A dimensional reduction of an (n+1) compartmental model

An (n+1) dimensional compartmental model is studied. It is an n dimensional model sits in (n+1) dimensional space, consists of (n+1) variables and (n+1) equations. The model is a generalization of well-known SIR model. Reduced dimensional model is introduced. The reduced model consists of n variables and n equations. The equilibrium points of the original model and reduced model are discussed. The stability of the equilibrium points are analyzed and compared. Numerical simulation is applied for n = 5. The numerical result which is the evolution of the variables is presented.


Introduction
A system consisting of ( + 1) ordinary differential equations is considered. Study on such system is still an interesting topic, both from theoretical point of view and the applications. Study on differential equations was initialized after the invention of calculus where mathematics at those times meant study of change, see [1] for a brief history, modern work can be found in the books [2,3]. Moreover, the applications have spread not only in science and engineering, but also from public health to economics.
For the applications on public health for disease spread, one among well-known work was SIR model, a model of disease spread among compartmental populations, see [4] for detail discussion. SIR stands for susceptible, infected, recovered. A simple model applied for complex transitions in epidemics was discussed in [5]. The approach of mathematical of change was applied to study time series modeling of childhood diseases [6], for disease spread among predator prey system [7]. More recently, study on a vector-borne disease model with age of vaccination was presented in [8]. An overview on the application of system dynamics for epidemics was well narrated in [9].
In economics and finance, the use of nonlinear dynamic models has expanded rapidly since the late of 1980s, [10]. Guegan [11] discussed the use of dynamical chaotic systems in economics and finance by employing methods that could be useful in practice to detect the existence of chaotic behavior inside real data sets. Jakimowicz [12] discussed model economic systems strived for a state called "the edge of chaos". He considered a case concerned an economy based on a two-stage accelerator, where the economic cycle adopted the form of chaotic hysteresis, a case concerned a Cournot-Puu duopoly model in which striving for the edge of chaos stems from profit maximization by entrepreneurs. It was observed that the evolution of systems at the edge of chaos could be sudden. Some recent works for application on business cycle for discrete and continuous time have been reported in [13,14]. Let ≥ 0, an integer > 2 and for = 1, ⋯ , + 1 be non negative functions of . Consider a system of equations in the form

A system of
(2) Observe that is an -dimensional set which sits in ( + 1)-dimensional ambient space. This problem is a generalization of the results discussed in [15], where its application is for knowledge dissemination [16] for = 3. The analysis was conducted to the system (1) directly, to the reduced system, and also the relation of the two systems. For the case = 3, 5 system (1) has also directly been studied in [17]. For the case = 2, see [15], the system is in the form (1)  The proof is obvious.

Proof
This Jacobian matrix of (3) evaluated at is On the other hand, let ( * : = 1, ⋯ , − 1, + 1) be any point in . The Jacobian matrix of the system (3) evaluated at is Eigenvalue , guaranties that the equilibrium manifold is not stable. Hence, is a stable point and is an unstable manifold. The proof is obvious, since ̇ is not negative.

Numerical simulation
For numerical simulation purposes, the case of = 5 is considered. Four cases are taken into account. Each case is related to the parameters listed in the Table 1. The sum of the state variables is normalized ( = 1). The initial conditions of each variable are listed in Table 2. Variables , for = 1,2,3,4,6 are computed using (3). Variable is computed by applying    Figure 1 shows the evolution of for = 1,2, ⋯ ,6 graphically; (a) for case 1, (b) for case 2, (c) for 3 and (d) for case 4. As = = 0 in case 1, hence and are constant. Variables and decrease, and decreases very slowly because the small positive value of . On the other hand, significantly increases. This is presented in Figure 1(a). In Figure 1(b) where = 0 and other parameters are positive, varible is constant. State variables and increase, while , and decrease. Observe that the evolution of is much different than in case 1, Figure 1(a). In Figure 1(c) = 0 and other parameters are positive. State variables and increase, while , and decrease. Observe that the evolution of decrease slowly as also indicated in case 1, Figure 1(a). In Figure 1(c) all parameters are positive. State variables and increase, while , , and decrease.

Conclusion and further research
An dimensional and compartmental model that sits in + 1 dimensional space has been discussed. The model is a dynamical system that consists of + 1 variables. The system has been reduced to a system of variables. The reduced system has an equilibrium point which is stable, and an equilibrium manifold that is unstable. Future work will be on the analysis of the equilibrium computed directly from + 1 dimensional compared to the one from its reduced system.