5.3 Analysis
In this section, we compare the results for each method and discuss the benefits and potential problems of each method as well as initial guess selection.
Firstly, let us discuss about the efficiency of those four optimization methods on our **tumor** **growth** **model** by comparing CPU use time of each method. Without any doubt, the Nelder-mead method is the most efficient among those four methods while conjugated gradient trust region method also does well in efficiency. However, pattern search method and Gradient projection method spend over over ten times time on getting the estimated data. In addition, it is obvious that each method spends more time on getting the results with the increase of time steps and grid points.

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in the next section.
2.4. Age-Structured Models
We now present a typical PDE **model** used in several applications. Indeed, a cell starts from the hematopoietic stem cell and then its progeny go through a number of stages before being released into the circulation. One could **model** this process by associating a partial differ- ential equation for the cell density function with each stage, which describes the population in the compartment as a function of the variables age a and time t. The **model** also contains feedback control elements that regulate the release of cells from one compartment to the other. The number of compartments depends on the data available which determines the maximum level of detail appropri- ate for the **model**. For instance, a **model** of erythropoiesis could have one compartment for each recognizable stage of erythrocytes precursors, or alternatively merge some of the compartments together and thus reduce the **model** dimensions. In the following, we will present some re- sults using only a generic compartment. The treatment for a larger **model** is the same. We then show that by partial integration we can express this problem as a delay differential equation **model**. Age-structured models pro- vide a means of understanding the regulation of hemato- poiesis. Examples in the literature can be found in [2].

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extended discussion, we refer the reader to the reviews in [14, 10, 153, 33, 158, 24, 164, 3].
In general, statistical techniques can be applied to experimental data to reveal correlations between observable phenomena. Then, it is necessary to postulate hy- potheses to establish the reasons underlying these correlations, stating which physical processes are involved and how they interact [24]. Biological experiments for testing these hypotheses may be extremely time-consuming, expensive, or even impossible with the current technologies. In such cases, **mathematical** modeling can play an intermediate role, providing an independent check for the consistency of the hypothe- ses. If a **model** derived from such hypotheses is not able to reproduce the observed phenomena, then the original statements have to be modified before carrying on the work. Moreover, **mathematical** models can improve the design of experiments by highlighting which measurements are required to test a particular theory, or whether supplementary information can be obtained by collecting additional data. Finally, the parameters that feed the equations in the models can be varied over a large set, providing a thorough characterization of the system. These ideas are summarized in Figure 2.7, where the different stages involved in the formulation of the **mathematical** **model** are represented. Actually, **mathematical** modeling is an iterative process and

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While it is often simple to construct an algebraic expression for R 0 in terms of epidemiological parameters, one or more of these values is typically not obtainable by direct methods. Instead, their values are usually estimated indirectly by fitting a **mathematical** **model** to incidence or prevalence data (see, for example, [3, 11, 31, 35, 38, 39]), obtaining a set of parameters that provides the best match, in some sense, between **model** output and data. It is, therefore, crucial that we have a good understanding of the properties of the process used to fit the **model** and its limitations when employed on a given data set. An appreciation of the uncertainty accompanying the **parameter** estimates, and indeed whether a given **parameter** is even individually identifiable based on the available data and **model**, is necessary for our understanding.

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In carrying out the inverse problems, we arrive at a set of parameters for a mathemat- ical **model** (in this case, the ATN **model**) that allow us to best-fit the observed data.
However, the first attempt to solve an inverse problem naturally invites us to question the accuracy of the data, including the properties its observation errors might possess, as well as how much information about our system could be possibly represented in a given data set. In evaluating the performance of the inverse problems, we often find that the **mathematical** and/or statistical models require reformulation. Thus we usually must statistically address the nuances of inherent errors in data collection. Arriving at these statistical and **mathematical** **model** reformulations yields new inverse problems to solve, and so we repeat the modelling process until a sufficient theoretical framework has been developed to compare with experimental data. In this way, the inverse problems represent an experiment in themselves; we have hypothesized a **mathematical** and a sta- tistical **model** and seek to investigate their validity and shortcomings within the context of our study system.

