## 4 From Exhaustive Search to Non-Linear Analysis
### Recurrence and Periodic Orbits Without Pulsed Immunotherapy  
We shall continue the analysis using the system (5)-(8), which has the Jacobian matrix  
\[\left(\begin{array}{cccc}\zeta_{3}&\zeta_{2}&\zeta_{1}&0\\ 0&\frac{D}{D+1}-r_{5}&0&r_{4}+\frac{C_{a}}{D+1}-\frac{C_{a}D}{(D+1)^{2}}\\ r_{6}&0&-r_{7}&0\\ 0&0&0&-r_{8}\end{array}\right) \tag{9}\] \[\zeta_{1} =\frac{C_{a}(a_{T_{\beta}}F_{\beta}+1)Tr_{3}}{(F_{\beta}+1)^{2}(T +1)}-\frac{a_{T_{\beta}}C_{a}Tr_{3}}{(F_{\beta}+1)(T+1)}\] (10) \[\zeta_{2} =-\frac{(a_{T_{\beta}}F_{\beta}+1)Tr_{3}}{(F_{\beta}+1)(T+1)} \tag{11}\]\[\zeta_{3}=-r_{2}Tr_{1}+(1-r_{2}T)r_{1}+\frac{(a_{T_{\beta}}F_{\beta}+1)Tr_{3}C_{a} }{(F_{\beta}+1)(T+1)^{2}}-\frac{C_{a}(a_{T_{\beta}}F_{\beta}+1)r_{3}}{(F_{\beta} +1)(T+1)}. \tag{12}\]  
The system (5)-(8) has two fixed points: \(E_{1}=(0,0,0,0)\) and \(E_{2}=(\frac{1}{r_{2}},0,\frac{r_{6}}{r_{2}r_{2}},0)\). The eigenvalues of (9) evaluated in \(E_{1}\) give us the eigenvalues \(\lambda_{1}=-r_{7}\), \(\lambda_{2}=r_{1}\), \(\lambda_{3}=-r_{5}\), \(\lambda_{4}=r_{8}\) which implies that \(E_{1}\) is a saddle point since all parameters are strictly positive. The eigenvalues in \(E_{2}\) are \(\lambda_{1}=-r_{7}\),\(\lambda_{2}=-r_{1}\),\(\lambda_{3}=-r_{5}\),\(\lambda_{4}=-r_{8}\), therefore, \(E_{2}\) is asymptotically stable. We now proceed to show that for our parameter configuration there are no periodic orbits in this system (5)-(8). The instability of the fixed point \(E_{1}\) shows that it is not likely to prevent a relapse of tumor population without pulsed immunotherapy. On the contrary, the stability of \(E_{2}\) exhibits the tendency of the tumor population to go to its carry capacity: \(\frac{1}{r_{2}}=\frac{K}{R_{T}}\). Another option is to observe if the tumor can stay in a periodic orbit where a threshold of lethality is not surpassed. For instance, a tumor mass of \(10^{5}-10^{9}\) cells is clinically not detectable; with \(10^{9}-10^{12}\) cells it is detectable, and over \(10^{12}\) it is lethal [22]. The next result shows that without pulsed immunotherapy it is not expected that the tumor stays in a periodic orbit under a non-lethal threshold because periodic orbits do not exist for our set of parameters. To show this, we used [PERSON]'s criterion which can be consulted in reference [23].  
The system (5)-(8) has no periodic orbit if \(\mu_{\beta}+\mu_{\mathcal{D}}\theta_{D}+2\frac{Tr_{1}}{K}T>r_{T}\).  
Proof.: We proceed using [PERSON]'s criterion in the region \(G=\{(T,C_{a},F_{\beta},D)|T\geq 0,C_{a}>0,\)\(F_{\beta}\geq 0,D\geq 0\}\). Let \(g=\frac{1}{C_{a}}\) and \(x=(\frac{dT}{dt},\frac{dC_{a}}{dt},\frac{dF_{\beta}}{dt},\frac{dD}{dt})\) in (5)-(8) then  
\[\
abla\cdot(gx)=-\Bigg{(}\frac{r_{4}}{C_{a}^{2}}+\frac{r_{7}+r_{8}-r_{1}+2r_{ 2}Tr_{1}}{C_{a}}+\frac{(1+a_{T_{\beta}}F_{\beta})r_{3}}{(1+F_{\beta})(1+T)^{2 }}\Bigg{)}.\]  
So if \(r_{7}+r_{8}-r_{1}+2r_{2}Tr_{1}>0\) then \(\
abla\cdot(gx)<0\) on \(G\). This in the original parameters is \(\mu_{\beta}+\mu_{\mathcal{D}}\theta_{D}+2\frac{r_{2}Tr_{1}}{K}T>r_{T}\).  
Using the parameters of Table 4 gives us \(\mu_{\beta}=2.7\) and \(r_{T}=0.001060289\) then the condition of (13) is consummate for this paper.