Year: 2015

Mathematical models allow us to extrapolate from current information about the state and progress of an outbreak, to predict the future and, most importantly, to quantify the uncertainty in these predictions. Many sources of data are used in mathematical modeling, with some forms of model requiring vastly more data than others. However, a good estimation of the number of cases is vitally important. We stretch existing knowledge on model formulation and develop an enhanced mathematical model to study the pattern of spread of infectious diseases. The SIR model formalism was used to compartmentalize the population and the resulting model equations were solved numerically. The disease free equilibrium and endemic equilibrium of the system were established and analyzed for stability. A graph representation of the sub groups is presented and discussed based on the results from simulation.

In many countries of the developed World, infectious diseases of the childhood in particular have been generally conquered. This however, is far from being the case in developing countries, where they are responsible for 45% of all death. Coming down home, the data of infectious disease prepared by the Department of Health Planning and Research, Federal Ministry of Health at our disposal shows the same trend in Rivers State.
A mathematical model is a set of equations, which are the mathematical translation of hypothesis (or assumptions) when interpreting model prediction. It is thus important to bear in mind the underlying assumptions. An assumption, by our definition, is an unverified proposition, tentatively accepted to explain certain facts or to provide a basis for further investigation.
Following analysis, modeling infectious diseases demands that we investigate whether the disease spread could attain a pandemic level or it could be wiped out. The equilibrium analysis helps to achieve this.

Fixed point theory occupies a prominent place in the study of metric spaces. One of the important questions that may arise in this connection is whether metric spaces really provide enough space for this theory or not. Recently, in Huang and Zhang (2007), the authors rather implied that the answer is no. Actually, they introduced the notion of cone metric space, and gave an example of a function which is a contraction in the category of cone metric spaces but not a contraction if considered over metric spaces and hence, by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point. After that a series of articles about cone metric spaces started to appear. Some of those articles dealt with fixed point theorems in those spaces, specially in complete ones, and others dealt with the structure of the spaces themselves.
On the other hand Cone metric spaces were introduced in Huang and Zhang (2007). The authors there described convergence in cone metric spaces, and introduced completeness. Then they proved some fixed point theorems of contractive mappings on cone metric spaces, where they also provided an example of a contractive mapping on a cone metric space with E = R2 and P = (x, y) ∈ R2: x, y ≥ 0. However this mapping is not contractive in the Euclidean metric space. Then recently, in (Ili and Rako, 2008; Abbas, and Jungck, 2008) some common fixed point theorems were proved for maps on cone metric spaces. In this paper we introduce some basic topological concepts and definitions in cone metric spaces. Actually, we prove that every cone metric space is a topological space. Then we generalize the concepts diametrical contractivity and asymptotical diametrical contractivity to cone metric spaces. Finally we obtain some fixed point theorems for such mappings. By this we generalize some results from Xu(2004) to cone metric spaces.