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Keywords: **Mathematical** models; **tumor** **growth**; chemotherapy; diffusion; optimal control.
AMS Subject Classification 2010 : 35K61, 37N25, 49J15, 92D25.
1 Introduction
**Mathematical** modelling has been playing a great role in exploring a variety of information not only in different branches of physical and engineering sciences, but also in biological and medical sciences. A number of theoretical studies have been performed by researchers on different topics of life sciences. Among others, during the last five decades Misra and his co- workers have carried out extensive studies on a variety of topics in Biomedical Mathematics and Physiological Fluid Dynamics (e.g. [1]-[12]). The method of **mathematical** modelling was employed in [1] to study the dynamic of arterial wall tissues and in [2] to determine the left ventricular wall stresses of a human-sized heart. A **mathematical** **model** was developed

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Introduction
Heat is a form of energy in transit due to a temperature difference (Rathore & Raul R. A.
Kapuno, 2011). This variation in temperature is governed by the principle of energy conservation, which when applied to a control volume or a control mass, states that the sum of the flow of energy and heat across the system, the work done on the system, and the energy stored and converted within the system, is zero. Numerical method is a technique for obtaining approximate solutions. Numerical method is well known due to its applications to the industries especially in sciences and engineering field. In the reality the analytical solutions a not exist. Thus, this method is being used to calculate the approximation to the analytical solutions. Finite Element Method (FEM) and Finite Difference Method (FDM) are the examples of numerical methods that can be used to solve this kind of problems. FEM is useful for problems with complicated geometries, loadings and material properties where analytical solutions cannot be obtained. It is a numerical technique in finding approximate solutions to boundary value for differential equations problems of engineering and **mathematical** physics. This research is focusing on solving a two-dimensional irregular geometry heat transfer equation by applying the finite element method. Computations of this method will be performed using MATLAB program due to large scale problem. It is also to assure the accuracy of the solution.

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In this review we can clearly not describe every **model** of avascular **tumor** **growth**. Most models fall into two categories: (1) continuum **mathematical** models that use space averaging and thus consist of partial diﬀerential equations; and (2) discrete cell population models that consider processes that occur on the single cell scale and in- troduce cell-cell interaction using cellular automata–type computational machinery. We will discuss each of the two approaches in detail and we aim to present a compre- hensive list of references for each category. In addition, in each of the categories, we will discuss one or two models in depth. We have chosen these representatives because they were published in or are frequently referenced in biological journals read by exper- imentalists in the area of cancer biology and in general make quantitative predictions that have been validated experimentally. This approach to choosing the representative models ensures that we are by default selecting examples that have had real impact thus far on the scientiﬁc research of cancer. However, such an approach does mean that we riskignoring papers that have made crucial conceptual advances, or which have put forward important hypotheses which are too diﬃcult to test experimentally at the moment, or whose signiﬁcance has not yet been recognized by the experimental community. Therefore we broaden our review by summarizing these models also.

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Abstract
Although there are several **mathematical** models present for baculovirus infection, the specific functions for insect cell **growth** and cell death during infection processes remain unknown.
Specifically, it is challenging to identify the most suitable **model** from a large set of plausible models and estimate the kinetic parameters to account for the day to day variability present in the infection experiments. In this context, identification of an unstructured **model** that can predict the day to day variability in cell **growth** and cell viability can be useful in determining the optimal operating conditions in fermenters at industrial scale. The major objectives of the present work were to develop a **model** screening framework that can be used to select the best **model** and identify the **growth** and death mechanisms during viral infection through non-linear programming. We then constructed a series of plausible models based on system of ordinary differential equations and performed the **model** selection using experimental data obtained from shaker flasks. The proposed scheme was tested for selecting the **model** for uninfected cell **growth** profiles. The objective function used was the root mean square error between the predicted values and experimental data points obtained from triplicate dataset. The computational scheme was validated using two types of virus, the WT AcMNPV and stabilized AcMNPV. Additionally, we propose a numerical scheme to simulate the cell **growth** and cell viability during viral passaging. The kinetic parameters were estimated in case of **growth** of uninfected cells, cells infected with WT virus as well as stabilized AcMNPV. The result shows that Monods equation fits the best for insect cell **growth** without infection and infection with WT AcMNPV. Whereas, the Contois **model** fits the best for the stabilized virus. The simulated results also indicate that the day to day variability in cell **growth** and cell viability profile can be explained through the variation in the specific **growth** rate and the death rate. The estimated kinetic parameters indicate that the **growth** and death parameters undergo specific modifications during the passaging of viruses associated to infection process.

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Solving this equation to determine **tumor** **growth** requires knowledge of how **tumor** metabolic rate, B T , depends on its viable mass, m v , to which we now turn.
**Model** for **tumor** vascular system and the prediction of metabolic rate. **Tumor** metabolic rate, B T , is proportional to the sum of the rates of cellular fermentation and aerobic respiration. For avascular tumors, B T depends on the diffusion rate of nutrients and oxygen from the surrounding environment [18]. For vascular tumors, B T is proportional to the total blood volume flow rate to the **tumor**, _ Q Q T , consistent with observations that glucose and oxygen consumption rates vary linearly with blood flow rate [19]. The dependence of _ Q Q T on m v and host mass, M, is determined by the structure, dynamics and interaction of the **tumor** and host vasculatures. Here, we develop a complete analytical **model** of **tumor** vascular networks applicable throughout different phases of development by deriving the allometric scaling of **tumor** rates and times with host body size and capillary density. Although the importance of the vascular interface between the **tumor** and the host has been previously recognized, our work is a novel attempt to mechanistically **model** its role in **tumor** **growth** [10–12,20].

