MATHEMATICAL MODELING FOR PREDICTING POPULATION OF MALARIA (CASE STUDY AMINU KANO TEACHING HOSPITAL)

 

Malaria is one of the most fatal diseases in the world. The symptoms that characterizes malaria may have been observed as far back as the prehistoric period, through the classical era but it was not until the European renaissance period that the name malaria was derived from the Medieval Italian word, mal aria meaning \bad air", thinking that the foul vapours emanating from the stagnate water and swamps was the cause of fever, a major symptom of the disease.

A brief historical overview of the disease shows that some descriptions of what seemed to be the disease symptoms are given in the historical records of some early civilizations. The Chinese record, Huangdi Neijing describes the disease as repeated fever paroxysm that causes enlargement of the spleen with the potential of generating an epidemic. Ateminisinin combination treatment, a front line drug adopted by the World Health Organization for the treatment of malaria came from a Chinese plant, Qing-hao. This was discovered about 2300 years ago when it was used to treat acute intermittent fever episodes. An account of the disease is also given in the ancient Egyptian medical Papyri. For instance, the ancient Hindus of India ascribe the disease to the bite of a certain insect. Ancient Greeks, including Homer, Empedocles and Hippocrates also referred to the disease as having characteristics of intermittent fever causing enlarged spleens seen in people living in marshy places. It is believed by some researchers that malaria must have been responsible for the fall of the Roman Empire following an archaeological discovery of the presence of malaria in the bones of a Roman child who died 1500 years ago. The cause of malaria was not known from the down of history until later part of the 19th century when Charles Laveran discovered the malaria parasite in human blood in Africa. Few years later, Giovanni Grassi and Raimondo Filetti used the word plasmodium to name the malaria parasite and in 1897, Ronald Ross demonstrated that plasmodium parasite can be transmitted from infected human to mosquitoes.

Malaria forecasting can be an invaluable tool for malaria control and elimination efforts. A public health practitioner developed a simple forecasting method, which led to the first early-warning system of malaria. Forecasting methods for malaria have advanced since that early work, but the utility of more sophisticated models for clinical and public health decision making is not always evident. The accuracy of forecasts is a critical factor in determining the practical value of a forecasting system. The variability in methods is the strength of malaria forecasting, as it allows for tailored approaches to specific settings and contexts. There should also be continued effort to develop new methods although common forecasting accuracy measures are essential as they will help determine the optimal approach with existing and future methods. 

When performing forecasting, it is important to understand the assumptions of forecast models and to understand the advantages and disadvantages of each. Forecast accuracy should always be measured on reserved data and common forecasting measures should be used to facilitate comparison between studies. One should explore non-climate predictors, including transmission reducing interventions, as well as different forecasting approaches based upon the same data. 

This project work is mainly concerned with the use of first ordinary differential equation in modeling and predictions of population of malaria patients. Many people are interested in a way populations grow and in determining what factors influence their growth. Knowledge of this kind is important in studies of bacteria growth, wildlife management, ecological and harvesting. Some models for population growth are very simple while others can be sophisticated.

Consider a population of bacteria by a simple cell division. We assume that the growth is proportional to the population present. It is natural to think of the population changing from season to season, as such, the state government needs to know what would be the total population of the state, local government in the next n-years ahead for effective and efficient planning in terms of budgeting, resources allocation, social institution, social welfare and good governance.

It is important to explain not only what population modeling is, but also why it is worth doing.

The objective is to provide an approach to formulating and tackling problems in terms of mathematics. The process of building an effective mathematical model take skills, imagination and objective evaluation of mathematical tools, so that all the problem can be obtained with good model which is then translated into useful solution to real problem.

Mathematical model is an equation or system of equations that describes a real life system, In this description, a mathematical model could be a system of equations, process (random or probabilistic behavior), a geometric, or algebraic structures of algorithm or even just a set of number. The term real world could refer to a physical system, a social system, an ecological system, or essentially any system whose behavior can be observed. Mathematical model is an attempt to study the situation in the real world.

