Mutations make pandemics worse or better: modeling SARS-CoV-2 variants and imperfect vaccination

Abstract

COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, $R_0$, which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when $R_0<1$, that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate $\nu _1$ leads to oscillations. When $\nu _1$ increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation.

Appendix A

$$\begin{aligned} l_1&= - (a_{11}+b_{12}+c_{13}+d_{13}+e_{12}+f_{13}+g_{13}), \\ l_2&= -a_{12} b_{11}+c_{13} d_{13}+c_{13} e_{12}-c_{14} e_{11}+d_{13} e_{12}-d_{14} f_{11} +(c_{13} +d_{13}+e_{12}) f_{13} \\&+ g_{13} (c_{13}+d_{13}+e_{12}+f_{13}) +b_{12} (c_{13}+d_{13}+e_{12}+f_{13}+g_{13}) \\&+a_{11} (b_{12}+c_{13}+d_{13}+e_{12}+f_{13}+g_{13}), \\ l_3 =&a_{11} (-b_{12} (c_{13}+d_{13}+e_{12}+f_{13}+g_{13})-c_{13} (d_{13}+e_{12}+f_{13}+g_{13})+c_{14} e_{11} \\&-g_{13} (d_{13}+e_{12}+f_{13})-d_{13} e_{12}-d_{13} f_{13}+d_{14} f_{11}-e_{12} f_{13})\\&+a_{12} b_{11} (c_{13}+d_{13}+e_{12}+f_{13}+g_{13})- g_{13} (b_{12} (c_{13}+d_{13}+e_{12}+f_{13})\\&+c_{13} (d_{13}+e_{12}+f_{13})-c_{14} e_{11}+f_{13} (d_{13}+e_{12})+d_{13} e_{12}-d_{14} f_{11})\\&- e_{11} (a_{13} c_{11}+b_{13} c_{12}) - c_{13} d_{13} e_{12} + c_{13} d_{14} f_{11} - c_{13} d_{13} f_{13}- c_{13} e_{12} f_{13} \\&- f_{11} (a_{14} d_{11}+b_{14} d_{12}) + d_{14} e_{12} f_{11}-d_{13} e_{12} f_{13} + c_{14} e_{11} (d_{13}+f_{13}) - d_{15} f_{11} g_{12} \\&- b_{12} (c_{13} (d_{13}+e_{12}+f_{13})-c_{14} e_{11}+f_{13} (d_{13}+e_{12})+d_{13} e_{12}- d_{14} f_{11}), \end{aligned}$$

$$\begin{aligned} l_4 =&a_{13} e_{11} (c_{11} (b_{12}+d_{13})-b_{11} c_{12})+d_{13} (b_{12} c_{13} e_{12}-b_{12} c_{14} e_{11}+b_{13} c_{12} e_{11}) \\&+ f_{11} (a_{14} (d_{11} (b_{12}+c_{13}+e_{12})-b_{11} d_{12})-d_{14} (e_{12} (b_{12}+c_{13})+b_{12} c_{13}-c_{14} e_{11})\\&+b_{14} d_{12} (c_{13}+e_{12})) - a_{12} (b_{11} (c_{13} (d_{13}+e_{12})-c_{14} e_{11}+d_{13} e_{12}\\&-d_{14} f_{11})+b_{13} c_{11} e_{11}\\&+b_{14} d_{11} f_{11}) - e_{11} f_{12} (a_{14} c_{11}+b_{14} c_{12})\\&+ b_{12} c_{13} d_{13} f_{13} + f_{13} (a_{13} c_{11} e_{11} \\&+e_{12} (d_{13} (b_{12}+c_{13})+b_{12} c_{13})-c_{14} e_{11} (b_{12}+d_{13})+b_{13} c_{12} e_{11})\\&- a_{12} b_{11} f_{13} (c_{13}+d_{13}+e_{12}) + d_{15} f_{11} g_{12} (b_{12}+c_{13}+e_{12}) - a_{12} b_{11} g_{13} (c_{13}\\&+d_{13}+e_{12}+f_{13}) + g_{13} (a_{13} c_{11} e_{11}+a_{14} d_{11} f_{11}+b_{12} (c_{13} (d_{13}+e_{12}+f_{13})\\&-c_{14} e_{11}+f_{13} (d_{13}+e_{12})+d_{13} e_{12}-d_{14} f_{11})+b_{13} c_{12} e_{11}+b_{14} d_{12} f_{11}\\&+c_{13} d_{13} e_{12}+c_{13} d_{13} f_{13}-c_{13} d_{14} f_{11}+c_{13} e_{12} f_{13}-c_{14} e_{11} (d_{13}+f_{13})\\&+d_{13} e_{12} f_{13}-d_{14} e_{12} f_{11}) + a_{11} (b_{13} c_{12} e_{11}+b_{14} d_{12} f_{11}+c_{13} (f_{13} (d_{13}+e_{12})\\&+d_{13} e_{12}-d_{14} f_{11})+d_{13} e_{12} f_{13}-d_{14} e_{12} f_{11}+d_{15} f_{11} g_{12}) + a_{11} (b_{12} (c_{13} (d_{13}\\&+e_{12}+f_{13}+g_{13})-c_{14} e_{11}+d_{13} (e_{12}+f_{13}+g_{13})-d_{14} f_{11}+g_{13} (e_{12}+f_{13})\\&+e_{12} f_{13})+g_{13} (c_{13} (d_{13}+e_{12}+f_{13})+d_{13} (e_{12}+f_{13})-d_{14} f_{11}+e_{12} f_{13})\\&-c_{14} e_{11} (d_{13}+f_{13}+g_{13})), \end{aligned}$$

