NHM
The impact of cell crowding and active cell movement on vascular tumour growth
Russell Betteridge Markus R. Owen H.M. Byrne Tomás Alarcón Philip K. Maini
Networks & Heterogeneous Media 2006, 1(4): 515-535 doi: 10.3934/nhm.2006.1.515
A multiscale model for vascular tumour growth is presented which includes systems of ordinary differential equations for the cell cycle and regulation of apoptosis in individual cells, coupled to partial differential equations for the spatio-temporal dynamics of nutrient and key signalling chemicals. Furthermore, these subcellular and tissue layers are incorporated into a cellular automaton framework for cancerous and normal tissue with an embedded vascular network. The model is the extension of previous work and includes novel features such as cell movement and contact inhibition. We present a detailed simulation study of the effects of these additions on the invasive behaviour of tumour cells and the tumour's response to chemotherapy. In particular, we find that cell movement alone increases the rate of tumour growth and expansion, but that increasing the tumour cell carrying capacity leads to the formation of less invasive dense hypoxic tumours containing fewer tumour cells. However, when an increased carrying capacity is combined with significant tumour cell movement, the tumour grows and spreads more rapidly, accompanied by large spatio-temporal fluctuations in hypoxia, and hence in the number of quiescent cells. Since, in the model, hypoxic/quiescent cells produce VEGF which stimulates vascular adaptation, such fluctuations can dramatically affect drug delivery and the degree of success of chemotherapy.
keywords: tumour growth chemotherapy Multiscale modelling cell movement.
MBE
A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence
Andrei Korobeinikov Philip K. Maini
Mathematical Biosciences & Engineering 2004, 1(1): 57-60 doi: 10.3934/mbe.2004.1.57
Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I^p S^q$ for the case $p \leq 1$ are constructed. Global stability of the models is thereby established.
keywords: nonlinear incidence. endemic equilibrium state global stability Direct Lyapunov method
DCDS-B
Diffusion-aggregation processes with mono-stable reaction terms
Philip K. Maini Luisa Malaguti Cristina Marcelli Serena Matucci
Discrete & Continuous Dynamical Systems - B 2006, 6(5): 1175-1189 doi: 10.3934/dcdsb.2006.6.1175
This paper analyses front propagation of the equation

$\upsilon_{\tau}=[D(\upsilon)\upsilon_{x}]_{x}+f(\upsilon) \tau\ge 0, x\in R,$

where $f$ is a monostable (i.e. Fisher-type) nonlinear reaction term and $D(\upsilon)$ changes its sign once, from positive to negative values, in the interval $\upsilon \in [0, 1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $\upsilon$ at time $\tau$ and position $x$. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states $\upsilon \equiv 0$ and $\upsilon \equiv 1$. In the degenerate case, i.e. when $D(0) = 0$ and/or $D(1) = 0$, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.

keywords: finite speed of propagation. population dynamics Diffusion-aggregation processes sharp profiles front propagation
DCDS-B
Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation
R.A. Satnoianu Philip K. Maini F.S. Garduno J.P. Armitage
Discrete & Continuous Dynamical Systems - B 2001, 1(3): 339-362 doi: 10.3934/dcdsb.2001.1.339
We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium Bacillus subtilis on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes, via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis.
keywords: phase plane analysis bacterial chemotaxis travelling waves Degenerate diffusion nonlinear coupled parabolic equations.
MBE
HIV/AIDS epidemic in India and predicting the impact of the national response: Mathematical modeling and analysis
Arni S. R. Srinivasa Rao Kurien Thomas Kurapati Sudhakar Philip K. Maini
Mathematical Biosciences & Engineering 2009, 6(4): 779-813 doi: 10.3934/mbe.2009.6.779
After two phases of AIDS control activities in India, the third phase of the National AIDS Control Programme (NACP III) was launched in July 2007. Our focus here is to predict the number of people living with HIV/AIDS (PLHA) in India so that the results can assist the NACP III planning team to determine appropriate targets to be activated during the project period (2007-2012). We have constructed a dynamical model that captures the mixing patterns between susceptibles and infectives in both low-risk and high-risk groups in the population. Our aim is to project the HIV estimates by taking into account general interventions for susceptibles and additional interventions, such as targeted interventions among high risk groups, provision of anti-retroviral therapy, and behavior change among HIV-positive individuals. Continuing the current level of interventions in NACP II, the model estimates there will be 5.06 million PLHA by the end of 2011. If 50 percent of the targets in NACP III are achieved by the end of the above period then about 0.8 million new infections will be averted in that year. The current status of the epidemic appears to be less severe compared to the trend observed in the late 1990s. The projections based on the second phase and the third phase of the NACP indicate prevention programmes which are directed towards the general and high-risk populations, and HIV-positive individuals will determine the decline or stabilization of the epidemic. Model based results are derived separately for the revised HIV estimates released in 2007. According to revised projections there will be 2.08 million PLHA by 2012 if 50 percent of the targets in NACP III are reached. We perform a Monte Carlo procedure for sensitivity analysis of parameters and model validation. We also predict a positive role of implementation of anti-retroviral therapy treatment of 90 percent of the eligible people in the country. We present methods for obtaining disease progression parameters using convolution approaches. We also extend our models to age-structured populations.
keywords: behavioral interventions. anti-retroviral therapy epidemic modeling
DCDS-B
A non-linear degenerate equation for direct aggregation and traveling wave dynamics
Faustino Sánchez-Garduño Philip K. Maini Judith Pérez-Velázquez
Discrete & Continuous Dynamical Systems - B 2010, 13(2): 455-487 doi: 10.3934/dcdsb.2010.13.455
The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form $u_{t}=[D(u)u_{x}]_{x}+g(u)$. The model involves a density-dependent non-linear diffusion coefficient $D$ whose sign changes as the population density $u$ increases. For negative values of $D$ aggregation occurs, while dispersion occurs for positive values of $D$. We deal with a family of degenerate negative diffusion equations with logistic-like growth rate $g$. We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included.
keywords: Direct aggregation negative diffusion. traveling waves degenerate diffusion ill-posed problems
DCDS
Modelling collective cell behaviour
Deborah C. Markham Ruth E. Baker Philip K. Maini
Discrete & Continuous Dynamical Systems - A 2014, 34(12): 5123-5133 doi: 10.3934/dcds.2014.34.5123
The classical mean-field approach to modelling biological systems makes a number of simplifying assumptions which typically lead to coupled systems of reaction-diffusion partial differential equations. While these models have been very useful in allowing us to gain important insights into the behaviour of many biological systems, recent experimental advances in our ability to track and quantify cell behaviour now allow us to build more realistic models which relax some of the assumptions previously made. This brief review aims to illustrate the type of models obtained using this approach.
keywords: volume exclusion cell biology. moment dynamics continuum models Discrete models

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