Suppose that $S _ { 0 } > 0$. Show that the function $S$ of the solution triplet $( S , I , R )$ of $(F)$ satisfies the relation $$\left( - \frac { S ^ { \prime } } { S } \right) ^ { \prime } = - S ^ { \prime } + \frac { S ^ { \prime } } { S }$$ The system $(F)$ is: $$( F ) : \left\{ \begin{array} { l } S ^ { \prime } ( x ) = - I ( x ) S ( x ) \\ I ^ { \prime } ( x ) = I ( x ) S ( x ) - I ( x ) \\ R ^ { \prime } ( x ) = I ( x ) \\ S ( 0 ) = S _ { 0 } , \quad I ( 0 ) = I _ { 0 } , \quad R ( 0 ) = R _ { 0 } \end{array} \right.$$
Suppose that $S _ { 0 } > 0$. Show that the function $S$ of the solution triplet $( S , I , R )$ of $(F)$ satisfies the relation
$$\left( - \frac { S ^ { \prime } } { S } \right) ^ { \prime } = - S ^ { \prime } + \frac { S ^ { \prime } } { S }$$
The system $(F)$ is:
$$( F ) : \left\{ \begin{array} { l } S ^ { \prime } ( x ) = - I ( x ) S ( x ) \\ I ^ { \prime } ( x ) = I ( x ) S ( x ) - I ( x ) \\ R ^ { \prime } ( x ) = I ( x ) \\ S ( 0 ) = S _ { 0 } , \quad I ( 0 ) = I _ { 0 } , \quad R ( 0 ) = R _ { 0 } \end{array} \right.$$