A review of mathematical model-based scenario analysis and interventions for COVID-19

Abstract

Mathematical model-based analysis has proven its potential as a critical tool in the battle against COVID-19 by enabling better understanding of the disease transmission dynamics, deeper analysis of the cost-effectiveness of various scenarios, and more accurate forecast of the trends with and without interventions. However, due to the outpouring of information and disparity between reported mathematical models, there exists a need for a more concise and unified discussion pertaining to the mathematical modeling of COVID-19 to overcome related skepticism. Towards this goal, this paper presents a review of mathematical model-based scenario analysis and interventions for COVID-19 with the main objectives of (1) including a brief overview of the existing reviews on mathematical models, (2) providing an integrated framework to unify models, (3) investigating various mitigation strategies and model parameters that reflect the effect of interventions, (4) discussing different mathematical models used to conduct scenario-based analysis, and (5) surveying active control methods used to combat COVID-19.

Keywords: Active control; COVID-19; Mathematical models; Unified framework.

Fig. 1

Schematic diagram representing the general model given by (1). In this figure, the dark inward arrow denote the import of population from other region/subpopulation.

Fig. 2

Simple SEIR (susceptible-exposed-infected-recovered) model with time constants and inter-compartmental transmission rates , , , .

Fig. 3

Model parameters that are affected by various interventions are given in parenthesis. Border closure, lockdown, social distancing, hygiene habits, testing, contact tracing, and isolation reduce the exposure rate $\beta$ and infection rate $\gamma$. Hygiene habits increase the protection rate $\alpha$. Reverse quarantine of elderly and chronically ill population, compliance with regulations, and vaccination also increase protection rate $\alpha$. Hospitalization and use of supportive medicines increase recovery rate $\lambda$. All the interventions together contribute to the journey towards the containment of the epidemics.

Fig. 4

Model 1: Mathematical model of COVID-19 that describes the mobility of individuals from severely infected region to mildly infected one. $N{\left(0\right)}_{\mathrm{A},\mathrm{B}}=8\times {10}^6$, $I_{\text{sA}}\left(0\right)=I_{\text{aA}}\left(0\right)=2500$, $I_{\text{sB}}\left(0\right)=I_{\text{aB}}\left(0\right)=250$, $R_{0\mathrm{A}}=\frac{{\gamma }_A{\chi }_{\mathrm{A}}}{({\mu }^{\prime }+\omega )({\mu }_{\mathrm{a}}+{\mu }^{\prime }+{ϵ}_{\mathrm{A}}+{\lambda }_{\mathrm{a}}+\omega )}$, and $R_{0\mathrm{B}}=\frac{{\gamma }_{\mathrm{B}}[{\chi }_{\mathrm{B}}+\frac{\omega {\chi }_{\mathrm{A}}}{{\mu }^{\prime }+\omega }]}{{\mu }^{\prime }({\mu }_{\mathrm{a}}+{\mu }^{\prime }+{ϵ}_{\mathrm{B}}+{\lambda }_{\mathrm{a}})}$, ${\lambda }_{\mathrm{a}}=1/8$ (per unit time), ${\lambda }_{\mathrm{s}}=1/14$ (per unit time), ${\omega }_{\max }=0.4/(365\times 8\times {10}^6)$, ${\mu }_{\mathrm{s}}=0.02/11$, ${\mu }_{\mathrm{a}}=0.2/11$ (per unit time), ${\mu }^{\prime }=0.007/365$ (per unit time), ${ϵ}_i={ϵ}_0+k_{ϵ}r$, ${ϵ}_0=1/11$, $k_{ϵ}=0.3$, and ${\chi }_i=260$, $i=\mathrm{A},\mathrm{B}$ (newborns per unit time) . Control intervention $u_i\left(t\right)$ is assumed to alter ${\gamma }_i\left(t\right)$, where ${\gamma }_i\left(t\right)={\gamma }_0{(1-u_i\left(t\right))}^2{S}_i\left(t\right)I_i\left(t\right)$, $i=A,B$, and ${\gamma }_0=6.25\times {10}^{-8}$ (per unit time). .

