The aim of this project is to develop and apply a predictive computational model for the Covid-19 epidemic based on a high-dimensional population balance modelling.

Let $T_\infty$ be a given final time and $\Omega:=\Omega_x\otimes\Omega_\ell$ be the computational domain of interest. Here, $\Omega_x\subset\mathbb{R}^2$ is the spatial domain defining the geographical region of interest and $\Omega_\ell:=L_v\times L_d\times L_a$ is an internal domain, where $L_v=[0, 1]$ denotes the Covid-19 infection severity interval, $L_d=[0,d_{\max}]$ denotes the time interval since contracting the disease, and $L_a=[0, 1]$ denotes the non-dimensional age interval with the non-dimensionalization constant being 125 years. The infection index $\ell_v\in L_v$ quantifies the severity of the population being infected by Covid-19. Specifically, the population with infection index $\ell_v=0$ has recovered completely from Covid-19, with $\ell_v = 1$ has died due it, with $\ell_v\ge v_{\rm sym}$ shows symptoms and those with $\ell_v\le v_{\rm sym}$ are asymptomatic. The time since contracting the disease index $\ell_d\in L_d$ quantifies the time since a population has been exposed to and contracted the disease. Specifically, the population that just contracted the disease has $\ell_d=0$. Typically, a person is asymptomatic until they reach $\ell_v=v_{\rm sym}$, and the duration elapsed $\ell_d$ is the incubation period in which the disease is sub-clinical and that population is actively spreading the disease. After recovery, a population doesn't necessarily go to $\ell_v=0$, rather they reach $\ell_v\le v_{\rm reco}$.

