## 2 Mathematical Model  
Human immune systems have a very powerful and effective way of eliminating viruses. Macrophages and dendritic cells (DCs) are activated by virus-infected cells through the presentation of virus antigen [18; 39]. Additionally, when virus-infected cells undergo lysis, cytokines and antigens are released, and these can activate DCs and macrophages [18]. These activated (mature) cells then stimulate helper T cells and killer T cells. Helper T cells also secrete specific cytokines (specifically type Th1) that provide continual stimulation of killer T cells. Previously, we developed a system of ODEs that assumed the primary driver of the immune response was virus-infected tumour cells [36]. The virus-infected tumour cells stimulated APCs which in turn activated killer T cells and helper T cells. This system was used to model an oncolytic adenovirus expressing IL-12 and GM-CSF (Ad/IL12/GMCSF) created by [PERSON] et al. [13]. Intratumoral doses of Ad/IL12 were shown to induce the activation and recruitment of helper T cells and killer T cells, whereas administration of Ad/GMCSF strongly recruited APCs to the tumour site. In this work, we consider the injection of Ad/IL12/GMCSF, combined with an injection of immature DCs. Due to the presence of additional immature DCs, we believe the stimulation of the immune system by uninfected tumour cells would be crucial, and we extend the system of ODEs in [PERSON] et al. [36] to consider this,  
\[\frac{dU}{dt} =r\log\left(\frac{L}{U}\right)U-\beta\frac{UV}{T}-\kappa\frac{ KI}{T}, \tag{1}\] \[\frac{dI}{dt} =\beta\frac{UV}{T}-d_{I}I-\kappa\frac{KI}{T},\] (2) \[\frac{dV}{dt} =u_{V}(t)-d_{V}V+\alpha d_{I}I,\] (3) \[\frac{dD_{S}}{dt} =(1-f)u_{DC}(t)-s_{AU}D_{S}U-s_{AI}D_{S}I-d_{S}D_{S},\] (4) \[\frac{dD_{L}}{dt} =fu_{DC}(t)-s_{AU}D_{L}U-s_{AI}D_{L}I-d_{L}D_{L},\] (5) \[\frac{dA_{I}}{dt} =r_{AI}I-s_{AU}A_{I}U-s_{AI}A_{I}I-d_{AI}A_{I},\] (6) \[\frac{dA_{M}}{dt} =s_{AU}U(A_{I}+D_{L}+D_{S})+s_{AI}I(A_{I}+D_{L}+D_{S})-d_{A}A_{M},\] (7) \[\frac{dH}{dt} =s_{H}A_{M}-d_{H}H,\] (8) \[\frac{dK}{dt} =s_{KH}H+s_{KA}A_{M}-d_{K}K, \tag{9}\]  
where \(t\) is time (days), \(U\) is the uninfected tumour population, \(I\) is the infected tumour population and \(V\) is the number of virus particles. As the model was developed for an adenovirus expressing IL-12 and GM-CSF, the populations of immune cells considered here are those most affected by these cytokines: activated (or mature) antigen-presenting cells (APCs), \(A_{M}\); helper T cells, \(H\); and killer T cells, \(K\). Additionally, as immature DCs are injected into the system, we now consider populations of short-lived and long-lived immature DCs that have been injected, \(D_{S}\) and \(D_{L}\), and immature APCs that are already present in the body, \(A_{I}\). The total cell population at the tumour site at any time \(t\) is given by \(T=U+I+D_{L}+D_{S}+A_{I}+A_{M}+H+K\). Figure 1 summarises the interactions modelled in Equations (1)-(9). The biological mechanisms described by each ODE in the system are described below. * In Equation (1), uninfected tumour cells, \(U\), are growing at a rate described by a Gompertz function with proliferation rate \(r\) and carrying capacity \(L\). Uninfected cells are infected by virus particles, \(V\), at a frequency-dependent rate with rate constant \(\beta\). Killer T cells, \(K\), induce apoptosis in uninfected cells at a frequency-dependent rate with rate constant \(\kappa\). * In Equation (2), tumour cells become infected cells, \(I\), at rate \(\beta UV/N\). Infected cells lyse at rate \(d_{I}I\), and, like uninfected cells, are killed at a frequency-dependent rate. * In Equation (3), virus particles enter the system at rate \(u_{V}(t)\), either by a single injection or release from the hydrogel. Virus particles decay at a rate \(d_{V}\), and \(\alpha\) new viruses are created through lysis. * In Equation (4), short-term immature DCs, \(D_{S}\), enter the system at a rate \((1-f)u_{DC}(t)\), either by a single injection or release from a hydrogel.