Fig. 5

Model 2: Mathematical model of COVID-19 to study the transmission of COVID-19 for 5 regions in China. ${\beta }_i\left(t\right)={\beta }_{0i}\frac{I_i\left(t\right)}{N_i}$, $i=1,\dots ,5$, for Mainland, Hubei province, Wuhan, Beijing, and Shanghai, respectively, $N_1\left(0\right)=1340\times {10}^6$, $N_2\left(0\right)=45\times {10}^6$, $N_3\left(0\right)=14\times {10}^6$, $N_4\left(0\right)=22\times {10}^6$, $N_5\left(0\right)=24\times {10}^6$, $E_1\left(0\right)=696$, $E_2\left(0\right)=592$, $E_3\left(0\right)=318$, $E_4\left(0\right)=27$, $E_5\left(0\right)=18$, $I_1\left(0\right)=652$, $I_2\left(0\right)=515$, $I_3\left(0\right)=389$, $I_4\left(0\right)=19$, $I_5\left(0\right)=23$, ${\beta }_{0i}=1.0$$\left(1/\text{day}\right)$, $i=1,2,3,5$, ${\beta }_{04}=0.99$$\left(1/\text{day}\right)$, ${\tau }_{\mathrm{l}}=\frac{1}{\gamma }=2$$\left(\text{days}\right)$ for the 5 regions, ${\tau }_{\mathrm{q}i}={\delta }_i^{-1}$, $i=1,2,3,4,5$, ${\delta }_1^{-1}=6.6$$\left(\text{days}\right)$, ${\delta }_2^{-1}=7.2$$\left(\text{days}\right)$, ${\delta }_3^{-1}=7.4$$\left(\text{days}\right)$, ${\delta }_4^{-1}=5.7$$\left(\text{days}\right)$, ${\delta }_5^{-1}=5.6$$\left(\text{days}\right)$, ${\alpha }_1=0.172$$\left(1/\text{day}\right)$, ${\alpha }_2=0.133$$\left(1/\text{day}\right)$, ${\alpha }_3=0.085$$\left(1/\text{day}\right)$, ${\alpha }_4=0.175$$\left(1/\text{day}\right)$, ${\alpha }_5=0.183$$\left(1/\text{day}\right)$, and $R_0=\beta \times {\delta }^{-1}\times {(1-\alpha )}^T$, where $T$ is the number of days .

Fig. 6

Model 3: Mathematical model of COVID-19 that accounts for disease severity and hospital saturation, where $N\left(0\right)=67\times {10}^6$, $I\left(0\right)=120$, $S\left(0\right)=N\left(0\right)-I\left(0\right)=66.99\times {10}^6$, $I_{\mathrm{m}0}=p{I}_0$, $I_{\mathrm{s}0}=(1-p)I_0$, $E_{\mathrm{m}}\left(0\right)=E_{\mathrm{s}}\left(0\right)=I_{\text{am}}\left(0\right)=I_{\text{as}}\left(0\right)=R_{\mathrm{m}}\left(0\right)=R_{\mathrm{s}}\left(0\right)=D\left(0\right)=0$, $R_0=2.5$, ${\mu }_{\mathrm{h}}^{\prime }=0$, if $I_{\mathrm{s}}\left(t\right)<H$, ${\mu }_{\mathrm{h}}^{\prime }={\mu }_{\mathrm{h}}$, if $I_{\mathrm{s}}\left(t\right)\ge H$, $\mu ={\mu }_{\min },\text{if}{I}_{\mathrm{s}}\left(t\right)<H$, $\mu ={\mu }_{\max }\text{if}{I}_{\mathrm{s}}\left(t\right)\ge H$, ${\mu }_{\min }=8.82\times {10}^{-3}$ ($1/\text{day}$), ${\mu }_{\max }=2{\mu }_{\min }$, ${\tau }_{\mathrm{l}}=0.238$$(1/\text{day})$, ${\tau }_{\mathrm{a}}=1$, ${\lambda }_1=\frac{1}{17}$$(1/\text{day})$, ${\lambda }_2=\frac{1}{20}$$(1/\text{day})$, $p=.9$, $m=0.2$, ${\mu }_{\mathrm{h}}={10}^{-5}$, $\rho =2$, and $H=12000$. ${\sum }_{i=1}^9{\mu }_{\mathrm{h}}^{\prime }{x}_i\left(t\right)$ into $D\left(t\right)$ denotes that the population that move out of the other 9 compartments due to hospital saturation are added to the death compartment . Control intervention $u\left(t\right)$ is assumed to alter $\beta \left(t\right)$, where $\beta \left(t\right)=(1-u\left(t\right))({\beta }_0(I_{\text{am}}\left(t\right)+I_{\text{as}}\left(t\right))+{\beta }_0(I_{\mathrm{m}}\left(t\right)+m{I}_{\mathrm{s}}\left(t\right)))$, ${\beta }_0=2.24\times {10}^{-9}$.

Fig. 7

Model 4: Mathematical model of COVID-19 that models differential disease transmission in frontliners and general public, ${\beta }_1\left(t\right)={\beta }_{10}\frac{I_{\mathrm{g}}\left(t\right)}{N\left(t\right)}$, ${\beta }_2\left(t\right)={\beta }_{20}\frac{I_{\mathrm{f}}\left(t\right)}{N\left(t\right)}$, ${\beta }_{i0}=\frac{R_{0i}(S_{\mathrm{g}}\left(0\right)+E_{\mathrm{g}}\left(0\right)+I_{\mathrm{g}}\left(0\right)+S_{\mathrm{f}}\left(0\right)+E_{\mathrm{f}}\left(0\right)+I_{\mathrm{f}}\left(0\right))}{\tau (S_{\mathrm{g}}\left(0\right)+S_{\mathrm{f}}\left(0\right))}$, $i=1,2$, $R_{01}=2.5$, $R_{02}=10$, ${\tau }_{\text{in}}=14\text{days}$, $S_{\mathrm{g}}\left(0\right)=100,000$, $S_{\mathrm{f}}\left(0\right)=1000$, $E_{\mathrm{f}}\left(0\right)=100$, $I_{\mathrm{g}}\left(0\right)=10$, $E_{\mathrm{g}}\left(0\right)=I_{\mathrm{f}}\left(0\right)=0$, $H_i\left(0\right)=R_i\left(0\right)=0$, ${\alpha }_i=0-0.9$, ${\gamma }_i=10/14$, ${\delta }_i=0.01/14$, ${\rho }_i=0.1$, ${\mu }_i=0.03/14$, ${\sigma }_i=0.1/30$, ${\lambda }_{\mathrm{h}i}=0.98/14$, ${\lambda }_i=0.96/14$, and ${\mu }_{\mathrm{h}i}=0.02/14$, $i=1,2$.