$$\begin{aligned} l_5&= a_{12} (b_{11} (g_{13} (c_{13} (d_{13}+e_{12}+f_{13})+f_{13} (d_{13}+e_{12})+d_{13} e_{12}- d_{14} f_{11})+c_{13} d_{13} e_{12}\\&+c_{13} d_{13} f_{13}- c_{13} d_{14} f_{11}+ c_{13} e_{12} f_{13}- c_{14} e_{11} (d_{13}+f_{13}+g_{13})+d_{13} e_{12} f_{13}\\&- d_{14} e_{12} f_{11}+ d_{15} f_{11} g_{12})+b_{13} c_{11} e_{11} (d_{13}+f_{13}+g_{13})\\&+b_{14} (d_{11} f_{11} (c_{13}+e_{12}+g_{13})\\&-c_{11} e_{11} f_{12})) - a_{13} e_{11} (-b_{11} c_{12} (d_{13}+f_{13}+g_{13})+b_{12} c_{11} (d_{13}+f_{13}+g_{13})\\&+c_{11} (g_{13} (d_{13}+f_{13})+d_{13} f_{13}- d_{14} f_{11})) - g_{13} (f_{11} (a_{14} (d_{11} (c_{13}+e_{12})- b_{11} d_{12}) \\&+b_{14} d_{12} (c_{13}+e_{12})-c_{13} d_{14} e_{12}+ c_{14} d_{14} e_{11})- b_{12} (-a_{14} d_{11} f_{11}- c_{13} d_{13} e_{12} \\&+ d_{14} f_{11} (c_{13}+ e_{12})+c_{14} d_{13} e_{11})-e_{11} f_{12} (a_{14} c_{11}+b_{14} c_{12})+ b_{13} c_{12} d_{13} e_{11}\\&+ f_{13} (b_{12} (c_{13} (d_{13}+e_{12})-c_{14} e_{11}+ d_{13} e_{12})+ b_{13} c_{12} e_{11}+c_{13} d_{13} e_{12}- c_{14} d_{13} e_{11}) \\&+ a_{11} (b_{12} (c_{13} (d_{13}+e_{12}+f_{13})-c_{14} e_{11}+ f_{13} (d_{13}+e_{12})+d_{13} e_{12}-d_{14} f_{11})\\&+b_{13} c_{12} e_{11}+b_{14} d_{12} f_{11}+c_{13} d_{13} e_{12}+c_{13} d_{13} f_{13}-c_{13} d_{14} f_{11}+c_{13} e_{12} f_{13}\\&-c_{14} e_{11} (d_{13}+f_{13})+d_{13} e_{12} f_{13}-d_{14} e_{12} f_{11})) + a_{14} e_{11} f_{12} (c_{11} (b_{12}\\&+d_{13})- b_{11} c_{12})\\&+a_{14} f_{11} (b_{11} d_{12} (c_{13}+e_{12})-b_{12} d_{11} (c_{13}+e_{12})-c_{13} d_{11} e_{12}+c_{14} d_{11} e_{11})\\&-d_{15} f_{11} g_{12} (b_{12} (c_{13}+e_{12})+c_{13} e_{12}-c_{14} e_{11})+b_{14} c_{12} d_{13} e_{11} f_{12} \\&+ f_{11} (d_{14} (b_{12} c_{13} e_{12}\\&-b_{12} c_{14} e_{11}+b_{13} c_{12} e_{11})+b_{14} d_{12} (c_{14} e_{11}-c_{13} e_{12})) -d_{13} f_{13} (b_{12} c_{13} e_{12}\\&- b_{12} c_{14} e_{11} +b_{13} c_{12} e_{11}) + a_{11} (-f_{13} (b_{12} (c_{13} (d_{13}+e_{12})-c_{14} e_{11}+d_{13} e_{12})\\&+b_{13} c_{12} e_{11}+ c_{13} d_{13} e_{12}\\&-c_{14} d_{13} e_{11})-b_{12} c_{13} d_{13} e_{12}+ b_{12} c_{13} d_{14} f_{11}-d_{15} f_{11} g_{12} (b_{12}\\&+c_{13}+e_{12})+b_{12} c_{14} d_{13} e_{11} \\&+b_{12} d_{14} e_{12} f_{11}- b_{13} c_{12} d_{13} e_{11}+b_{14} c_{12} e_{11} f_{12}-b_{14} c_{13} d_{12} f_{11}\\&-b_{14} d_{12} e_{12} f_{11}\\&+c_{13} d_{14} e_{12} f_{11}- c_{14} d_{14} e_{11} f_{11}), \end{aligned}$$