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be responsible for the differences observed in the vascular patterns of different tissues and pathological conditions.
These results provide a novel perspective concerning the causes that determine the various vascular patterns, which have been normally considered to be determined only by soluble and matrix- bound **growth** factors (such as VEGF) [23]. We verify that the constitution of the ECM has a direct implication on the ability of the endothelial cells to move through it, thus regulating their velocity. Also a different composition of the angiogenic chemical cocktail may lead to different proliferation rates of the endothelial cells. We conclude that the resulting capillary network morphology is not only a result of the levels and distribution of angiogenic factor, but also of the extracellular matrix properties and the exact composition of the protein cocktail.

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Non-Malignant tumors are not of themselves dangerous to life. They may however cause a great deal of pain and even death, when situated in some delicate organs. For instance, a small **tumor** may cause intense pain by pressing on a nerve or dropsical swelling of a limb by obstructing a vein or death from suffocating by blocking the larynx. Nevertheless, it remains true that any disastrous effect are not due to the nature of the **tumor**, but to its situations, whereas once it becomes cancerous can lead to death despite its position. They (Benign) do not have the capacity to reproduce themselves in distant parts of the body. More than one may be present in the same individual, but each occurs independently, and the widespread dissemination so typical of a cancerous **growth** is never seen. The boundaries of the **growth** are so well defined that complete removal is usually easy, and the processes (operation) is a simple and satisfactory procedure [4]. On the other hand, malignant tumors are cancerous. They emerge out of the cavity, by breaking out through the basal membrane and then metastasizing into extra-cellular matrix. Its ability to metastasize is based on the process called “ANGIOGENESIS-the arterial nerves grows into the **tumor** allowing the passage of nutrients and thus, the **tumor** fluid been carried to other parts of the body” [11]. Their boundaries of **growth** are not well defined, hence complicating removal atimes. They make their way in all directions into the surrounding parts. They are of two kinds: Sarcomata-evolving from the connective tissues and Carcinomata-emanating from the epithelial tissues. They can spread to lymph nodes through the lymphatic vessels by producing **growth** factors that stimulate lymphatic vessel **growth** and spread directly or indirectly. It actively invades and destroys surrounding tissues and also gives rise to cells which often spread to produce foci of **growth** at distant sites [4].

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uals, and the estimates will be inaccurate. Finally, the usage of anti-retroviral therapies reduces the viral load and transmission potential of infected individuals.
The diculty in studying this system mathematically lies in **parameter** estimates. A minimal **model** of this system requires at least three parameters: transmission of the dis- ease, diagnosis of the disease, and death due to the disease. Since knowledge about the undiagnosed population is restricted to those who have been diagnosed, estimates of these parameters are generally forced to assume that this population is representative of the whole. Our motivation to **model** the system stochastically arises from heterogeneity due to reporting delays associated with population-level data [31]. Stochastic modeling will allow us to better understand both the eectiveness of our estimates and the quality of **model** t.

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National Science Foundation of US, Grant/Award Number: DMS-1446139;
Foundation for Innovation at HIT (WH)
A two-dimensional **tumor**-immune **model** with the time delay of the adaptive immune response is considered in this paper. The **model** is designed to account for the interaction between cytotoxic T lymphocytes (CTLs) and cancer cells on the surface of a solid **tumor**. The **model** considers the surface **growth** as a major **growth** pattern of solid tumors in order to describe the existence of necrotic kernels. The qualitative analysis shows that the immune-free equilib- rium is unstable, and the behavior of positive equilibrium is closely related to the ratio of the immune killing rate to **tumor** volume **growth** rate. The positive equilibrium is locally asymptotically stable when the ratio is smaller than a crit- ical value. Otherwise, the occurrence of the delay-driven Hopf bifurcation at the positive equilibrium is proved. Applying the center manifold reduction and nor- mal form method, we obtain explicit formulas to determine the properties of the Hopf bifurcation. The global continuation of a local Hopf bifurcation is investi- gated based on the coincidence degree theory. The results reveal that the time of the adaptive immune system taken to response to tumors can lead to oscilla- tion dynamics. We also carry out detailed numerical analysis for parameters and numerical simulations to illustrate our qualitative analysis. Numerically, we find that shorter immune response time can lead to longer patient survival time, and the period and amplitude of a stable periodic solution increase with the increas- ing immune response time. When CTLs recruitment rate and death rate vary, we show how the ratio of the immune killing rate to **tumor** volume **growth** rate and the first bifurcation value change numerically, which yields further insights to the **tumor**-immune dynamics.

