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)