## 3 Calibrating Parameters to In Vitro and In Vivo Time-Series Measurements
### Tumour, Immune and Viral Parameters
As there was no injection of immature DCs in this experiment, we assumed the immune stimulation would be driven solely by the injected virus with the effects of endogenous DCs negligible, i.e., we fixed \(D_{S}=D_{L}=A_{I}=0\). Under this assumption, we removed any activation of APCs by uninfected tumour cells, and modelled APC stimulation as directly relating to infected cells \(I\), i.e., \(s_{AI}I(A_{I}+D_{L}+D_{S})=s_{AI}I\). The remaining parameters of the model describing the infection rate of the virus, \(\beta\); the killing rate of the killer T cells, \(\kappa\); and the initial tumour size, \(U_{0}\), were optimised, see Figure 3b. [PERSON] et al. (2010) then measured tumour growth under a single injection of \(2.5\times 10^{6}\) immature DCs. As there was no virus present, we fixed \(I=V=0\). The decay rate for immature APCs, \(d_{AI}\), was fixed to the short-term DC decay rate, \(d_{S}\), as we assumed that any recruited immature APCs would not reside at the tumour site for long. The parameters describing the stimulation rate of immature APCs, \(s_{AUI}\); the killing rate of killing cells, \(\kappa\); and the initial tumour size were optimised and fixed for subsequent optimisations, see Figure 3c. Following this, [PERSON] et al. (2010) combined a single injection of Ad/IL12/GMCSF with a single injection of immature DCs. Fixing the previous fit parameters, we optimised the kinetics for the activation of immature APCs and DCs using the full model in Equations (1)-(9). Optimising the rate that infected cells recruit inactivated APCs \(r_{AI}\), the stimulation of immature APCs \(s_{AI}\) and initial tumour size \(U_{0}\) gave the model curve in Figure 3d. The final experiment of [PERSON] et al. (2010) measured the tumour volume after a gel loaded with Ad/IL12/GMCSF and immature DCs was injected adjacent to the tumour. Using these measurements, we obtained the release profile for the virus from the gel \(u_{V}(t)\). Fixing the release profile of the immature DCs from the gel, \(u_{DC}(t)\), to the profile determined in Section 3.1 and all parameters to the values for the DC+Ad/IL12/GMCSF experiment, we assumed the release rate for the virus would be linear, and the time of release \(t_{r}\) would be equivalent to that obtained for the DCs, \(t_{r}=5.945\) days,  
\[u_{V}(t)=\left\{ \begin{array}{cl}a_{V}t+b_{V},&\text{for }t\leq t_{r}\\ 0&\text{otherwise}\end{array} \right.\]  
where \(a_{V}>0\) and \(b_{V}\geq 0\). The value of \(b_{V}\) was obtained by fixing the initial amount of virus as \(V_{0}=2\times 10^{10}\) and integrating \(u_{V}(t)\). Additionally, we obtained an upper and lower bounds of \(a_{V}\) using the condition that \(u_{V}(t)\) is an increasing function. Optimising the release function parameter \(a_{V}\) was not sufficient, and a fit could only be obtained when the activation rate of APCs by uninfected cells was also allowed to vary. The resulting fit is in Figure 3e and the parameter values are in Table 3. The model dynamics for this experiment are plotted adjacent in Figure 3f. All parameter optimisations were undertaken using a least-squares nonlinear fitting algorithm \"lsqnonlin\" in MATLAB2017a. The termination tolerance was \(1\times 10^{-6}\). The maximum number of function evaluations was fixed as \(100\times\)number of parameters. The maximum number of interactions for each fit was \(400\). The solver ode45 was used to solve the model. The model was fit simultaneously to the set of individual mouse data for a particular experiment. When solving Equations (1)-(9) numerically, \(T\) was replaced by \(T+\epsilon\) for \(\epsilon=0.001\) to avoid a singularity as \(T\to 0\). The goodness of fit statistics have been reported in Table 4.  
It is evident visually in Figure 3 and from the goodness of fit measurements (Table 4) that our model and the sequential fitting algorithm we used in Table 2 provide a reliable representation of the data.