Here we present a new mathematical model that explicitly incorporates the ability of HIV-1 to use hybrid spreading mechanisms and evaluate the consequences for HIV-1 pathogenenesis.

In this paper, we introduce a mathematical model of HIV dynamics that explicitly incorporates hybrid spreading.

Mathematical models provide an important tool for understanding and predicting the course of natural HIV-1 infection that complements clinical studies.

Mathematical models have proved of value in the past, but have suffered from omitting important biological processes, thus compromising their ability to accurately recapitulate clinical observations.

With increased sophistication, and hence ability to accurately model the known biological drivers of disease progression, mathematical models can become increasingly important in preclinical testing of modified or novel HIV therapies.

The cellular and viral changes which drive each phase of this compleX infection have been the subject of intense debate, in which mathematical models have played an important role in delineating HIV-1 pathogenesis and informing antiretroviral therapy [1—3].

The rich literature appertaining to mathematical modeling of intra host HIV dynamics has been reviewed several times recently [1, 4—6].

In this paper we develop a new mathematical model which incorporates the basic principles of previous host-centric models including a virus-dependent immune response [8] , viral latency and a progressive increase in cell activation [26, 27].

Here, we report a novel hybrid mathematical modeling strategy to systematically unravel hepatocyte growth factor (HGF) stimulated phosphoinosi-tide-3-kinase (PI3K) and mitogen activated protein kinase (MAPK) signaling, which critically contribute to liver regeneration.

Here, we present a novel hybrid mathematical modeling strategy taking advantage of qualitative and quantitative modeling approaches.

Several other mathematical modeling approaches also deal with a family of candidate models aiming at the identification of the correct wiring.

By employing mathematical modeling , a study of the MAPK and PI3K pathways crosstalk showed that both compensate for each other [28].

To this aim, mathematical models provide unique tools to disentangle complexity and to predict the impact of perturbations.

Mathematical models of the MAPK signaling pathway have been developed that only consider negative feedback [22] , negative and positive feedback loops [5] or that analyze the signal-to-response relation [23].

Mathematical models describing both PI3K and MAPK signaling pathways upon single or combinatorial stimuli reveal the presence of crosstalk mechanisms between MAPK and PI3K pathways [24—26] or differences in the stimulus specific network topology [27, 28].

Then, using mathematical modeling , we show how these processes contribute to the densi-ty-dependent and biphasic survival kinetics observed.

Using mathematical modeling , we further reveal key underlying processes responsible for the perseverance.

Mathematical modeling of these processes accurately accounts for the density-dependent, biphasic survival kinetics.

Using mathematical modeling , we showed how these processes contribute to the intricate survival patterns observed.

Mathematical modeling of the known processes accounts for the kinetics observed

To examine whether the biological processes described above can quantitatively account for the survival kinetics observed in our experiments, we constructed a mathematical model based on them.

Supporting text for formulation of our mathematical model .

In this study, we use all available U.S. surveillance data to: fit a set of mathematical models and determine which best explains these data and determine the epidemiological and vaccine-related parameter values of this model.

We constructed a compartmental mathematical model of the natural history and population transmission of B. pertussis infection in the United States, in which the population transitions between three states of infection: susceptible, infected, and recovered from a prior infection.

The mathematical model we constructed has an age-structured susceptible-infected-recovered (Si, 111-, R1) structure (the subscript 1', corresponds to one of 35 age-groups), but With the addition of a second infected compartment (121-) to account for those Who have been previously infected.

Pertussis case count data from NNDSS were aggregated into annual counts for each age group so that yl- (t) is the number of pertussis disease cases for age group i in year t. Our mathematical model outputs were also aggregated into annual counts for each age group i, so that x,- (t) was the model-derived case count for age group 1', during year t. These model-derived case counts are functions of the model structure and parameters, so that they might be better expressed as x,- (t|0, M), where 0 represents the parameter vector for model M.

First, we estimate R0 to be in the range of 9—12 which is closer to the values found by previous mathematical models [fi,fl] than the often-quoted range of 12—17 [E].

In this study, we use a mathematical model of Dorsal dynamics, fit to experimental data, to determine the ability of the Dorsal gradient to regulate gene expression across the entire dor-sal-ventral axis.

Using a mathematical model of the Drosophila embryo, we have proposed a solution to this outstanding problem: namely that Cactus, the inhibitor to Dorsal, is present with Dorsal in nuclei across the embryo, which creates a disparity between the gradient measured by fluorescence and the gradient measured by gene expression.

By using distinct model equations for active and inactive pools of Dorsal, we were able to recreate the dynamics of the Dorsal gradient and the eXpression patterns of its target genes with a high level of accuracy, showing that mathematical models may be critical for properly interpreting fluorescence data.

We use mathematical modelling to show how control of such disease outbreaks can be optimised.

Here we use mathematical modelling to investigate epidemiological principles underlying successful control.

As defaults we therefore use illustrative parameters informed by the biology [29,30] and adapted from previous models of citrus canker [11,12,24,25] (Table 1) to drive our mathematical model .