been previously identified. Current mathematical models regularly make simplifying assumptions that lack empirical support but inadvertently preclude the observation of hormesis. In addition, due to the inherent population heterogeneities, the presence of hormesis is likely to be masked in empirical population-level studies. Therefore, examining hormetic responses at single-cell level coupled with improved mathematical models could substantially enhance detection and mechanistic understanding of hormesis.

A model for spatio-temporal dynamics in a regulatory network for cell polarity.

Cell polarity in Myxococcus xanthus is crucial for the directed motility of individual cells. The polarity system is characterised by a dynamic spatio-temporal localisation of the regulatory proteins MglA and MglB at opposite cell poles. In response to signalling by the Frz chemosensory system, MglA and MglB are released from the poles and then rebind at the opposite poles. Thus, over time MglA and MglB oscillate irregularly between the poles in synchrony but out of phase. A minimal macroscopic model of the Mgl/Frz regulatory system based on a reaction-diffusion PDE system is presented. Mathematical analysis of the steady states derives conditions on the reaction terms for formation of dynamic localisation patterns of the regulatory proteins under different biologically-relevant regimes, i.e. with and without Frz signalling. Numerical simulations of the model system produce either a stationary pattern in time (fixed polarity), periodic solutions in time (oscillating polarity), or excitable behaviour (irregular switching of polarity).

A model of oscillatory protein dynamics in bacteria.

Spatial oscillations of proteins in bacteria have recently attracted much attention. The cellular mechanism underlying these oscillations can be studied at molecular as well as at more macroscopic levels. We construct a minimal mathematical model with two proteins that is able to produce self-sustained regular pole-to-pole oscillations without having to take into account molecular details of the proteins and their interactions. The dynamics of the model is based solely on diffusion across the cell body and protein reactions at the poles, and is independent of stimuli coming from the environment. We solve the associated system of reaction-diffusion equations and perform a parameter scan to demonstrate robustness of the model for two possible sets of the reaction functions.