$$\begin{aligned} l_6&= a_{13} e_{11} (-g_{13} (b_{11} c_{12} (d_{13}+f_{13})-c_{11} d_{13} f_{13}+c_{11} d_{14} f_{11})\\&-b_{11} c_{12} d_{13} f_{13}+b_{11} c_{12} d_{14} f_{11}\\&+b_{12} c_{11} (g_{13} (d_{13}+f_{13})+d_{13} f_{13}-d_{14} f_{11})+b_{14} f_{11} (c_{11} d_{12}-c_{12} d_{11})\\&+c_{11} d_{15} f_{11} g_{12}) \\&+ a_{14} (b_{11} c_{12} d_{13} e_{11} f_{12}-b_{11} c_{13} d_{12} e_{12} f_{11}+b_{11} c_{14} d_{12} e_{11} f_{11}-b_{12} (c_{11} d_{13} e_{11} f_{12} \\&-c_{13} d_{11} e_{12} f_{11}+c_{14} d_{11} e_{11} f_{11})+b_{13} e_{11} f_{11} (c_{12} d_{11}-c_{11} d_{12})-c_{11} d_{15} e_{11} f_{11} g_{11} \\&+ g_{13} (f_{11} (-b_{11} d_{12} (c_{13}+e_{12})+b_{12} d_{11} (c_{13}+e_{12})+c_{13} d_{11} e_{12}-c_{14} d_{11} e_{11}) \\&-e_{11} f_{12} (c_{11} (b_{12}+d_{13})-b_{11} c_{12}))) - g_{13} (a_{11} (b_{12} d_{13} (c_{14} e_{11}-c_{13} e_{12}) \\&+b_{12} d_{14} f_{11} (c_{13}+e_{12})-b_{13} c_{12} d_{13} e_{11}+b_{14} c_{12} e_{11} f_{12}-f_{11} (b_{14} d_{12} (c_{13}+e_{12}) \\&-c_{13} d_{14} e_{12}+c_{14} d_{14} e_{11}))+d_{14} f_{11} (b_{12} c_{13} e_{12}-b_{12} c_{14} e_{11}+b_{13} c_{12} e_{11}) \\&+b_{14} (c_{12} d_{13} e_{11} f_{12}-c_{13} d_{12} e_{12} f_{11}+c_{14} d_{12} e_{11} f_{11}) \\&+ a_{12} (b_{11} (c_{13} (f_{13} (d_{13}+e_{12})\\&+d_{13} e_{12}-d_{14} f_{11})-c_{14} e_{11} (d_{13}+f_{13})+d_{13} e_{12} f_{13}-d_{14} e_{12} f_{11})\\&+b_{13} c_{11} e_{11} (d_{13}+f_{13})\\&-b_{14} c_{11} e_{11} f_{12}+b_{14} d_{11} f_{11} (c_{13}+e_{12})) - f_{13} (a_{11} (b_{12} (e_{12} (c_{13}+d_{13})\\&+c_{13} d_{13}-c_{14} e_{11}) \\&+b_{13} c_{12} e_{11}+c_{13} d_{13} e_{12}-c_{14} d_{13} e_{11})+d_{13} (b_{12} c_{13} e_{12}\\&-b_{12} c_{14} e_{11}+b_{13} c_{12} e_{11}))) \\&+ a_{11} (-(d_{14} f_{11}-d_{13} f_{13}) (b_{12} c_{13} e_{12}-b_{12} c_{14} e_{11}+b_{13} c_{12} e_{11})\\&+d_{15} f_{11} g_{12} (e_{12} (b_{12}+c_{13}) \\&+b_{12} c_{13}-c_{14} e_{11})-b_{14} (c_{12} d_{13} e_{11} f_{12}-c_{13} d_{12} e_{12} f_{11}\\&+c_{14} d_{12} e_{11} f_{11}))+a_{12} ((d_{14} f_{11} \\&-d_{13} f_{13}) (b_{11} c_{13} e_{12}-b_{11} c_{14} e_{11}+b_{13} c_{11} e_{11})-b_{11} d_{15} f_{11} g_{12} (c_{13}+e_{12}) \\&+b_{14} (c_{11} d_{13} e_{11} f_{12}-c_{13} d_{11} e_{12} f_{11}+c_{14} d_{11} e_{11} f_{11}))\\&+d_{15} f_{11} (g_{12} (b_{12} c_{13} e_{12}-b_{12} c_{14} e_{11}\\&+b_{13} c_{12} e_{11})-b_{14} c_{12} e_{11} g_{11}), \end{aligned}$$

