http://math.arizona.edu/~cushing/DeterminingImportantParametersInTheSpreadofMalaria%20BMB2008.pdf

Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model.

Centers for Disease Control and Prevention (CDC) ホームページの活用法

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

https://iopscience.iop.org/article/10.1088/0951-7715/23/1/R01/pdf

Nonlinear modelling of cancer: Bridging the gap between cells and tumours

The archetypal feature of a viscoplastic fluid is its yield stress: If the material is not sufficiently stressed, it behaves like a solid, but once the yield stress is exceeded, the material flows like a fluid. Such behavior characterizes materials common in industries such as petroleum and chemical processing, cosmetics, and food processing and in geophysical fluid dynamics. The most common idealization of a viscoplastic fluid is the Bingham model, which has been widely used to rationalize experimental data, even though it is a crude oversimplification of true rheological behavior. The popularity of the model is in its apparent simplicity. Despite this, the sudden transition between solid-like behavior and flow introduces significant complications into the dynamics, which, as a result, has resisted much analysis. Over recent decades, theoretical developments, both analytical and computational, have provided a better understanding of the effect of the yield stress. Simultaneously, greater insight into the material behavior of real fluids has been afforded by advances in rheometry. These developments have primed us for a better understanding of the various applications in the natural and engineering sciences.

Yielding to Stress: Recent Developments in Viscoplastic Fluid Mechanics

Models in Ecology

A mathematical model for endemic malaria with variable human and mosquito populations

On the population dynamics of the malaria vector.

A mathematical model for endemic malaria involving variable human
and mosquito populations is analysed. A threshold parameter $R_0$ exists
and the disease can persist if and only if $R_0$ exceeds $1$. $R_0$ is seen to be a generalisation
of the basic reproduction ratio associated with the Ross-Macdonald
model for malaria transmission. The disease free equilibrium always exist and
is globally stable when $R_0$ is below $1$. A perturbation analysis is used to approximate
the endemic equilibrium in the important case where the disease
related death rate is nonzero, small but significant. A diffusion approximation
is used to approximate the quasi-stationary distribution of the associated stochastic
model. Numerical simulations show that when $R_0$ is distinctly greater
than $1$, the endemic deterministic equilibrium is globally stable. Furthermore,
in quasi-stationarity, the stochastic process undergoes oscillations about
a mean population whose size can be approximated by the stable endemic
deterministic equilibrium.

Modelling the dynamics of endemic malaria in growing populations

A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.

Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

A novel coronavirus emerged in December of 2019 (COVID-19), causing a pandemic that inflicted unprecedented public health and economic burden in all nooks and corners of the world. Although the control of COVID-19 largely focused on the use of basic public health measures (primarily based on using non-pharmaceutical interventions, such as quarantine, isolation, social-distancing, face mask usage, and community lockdowns) initially, three safe and highly-effective vaccines (by AstraZeneca Inc., Moderna Inc., and Pfizer Inc.), were approved for use in humans in December 2020. We present a new mathematical model for assessing the population-level impact of these vaccines on curtailing the burden of COVID-19. The model stratifies the total population into two subgroups, based on whether or not they habitually wear face mask in public. The resulting multigroup model, which takes the form of a deterministic system of nonlinear differential equations, is fitted and parameterized using COVID-19 cumulative mortality data for the third wave of the COVID-19 pandemic in the United States. Conditions for the asymptotic stability of the associated disease-free equilibrium, as well as an expression for the vaccine-derived herd immunity threshold, are rigorously derived. Numerical simulations of the model show that the size of the initial proportion of individuals in the mask-wearing group, together with positive change in behavior from the non-mask wearing group (as well as those in the mask-wearing group, who do not abandon their mask-wearing habit) play a crucial role in effectively curtailing the COVID-19 pandemic in the United States. This study further shows that the prospect of achieving vaccine-derived herd immunity (required for COVID-19 elimination) in the U.S., using the Pfizer or Moderna vaccine, is quite promising. In particular, our study shows that herd immunity can be achieved in the U.S. if at least 60% of the population are fully vaccinated. Furthermore, the prospect of eliminating the pandemic in the U.S. in the year 2021 is significantly enhanced if the vaccination program is complemented with non-pharmaceutical interventions at moderate increased levels of compliance (in relation to their baseline compliance). The study further suggests that, while the waning of natural and vaccine-derived immunity against COVID-19 induces only a marginal increase in the burden and projected time-to-elimination of the pandemic, adding the impacts of therapeutic benefits of the vaccines into the model resulted in a dramatic reduction in the burden and time-to-elimination of the pandemic.

/pdf/toward-achieving-a-vaccine-derived-herd-immunity-threshold-2if6k1xoag.pdf

Gideon A. Ngwa

Papers

A mathematical model for endemic malaria with variable human and mosquito populations

On the population dynamics of the malaria vector.

Modelling the dynamics of endemic malaria in growing populations

Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

Toward Achieving a Vaccine-Derived Herd Immunity Threshold for COVID-19 in the U.S.