... Let $I(t,x,\ell_v,\ell_d,\ell_a)$ be the size distribution function of the infected population. To describe the evolution of the active infected population size distribution, we propose the population balance equation \begin{equation}\label{model} \begin{array}{rcll} \displaystyle\frac{\partial I}{\partial t} + \nabla_x\cdot({\bf u} I) +\nabla_\ell\cdot({\bf G} I) +CI &=& F \quad &{\rm in } \quad (0,T_\infty]\times\Omega_x\times\Omega_\ell \,,\\ {\bf u}\cdot {\bf n} &=&g_n &{\rm in } \quad (0,T_\infty]\times\partial\Omega_{ x}\,, \\ I(0,x,\ell) & =& I_0 &{\rm in } \quad \Omega_{x}\times\Omega_\ell \,,\\ I(t,\mathbf{x},\ell_v,0,\ell_a) & =& B_{\rm nuc} \quad &{\rm in } \quad (0,T_\infty]\times\Omega_{x}\times L_v\times L_a \,,\\ I(t,\mathbf{x},0,\ell_d>0,\ell_a) & =& 0 \quad &{\rm in } \quad (0,T_\infty]\times\Omega_{x}\times L_d\times L_a \,. \end{array} \end{equation} Here, ${\bf u}$ denotes the advection vector that quantifies the spatial movement of the population in a differential neighbourhood of $\Omega_{x}$, $f(t,x,\ell_a)$ denotes the net addition of an infected population into $\Omega_{\rm x}$ from outside, ${\bf n}$ is the outward unit normal vector to $\Omega_x$, $g_n$ is the flux that quantifies the net addition of an infected population into $\Omega_{\rm x}$ from outside (the spatial movement of the population across the border of the domain $\partial\Omega_{\rm x}$), and $I_0$ is the initial distribution of infected population. Further, $ {\bf G} = (G_{\ell_v}, G_{\ell_d}, G_{\ell_a})^T $ is the internal growth vector. Here, the Covid-19 infection severity growth rate in an infected population is defined by \begin{equation} G_{\ell_v} = \frac{d \ell_v}{d t} = G_{\ell_v}(\ell_a, \beta, \gamma(\ell_a), \alpha(x) ), \end{equation} where $\beta$ is the immunity of the infected population, $\gamma$ is the pre-medical history of the infected population and $\alpha$ is the effective treatment index. Additionally, the time since infection index has a growth rate \begin{equation} G_{\ell_d} = \frac{d \ell_d}{d t} = 1\,. \end{equation} Moreover, the change of age of the active, infected population can be modelled through the growth rate \begin{equation} G_{\ell_a} = \frac{d \ell_a}{d t}. \end{equation} Nevertheless, the influence will be negligible when the study is performed for a short duration $(0, T_\infty]$, and thereby may be ignored. $G_{\ell_v}$ is the growth function of the Covid-19 infection index $l_v$, We next rate term \begin{equation} C = C_R + C_{ID}, \end{equation} where $C_R(t,\mathbf{x},\ell_v,\ell_d,\ell_a)$ is a recovery rate function that quantifies the rate of recovery of the population from Covid-19, $C_Q(t,\mathbf{x},\ell_v,\ell_d,\ell_a)$ is a quarantine rate that quantifies the removal of people into a quarantine facility and $C_{ID}(t,\mathbf{x},\ell_v,\ell_d,\ell_a)$ is the Covid-19 death rate. We also define a source term $F = C_T(t,\mathbf{x},\ell_v,\ell_d,\ell_a)$ that quantifies the point-to-point movement of infected population (e.g., by air, train etc) within $\Omega_x$ not explained by ${\bf u}$ and the source/sink of infected population travelled by air into $\Omega_x$ not explained by $g_n$. Moreover, $C_T$ and ${\bf u}$ need to be defined in such a way that the net internal movement of infected population within $\Omega_{x}$ is conserved. Moreover $B_{\rm nuc}$ is the nucleation function that quantifies how a susceptible population becomes infected and it is defined by \begin{equation} B_{\rm nuc}(t,\mathbf{x},\ell_v,\ell_a) = B_{\rm nuc}\left(\ell_a, \sigma, H, S_D, N_S(t), N_Q(t) I\right), \qquad \forall ~\ell_v,\ell_a\in L_v\times L_a. \end{equation} Here, $\sigma\in[0,1]$ is the interactivity index, $H\in[0,1]$ is the hygiene index, $S_D\in[0,1]$ is the social distancing index. Finally, the total population $N(t)$ at a given time $t\in(0,T_\infty]$ is defined by \begin{align*} N(t) &= N_S(t) + N_R(t) + N_I(t) +N_Q(t) - N_{ID}(t) - N_D(t), \\ N_I(t) &= \int_{\Omega} I(t,\mathbf{x},\ell_v,\ell_d,\ell_a)\,d\Omega\,,\\ N_Q(t) &= \int_{\Omega} \gamma_Q(t,\mathbf{x},\ell_v,\ell_d,\ell_a)I(t,\mathbf{x},\ell_v,\ell_d,\ell_a) \,dx\,d\ell\,,\\ N_{R}(t) &= \int_{\Omega} C_RI(t,\mathbf{x},\ell_v,\ell_d,\ell_a)\,d\Omega\ \,,\\ N_{ID}(t) &= \int_{\Omega} C_{ID}I(t,\mathbf{x},\ell_v,\ell_d,\ell_a)\,d\Omega \,,\\ N_S(t) &= N(t) - [N_I(t)+N_Q(t)+N_R(t)-N_{ID}(t)-N_D(t)] \,. \end{align*} Here, $N_S$, $N_B$, $N_R$, $N_I$, $N_Q$ $N_{ID}$ and $N_D$ are the number of susceptible, newborn, recovered, asymptomatic/symptomatic infectives, qurantined, infective death and natural death populations, respectively. The population in the interval $(v_{\rm reco},v_{\min})$ has recovered from Covid-19.

Finite Element Approximation

A finite element scheme ^[1] based on operator splitting ^[2-5] has been implemented in our in-house (open-source) finite element package to solve the high-dimensional population balance equation.

References

1. S. Ganesan, L. Tobiska, Finite Elements: Theory and Algorithms, Cambridge IISc Series, Cambridge University Press, 2016.
2. S. Ganesan, L. Tobiska, Operator-splitting finite element algorithms for computations of high-dimensional parabolic problems, Appl. Math. Comp. 219 (2013) 6182-6196.
3. S. Ganesan, L. Tobiska, An operator-splitting finite element method for the efficient parallel solution of multidimensional population balance systems, Chem. Eng. Sci. 69 (1) (2012) 59-68.
4. S. Ganesan, An operator-splitting Galerkin/SUPG finite element method for population balance equations: Stability and convergence, ESAIM: M2AN 46 (2012) 1447-1465.
5. P. S. kumar, S. Ganesan, Numerical simulation of nanocrystal synthesis in a microfluidic reactor, Chem. Eng. Sci. 96 (2017) 128-138.