$$\begin{aligned} l_7 =&a_{14} d_{15} e_{11} f_{11} g_{11} (b_{12} c_{11}-b_{11} c_{12}) + g_{13} (a_{11} ((d_{14} f_{11}-d_{13} f_{13})\\&(b_{12} c_{13} e_{12}-b_{12} c_{14} e_{11}\\&+b_{13} c_{12} e_{11})+b_{14} (c_{12} d_{13} e_{11} f_{12}-c_{13} d_{12} e_{12} f_{11}+c_{14} d_{12} e_{11} f_{11}))\\&-a_{12} (d_{14} f_{11} (b_{11} c_{13} e_{12}\\&-b_{11} c_{14} e_{11}+b_{13} c_{11} e_{11})+b_{14} (c_{11} d_{13} e_{11} f_{12}-c_{13} d_{11} e_{12} f_{11}+c_{14} d_{11} e_{11} f_{11}))\\&+a_{12} d_{13} f_{13} (b_{11} c_{13} e_{12}-b_{11} c_{14} e_{11}+b_{13} c_{11} e_{11})+a_{13} e_{11} ((b_{12} c_{11}-b_{11} c_{12}) (d_{14} f_{11} \\&-d_{13} f_{13})+b_{14} f_{11} (c_{12} d_{11}-c_{11} d_{12})))+d_{15} f_{11} (g_{12} (e_{11} (a_{11} b_{12} c_{14}-a_{11} b_{13} c_{12}\\&-a_{12} b_{11} c_{14}+a_{12} b_{13} c_{11}+a_{13} b_{11} c_{12}-a_{13} b_{12} c_{11})+c_{13} e_{12} (a_{12} b_{11}-a_{11} b_{12}))\\&+b_{14} e_{11} g_{11} (a_{11} c_{12}-a_{12} c_{11}))+a_{14} g_{13} (d_{13} e_{11} f_{12} (b_{12} c_{11}-b_{11} c_{12})+f_{11} (b_{12} d_{11}\\&-b_{11} d_{12}) (c_{14} e_{11}-c_{13} e_{12})+b_{13} e_{11} f_{11} (c_{11} d_{12}-c_{12} d_{11})). \end{aligned}$$

$$\begin{aligned} a_{11} =&-\beta _1 I_1^* - \beta _2 I_2^* - (\mu + p), \quad a_{12} = \gamma , \quad a_{13} = -\beta _1 S^*, \quad a_{14} = -\beta _2 S^*, \\ b_{11} =&p, \quad b_{12} = -\delta _1 \beta _1 I_1^* - \delta _2 \beta _2 I_2^* - (\mu + \gamma ), \quad b_{13} = -\delta _1 \beta _1 V^*,\\ b_{14} =&-\delta _2 \beta _2 V^*, \quad c_{11} = \beta _1 I_1^*, \quad c_{12} = \delta _1 \beta _1 I_1^*, \quad c_{13} = -(a_1 + \mu ),\\ c_{14} =&\beta _1 (S^* + \delta _1 V^*), \quad d_{11} = \beta _2 I_2^*, \quad d_{12} = \delta _2 \beta _2 I_2^*, \quad d_{13} = -(a_2 + \mu ),\\ d_{14} =&\beta _2 (S^* + \delta _2 V^*) + \delta _3 \beta _2 R_1^*, \quad d_{15} = \delta _3 \beta _2 I_2^*, \quad e_{11} = a_1, \\ e_{12} =&-(\alpha _2 + \mu + d_1 + \nu _1), \quad f_{11} = a_2, \quad f_{12} = \nu _1, \quad f_{13} = -(\alpha _2 + \mu + d_2),\\ g_{11} =&\alpha _1, \quad g_{12} = -\delta _3 \beta _2 R_1^*, \quad g_{13} = -(\delta _3 \beta _2 I_2^* + \mu ). \end{aligned}$$