  We observed existence of periodic orbit in immune network under transitive
solvable lie algebra. In this paper, we focus to develop condition of maximal
Lie algebra for immune network model and use that condition to construct vector
field of symmetry to study non linear pathogen model. We used two methods to
obtain analytical structure of solution, namely normal generator and
differential invariant function. Numerical simulation of analytical structure
exhibits correlated periodic pattern growth under spatio temporal symmetry
which is similar to linear dynamical simulation result. We used lie algebraric
method to understand correlation between growth pattern and symmetry of
dynamical system. We employ idea of using one parameter point group of
transformation of variables under which linear manifold is retained. In
procedure, we present the method of deriving Lie point symmetries, calculation
of first integral and invariant solution for the ODE. We show the connection
between symmetries and differential invariant solution of the ODE. The
analytical structure of the solution exhibits periodic behavior around
attractor in local domain, same behavior obtained through dynamical analysis.

Biological Physics;Chaotic Dynamics