 # Network Modelling on Tropical Diseases vs. Climate Change

G. Udhaya Sankar (Alagappa University, India) and C. Ganesa Moorthy (Alagappa University, India)
DOI: 10.4018/978-1-7998-2197-7.ch004
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## Abstract

This chapter has proposed a systematic method to design mathematical models. These models have been associated with counting of white blood cells, counting of red blood cells, population of mosquitoes, and counting of foreign bodies like virus, bacteria, and parasite in a human body. Interpretations for critical points or equilibrium points have been given for network systems of differential equations associated with models. A practical method of applying these interpretations in administrating medicines to get control over diseases has been given. Order of priority in three types of critical points, namely, stable critical points, unstable critical points, and asymptotically stable critical points, has been given. Conversions of differential equations of models into integral equations and applying Picard's iteration method to solve integral equations have been explained. A step-by-step approach has been used in designing models, solving models, and interpreting solutions of models for tropical diseases.
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## Introduction

Tropical countries are the countries which are near to the equator of our earth planet, which are supposed to have high temperature, which are homelands for many insects which spread diseases, and which have many diseases solely like malaria, dengue, chikungunya etc. There is a chance to have hot conditions and humid conditions in these regions and these conditions are good enough for infectious diseases. Clinical tests and experimental tests help to understand nature of diseases and nature of medicines. But, they are not sufficient to explain many things, because results of tests in one region may contradict results of tests in another region. Only mathematical models, equations and solutions can explain some more things, more specifically about confronting medical facts. But these models alone cannot explain everything, because models cannot be complete in all aspects, and parameters involved in models can be found only by means of experimental data. Even in clinical tests, one has to depend on statistical methods in terms of measures of central tendency, measures of dispersions, correlation coefficients, regression lines, estimations and hypotheses testing which depend on probability. Regression line methods are modified as curve fitting, by guessing the curves in terms of solutions of equations involved in models. Probabilistic methods are modified as stochastic methods to understand long term effects. There are articles in literature for all these things explained above, and researches continue, because tropic countries are most affected countries by climate changes which happen.

The purpose of this chapter is to develop procedures to design models (Figure. 1) for tropical diseases and explain tools to analyze these models. It all begins with known prey-predator equations just to understand the beginning of the art of designing a model. Without correct justifications the words prey and predator are changed to the words foreign bodies and white blood corpuscles just to explain suitable mathematical methods to solve differential equations involved in the model and to guess suitable curves for curve fitting from forms of solutions of differential equations. Picard’s iteration method is considered as the most promising method for solving differential equations in this chapter. Synchronization methods are considered as the most favorable methods when the models are dynamic ones. Then an exact simplified model for foreign body-white blood corpuscle is designed. Complications are considered in terms of variations in temperature due climate changes, in designing models. A model is designed for network relations connecting growth of mosquito-population, white blood cell-population (counting), and malaria parasite-population. A model is designed for relations of numbers of malaria parasites, white blood cells, and red blood cells which would be helpful in understanding interrelationship between anemia and malaria. All these things are towards exact interpretations for equilibrium points and their stability. Some possible interesting theoretical conclusions are derived in this chapter.

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## Models For Encounters Of Foreign Bodies With White Blood Corpuscles

### Background

There are many tropical diseases like: Chagas disease, Dengue, Helminths, African trypanosomiasis, Leishmaniasis, Leprosy, Lymphatic filariasis, Malaria, Onchocerciasis, Schistosomiasis, Hookworm, Trichuriasis, Treponematoses, Buruli ulcer, Dracunculiasis, Leptospirosis, Strongyloidiasis, Foodborne trematodiases, Neurocysticercosis, Scabies, Flavivirus infections, etc. There are many articles, Greenwood, M. (1916), Kermack, W. O. et al, (1927), Kermack, W. O. et al., (1932), Kermack, W. O. et al., (1933), Anderson, R. M., (1988), Hethcote, H. W. (2000), Bernoulli, D.et al., (2004), Eubank, S. et al., (2004), Keeling, M. J. et al., (2005), Shirley, M. D. et al, (2005), Chitnis, N. et al., (2018), Gervas, H. E. et al., (2018), Rock, K. S. et al., (2018), Bañuelos, S. et al., (2019), Chowell, G.et al., (2019), Musa, S. S. et al., (2019), which provide mathematical approaches to analyze diseases to enable us to get a control over diseases.

## Key Terms in this Chapter

Stem Cells: They are basic cells from which red blood corpuscles and white blood corpuscles are produced.

Equilibrium Points: They are points (x(s),y(s)) for solution curves x and y of a system of differential equations with independent variable t, at which x(s) and y(s) are equal.

Synchronization Methods: They are mathematical methods which consider some equation as a prime equation and the methods always adjust solutions for secondary equations as approximate solutions of the prime equation.

Mathematical Modeling: Converting real life situations into mathematical concepts and symbols and thereby converting real life problems into mathematical problems.

Stability: This is a required behavior of parametric solution curves in neighborhoods of an equilibrium points.

Fixed Points: They are points x* for a function f from a set X to itself such that x* are members of X, and such that f(x*) coincides with x*.

Heat Rashes: They are rashes which appear due to above normal heat in climate changes.

Network Modeling: It is a mathematical modeling in which the problems or procedures to solve the problems may be described in terms of a network comprising of nodes and paths connecting nodes.

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