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medium RO . Brown wrote that pollen grains exhibited a continuous swarming motion when viewed under the microscope. Brownian motion is the most important stochastic process, being the archetype of Gaussian processes, of continuous time martingales, and of Markov processes. It is fundamental to the study of stochastic differential equations, financial math- ematics, and filtering, for example. A formal **mathematical** construction of Brownian motion and its properties was first given by the mathematician Norbert Wiener beginning in 1918, based on Fourier series. Subsequently, martingale techniques have been employed to con- struct Brownian motion as well.

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Various outbreaks including Zika (7, 10), Foot and Mouth disease (8), Ebola (9), and HIV- AIDS (11) have been modelled using the GGM.
2.3. **Parameter** **estimation**
For assessing the parameters, we conduct parametric bootstrap analyses using LSQ and MLE methods. A previous study shows that one can evaluate **parameter** with a simple computational bootstrap-based method, by replicating several data sets through repeated sampling from the **model**. (17). When estimating parameters, the initial **parameter** values or “guesses” can impact the results due to local maxima or minima. Therefore, we utilize the bootstrap method several times with different initial **parameter** guesses to estimate the best initial parameters, or those with the lowest MSE, for the ‘best-fit’ **model**. With the initial parameters from the bootstrap method, we then use these values and employ the bootstrapping method to simulate 500 curves (M=500) from the best-fit **model**, and further, re-estimate the parameters for each of these new datasets. We then utilize the **parameter** estimate distributions to calculate 95% confidential intervals, root mean square error, prediction intervals, and Anscombe residuals.

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with guidance for **parameter** **estimation** given real data for novel crops and crop combinations. The goal of this procedure is to arrive at a reduced-order
**parameter** space that can be estimated entirely from input-output data.
Furthermore, the goal is for the reduced **model** to closely follow the outputs of the original system (with any feasible parameterization) when driven by any input in the input class. The procedure is demonstrated on the well-known Yield- SAFE predictive agroforestry **growth** **model**. The advantages of an input-output system identification approach may also carry over into field trial design or **model** structure revisions. Further, because **model** parameterization can be based only on readily accessible **model** outputs, relatively low-tech data collection strategies emphasizing on-farm participatory research become possible. Participatory approaches allow a broader range of useful data to be collected for evaluating complex crop combinations.

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2 Department of Mathematics, University of Ilorin, Ilorin, Nigeria
Abstract
Unemployment is one of the major vices in our contemporal society, which weigh greatly on the economy of such nation. It is also, a fact that knowing ones enemy before battle gives 50 per cent chance of victory; thus, this re- search aimed at providing understanding about dynamics of unemployment with consideration for retirement and possible control criterion. And the ob- jectives are; formulation of **mathematical** **model** using the concept of deter- ministic **model** and **mathematical** epidemiology; then, **model** analysis. The **model** analysis includes, a numerical semi-analytical scheme for investigating validity of analytical solutions. The result of the analysis were that: 1) the **model** was mathematically well-pose and biologically meaningful 2) two equi- libria points exist, and 3) a threshold for recruitment from the pool of unem- ployment, assuring victory in the fight against unemployment was also, ob- tained. The threshold is required to be well managed in order to win the bat- tle against the socio-vice (unemployment) in the contemporary society. In addition, variational iterative method (VIM) is the numerical semi-analytic scheme employed to solve the **model**; thus, the approximate solution gave a practical meaningful interpretation supporting the analytical results and proof of verdict of assumptions of the **model**. The article concluded with three points; everyone has roles to play to curtail the socio-menace, beseech government and policy makers to look kindly, and create policy(ies) to sus- tain population **growth**, and the retiree should also, plan live after service, because over dependence on pension scheme could be died before death be- cause of corruption in the scheme.

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1 Introduction
Explosive **growth** in the size of spatial databases has highlighted the need for spatial data analysis and spatial data mining techniques to mine the interesting but implicit spatial patterns within these large databases. Many classical data mining algorithms, such as linear regression, assume that the learning samples are independently and identically distributed (i.i.d.). This assumption is violated in the case of spatial data due to spatial autocorrelation [1, 21]. In such cases classical linear regression yields a weak **model** with not only low prediction accuracy [22] but also residual error exhibiting spatial dependence. Modeling spatial dependencies improves overall classification and prediction accuracies.

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