Dynamics of COVID-19 transmission with comorbidity: a data driven modelling based approach

Abstract

An outbreak of the COVID-19 pandemic is a major public health disease as well as a challenging task to people with comorbidity worldwide. According to a report, comorbidity enhances the risk factors with complications of COVID-19. Here, we propose and explore a mathematical framework to study the transmission dynamics of COVID-19 with comorbidity. Within this framework, the model is calibrated by using new daily confirmed COVID-19 cases in India. The qualitative properties of the model and the stability of feasible equilibrium are studied. The model experiences the scenario of backward bifurcation by parameter regime accounting for progress in susceptibility to acquire infection by comorbidity individuals. The endemic equilibrium is asymptotically stable if recruitment of comorbidity becomes higher without acquiring the infection. Moreover, a larger backward bifurcation regime indicates the possibility of more infection in susceptible individuals. A dynamics in the mean fluctuation of the force of infection is investigated with different parameter regimes. A significant correlation is established between the force of infection and corresponding Shannon entropy under the same parameters, which provides evidence that infection reaches a significant proportion of the susceptible.

Keywords: Backward bifurcation; COVID-19; Comorbidity; Model calibration; Sensitivity analysis; Shannon entropy.

Fig. 1

Schematic illustration of SCEAIHR model. The flow diagram exhibits the interaction of different stages of individuals in the model: susceptible (S), comorbidity (C), exposed (E), asymptomatic (A), infected (I), hospitalized (H) and recovered (R)

Fig. 2

The SCEAIHR model fitted to daily new confirmed COVID-19 cases in India. Observed data points are shown in black dots and the solid red line depicts the model simulated curve

Fig. 3

a and b PRCC indicating sensitivity indices to infected individual (I) and basic reproduction number $(R_0)$. PRCC values of various parameters with the level of significance 0.05. Sample size = 500 for each parameters is taken based on LHS approach with uniform probability distribution

Fig. 4

Contour plots indicating the nature of change in basic reproduction number($R_0$) of SCEAIHR model under parametric planes. a $R_0$ versus $(\rho ,{\varphi }_s)\in (0,1]\times (0,0.1]$. b $R_0$ versus $(\rho ,{\beta }_s)\in (0,1]\times (0,2]$. (c) $R_0$ versus $({\varphi }_s,{\beta }_s)\in (0,0.1]\times (0,2]$

Fig. 5

a $R_0$ versus ${\zeta }^{\ast }$ plot indicating backward bifurcation of SCEAIHR model in $\rho \in [0.01,0.8]$. b $R_0$ versus ${\zeta }^{\ast }$ plot illustrating transcritical bifurcation of SCEAIHR model in ${\varphi }_s\in [10\times {10}^{-6},2.7\times {10}^{-5}]$ with $\rho =0$. All the remaining parameters values are reported in Table 1

Fig. 6

Impacts of variation in ${\varphi }_s$, on backward bifurcation with $\rho \in [0.01,0.8]$, keeping all parameters value remained same as in Table 1. The diagram exhibits that the extent of backward bifurcation regime increases gradually with the increasing of ${\varphi }_s$

Fig. 7

a, b represent ${\varphi }_s$ versus ${\zeta }^{\ast }$ plot (with $\rho \in (0,0.1])$ and ${\beta }_s$ versus ${\zeta }^{\ast }$ plot (with ${\beta }_s\in (0,2])$. c represent ${\zeta }^{\ast }$ over $({\varphi }_s,{\beta }_s)$ matrix plot, where $({\varphi }_s,{\beta }_s)\in (0.1]\times (0,2]$. The corresponding color bar indicates values of ${\zeta }^{\ast }$. The values of the other parameters are taken as same, shown in Table 1

Fig. 8

a, b represent ${\varphi }_s$ versus $En({\zeta }^{\ast })$ plot (with ${\varphi }_s\in (0,0.1])$ and ${\beta }_s$ versus $En({\zeta }^{\ast })$ plot (with ${\beta }_s\in (0,2])$. (c) represent $En({\zeta }^{\ast })$ over $({\varphi }_s,{\beta }_s)$ matrix plot, where $({\varphi }_s,{\beta }_s)\in (0,0.1]\times (0,2]$. The corresponding color bar indicates values of $En({\zeta }^{\ast })$. The values of the other parameters are taken as same, shown in Table 1