The mathematical model is not only in natural science and engineering discipline (such as physic, biology, meteorology etc) but also in social sciences (such as economics, psychology, sociology and political science etc). The process of developing a mathematical model is known as mathematical modeling.

The following are brief guiding principles of mathematical modeling.

These are certain characteristics we look for when trying to evaluating a model. They are as follows:

Many phenomena can be described in general way by saying that rate of change of the variable involved depends on past and present values of these variables. In other words, the variables are time dependent and so these situation lead to models involving differential equations. In the case of this project, time is absolute, since we are dealing with already recorded data.

Models in the physical sciences are usually solved to give an exact solution, hence, such solution are called analytic solutions. The main limitation of this approach is that it may not be possible to solve the equations analytically, since the solution of most equation cannon be found except that quantitative differential equations are model that lead to differential equations that have explicit solution that can be tested for accuracy by making predictions ant testing the results against existing data.

The rate of change of variable (population) is time dependent and it leads to models involving differential equations. First order ordinary differential equation are very useful in modeling of population as it appears frequently in modeling the attempt to describe real situation of population. A model for population growth (exponential) will be presented since population change in integer form. However, if the population is relatively large, the change due to the addition or deletion of one individual is small when compared to the whole population. These small changes enable us in certain cases to approximate the population by continuous function

δP(t)/δt αP(t)

δP(t)/δt=rP(t)

Separating variables give

δP(t)/(P(t))=rδt

Integrating both sides

∫▒〖δP(t)/(P(t))=∫▒rδt〗

∫▒〖δP(t)/(P(t))=r∫▒δt〗

lnP(t)=rt+c

Taking exponential of both sides gives,

P(t)=e^(rt+c)

P(t)=e^rt.e^c

Put      e^c=P_O

P(t)=e^rt.P_o

P(t)=P_(o ) e^rt

Variables representation;

P_(o =) Initial population at time t

r= Growth rate

t = Time

P(t)= The size of the population at a given time t 

The Significance of this study is to  understanding  the population growth rate and also predicts the future population. 

Although there are many population projection models, as such the scope of this research work is limited only to predict  the future population of Sokoto state using only the simple exponential growth model from 2006-2015 and also to compare between male and female population. It only takes care of population growth. 

Statement of an equality between two expressions, used in almost all the branches of pure and applied mathematics and in the physical, biological and social sciences. An equation usually involves one or more unknown quantities, called variables or indeterminate. A it equation is a statement that says two algebraic expression are equal

A variable is a symbol or letter as x,y etc. that represents a number. It can also be seen as a symbol that can assume any of a prescribed set of values within the domain of discussion.

A differential equation is a relationship between an independent variable, x, dependent variable, y, and one or more derivatives of y with respect to x.e.g.

Differential equation represents dynamic relationship, i.e. quantities that change, and thus frequently occurring in scientific and engineering problems.

An equation containing only ordinary derivatives of one or more dependent variable with respect to a single independent variable, In general any function of x, y and the derivative of y up to any order such that:

        F (x,y,□(24&dy)/□(24&dx),d^2 y/〖dx〗^2,d^3 y/〖dx〗^3…)=0

Defines an ordinary differential equation for y (dependent variable) in terms of x (the independent variable) e.g.

Are all type of equations. 

The order of a differential equation is given by highest derivative involved in the differential equation.

Equation of this type can, in general be written as: 

dy/dx=(x,y) Or  f(x,y,y')=0 ,

Where f(x,y) is given function and a first order.

The derivative or differential of a variable say r with respect to the time t, is the change the variable r with respect to t. e.g. if r represent the population of people, then the change of the population with respect to time is given as: dr/dt.

This is the process of setting up models in terms of variables, function and equation, solving it mathematically and interpreting the result in physical terms.

This means the total number of people living  in a particular area, city, region, country, or even a continent at a given period [point] per time.

This is the rate of change of population per time.

This means representation of data [population] in form of a graph.