Methods
The conventional infectious disease model considers an exponentially rise in cases during the transition period.4 The ordinary differential equations are compatible to work with the exponential transition period model. The methodology of exponential time periodswasdoneusing the Gillespie algorithm.5 Since the time gaps are based on the patient’s disease status,we divided theminto states based on the treatment received and infeced with SARS-CoV-2 andno treatment received and not infected with SARS-CoV-2, so that a hazard model could be applied.6 Model structures were applied through agent-based stochastic procedure.7
We assumed that patients with progressive disease and those receiving active treatment could not avoid a hospital vist and would continue rendering further treatment. Any of the above treatments would decline their immunity level from 0-100%. Due to social distancing and stringentmobility criteria, we assumed that there were no follow-up visits and these patients would have a nil to minimal risk of contracting COVID-19. The states are expressed as\(\text{\ E}^{1},E^{2},E^{3}\)and \(E^{4}\)dipicted in a directed acyclic graph(DAG).8The corresponding time shift from one state to another is defined by time \(T^{1},T^{2},T^{3}\) and\(T^{4}\) respectively. (Figure 1)
A multistate model was used to specify the treatment initiation, disease progression, COVID-19 transmission and death. A cohort of stage IV oral cancer patients being treated or not were considered for analysis and their data was simulated. The transmission time was generated assuming that the transmission would occur during their hospital visit. The transmission probabilities and cumulative incidence were generated. The computation was performed using R software. The ”mstate” and ”muhaz” packages were used.
Multistate model
Multi-state models are often used to describe the life history of an individual. It defines several possible events for a single individual or the dependence between several individuals. Events are considered when there is a transition between the states. This model is useful to represent an extremely flexible approach that can model almost any kind of longitudinal failure time data.9 Our model was formulated with states and transition steps. The DAG is formulated to describe the transition and time period for the transmission (Figure 2).
Patients that continue receiving treatment were defined as \(E_{i}\), and there could be n number of patients theoretically. Another state that received no treatment is defined as\(E_{j}\).Here,\(E_{1}\),\(E_{2}\),\(E_{3}\),\(E_{4}\) shows the transition from states\(E_{i}\) to \(E_{j}\) while \(i<j\). We used the stochastic process defined as \(\left(X_{\left(t\right)}\right),t\geq 0\) to explain the different states. The superset of states was defined as\(\varepsilon=\{E_{1},\ldots.E_{n}\}\).\(S_{i}=\inf\left\{t\geq 0\middle|X_{t}{=E}_{i}\right\}.\)
We thenformulated the transition times of state \(E_{j}\)from\(E_{i}\)as\(T_{j}=S_{j}-S_{max\{k|k<j,S_{k}<\infty\}}\),assuming that\(S_{0}\)=0. The entire process was then defined by the transition time\(T_{j}\) to state \(E_{j}\).The hazard functions \(h_{\text{ij}}\) for the transition from \(i\) to j was defined as\(\text{\ T}_{\text{ij}}\sim F_{\text{ij}}=1-exp\{-\int_{0}^{t}{h_{\text{ij}}\left(u\right)du\}}\)and\(T_{j}=\min_{i\in\{1,\ldots.,J-1|T_{i}<\infty\}}T_{\text{ij}}\). The cumulative distribution was presented as\(\text{\ F}_{\text{ij}}\)for the transition from \(E_{i}\)to \(E_{j}\).Finally, all hazards were considered as constant X using the Markovian structure.
Hazards model
Assuming that the primary setups of the patients are presented with state\(\text{\ \ E}_{1}\), further two states would be formulated as\(\text{\ \ E}_{2}\) and \(\text{\ \ E}_{3}\). The intermidiate and absorbing state would be defined as \(E_{4}\).
The probability of transition from \(E_{i}\)at time s to state \(E_{j}\)at time t is presented as\(p\left(s,t\right)=P\left[X=E_{j}|X_{s}=E_{i}\right]for\ s\leq\ t\).
If\(i<j\ \) and \(\neq j,\) it may be formulated as\(p_{\text{ii}}\left(s,t\right)=exp\{-\int_{s-S_{i}}^{t-S_{i}}{h_{i4}(u)}du\}\). Transition from i=4 to i=4 was not possible, but i=1 to j=2,3,4 were possible choices. Similarly, i=2 to j=3,4 were the other possibilities, i.e.\(p_{i4}\left(s,t\right)=1-\exp\left\{-\int_{s-S_{i}}^{t-S_{i}}{h_{i4}\left(u\right)}\text{du}\right\}\)and\(p_{\text{ii}}\left(s,t\right)=0\).
Assuming that \(S_{1}\)=0, the transition probability can be obtained through integration
\begin{equation} p_{11}\left(s,t\right)=exp\{-\int_{s}^{t}{h_{12}\left(u\right)du-}\int_{s}^{t}{h_{13}\left(u\right)du-\int_{s}^{t}{h_{14}\left(u\right)\text{du}}}\nonumber \\ \end{equation}\begin{equation} p_{12}\left(s,t\right)=\int_{s}^{t}{p_{11}\left(s,u\right)h_{12}\left(u\right)p_{22}\left(u,t\right)\text{du}}\nonumber \\ \end{equation}\begin{equation} p_{13}\left(s,t\right)=\int_{s}^{t}{p_{11}\left(s,u\right)h_{13}\left(u\right)p_{23}\left(u,t\right)\text{du}}\nonumber \\ \end{equation}\begin{equation} p_{14}\left(s,t\right)=1-p_{11}\left(s,t\right)-p_{12}\left(s,t\right)-p_{13}(s,t)\nonumber \\ \end{equation}
This process was defined as X for state \(E_{1}\) shifting from time s to u. The states\(E_{2}\) or \(E_{3}\) will move from time \(t\). The transition probability \(p_{12}\) and \(p_{13}\) can be obtained by calculating the integration over u.This integration can be obtained by simulating the transition time linked to the probabilities (Figure 3).
Simulation of Hazards
We simulated the hazard function to understand the magnitude of mortality. The transition specific hazard function was formulated with the assumption that hazard withmean\(\mu=4\) will specify the constant function with \(h\left(t\right)=\frac{1}{\mu}\) having parameter\(\ \mu=4\). The time points \(T_{\text{ik}}\)by the hazard functions can be explained in the DAG (Figure 2). The minimum time period for shifting one state to another state is represented as prefixed k.It was possible to take a minimum of k. If the transition time is \(T_{j}\)and connecting state \(E_{j}\).Therefore, we can simulate X from the initial state \(T_{1k}\) to calculate the first transition within a minimum period. The simulation then could be obtained by the corresponding state \(E_{j}\). It was iterated until it became nil at the end of the simulation.
The hazard function was formulated as \(h(t)\) with piecewise constant function\(h_{\text{pc}}(t)\). Using the msm package for simulation, the transition probabilities from the first state at time \(t=0\) by the process X was calculated.10 We then simulated for N (Figure 4). Similarly, we used real data obtained from the website (https://ourworldindata.org/) and the data available on the coronavirus positive cases in India. This data is presented for the general population. The simulated portion for cancer patients’ data is included for comparison.11,12 (Figure 4)