Fig. 8

Model 5: Mathematical model of COVID-19 with reinfection, compliant subpopulation, and viral concentration on contaminated surfaces . $N\left(0\right)=59.2\times {10}^6$, $N\left(t\right)=S\left(t\right)+S_{\mathrm{c}}\left(t\right)+I_{\mathrm{a}}+I\left(t\right)+R\left(t\right)$, $\lambda =0.0499$$(1/\text{day})$, ${\lambda }_{\mathrm{a}}=0.0749$$(1/\text{day})$, $\theta =0.5999$, $\Gamma ={\mu }^{\prime }N\left(0\right)=2113.44$ ($\text{persons}/\text{day}$), $\varsigma =0.6499$ ($\text{intensity}/\text{day}$), $e\left(\gamma \right)=\frac{{\gamma }^n}{{\gamma }_0^n+{\gamma }^n}$, $b=0.6337$, $\sigma =2.74\times {10}^{-6}$$(1/\text{day})$, $a=0.6503$, $f=0.8669$, $f^{\prime }=0.1499$, ${\gamma }^{\prime }=0.2499$ ($1/\text{day}$), ${\gamma }_{\text{va}}=0.1019$ ($\text{pathogens}/\text{person}/\text{day}$), ${\gamma }_{\text{vs}}=0.4315$ ($\text{pathogens}/\text{person}/\text{day}$), $d=0.7525$ ($1/\text{day}$), $\mu =0.11$ ($1/\text{day}$), ${\mu }^{\prime }=3.57\times {10}^{-5}$$(1/\text{day})$, and $C_{50}=2091775$$\left(\text{pathogens}\right)$. Control intervention $u_1\left(t\right)$ (effect of social distancing) and $u_2\left(t\right)$ (effect of disinfecting the environment) are assumed to alter $\gamma \left(t\right)$, where $\gamma \left(t\right)=(1-u_1\left(t\right))\beta \left(\frac{I\left(t\right)+u_2\left(t\right)I_{\mathrm{a}}\left(t\right)}{N\left(t\right)}\right)+(1-u_2\left(t\right)){\beta }_{\text{ch}}\left(\frac{C\left(t\right)}{C\left(t\right)+C_{50}}\right)$, $\beta =0.27$$(\text{contacts}/\text{day})$, ${\beta }_{\text{ch}}=0.00101$$(\text{contacts}/\text{day})$, $u_1\left(t\right)=0-1\%$ and $u_2\left(t\right)=2.663(1.1-3)$.

Fig. 9

Model 6: Mathematical model of COVID-19 that accounts for social behavior of compliant subpopulation. $N\left(0\right)=3,000,282$, $S\left(0\right)=3\times {10}^6$, $S_{\mathrm{c}}\left(0\right)=200$, $E\left(0\right)=40$, $E_{\mathrm{c}}\left(0\right)=40$, $I_{\mathrm{a}}\left(0\right)=2$, and the initial condition for all other compartments is 0, $R_0=<1$, $\delta =1/7$$(1/\text{day})$, $\lambda =0.0714$$(1/\text{day})$), ${\lambda }^{\prime }=0.0714$$(1/\text{day})$, $\psi =0.002$$(1/\text{day})$, $\mu =0.02$$(1/\text{day})$, ${\mu }^{\prime }=1/(365\times 50)$$(1/\text{day})$, $\sigma =0.2$$(1/\text{day})$, $\Gamma =100$ (1/day), $\zeta =1$$(1/\text{day})$, $\nu =0.1$$(1/\text{day})$, $\theta =1/7$$(1/\text{day})$, ${\gamma }_{\mathrm{s}}=0.86834$$(1/\text{day})$, $a=0.8683$, and $b=0.4$. Control interventions $u_1\left(t\right)$ (effect of quarantine) and $u_2\left(t\right)$ (effect of isolation) are assumed to alter ${\beta }_1\left(t\right)=\frac{{\beta }_0(u_1\left(t\right)m_{\mathrm{q}}Q\left(t\right)+m_{\mathrm{a}}{I}_{\mathrm{a}}\left(t\right)+m_{\mathrm{s}}{I}_{\mathrm{s}}\left(t\right)+u_2\left(t\right)m_{\mathrm{h}}{H}_{\text{ICU}}\left(t\right))}{N\left(t\right)-(1-u_1\left(t\right))Q\left(t\right)-(1-u_2\left(t\right))H_{\text{ICU}}\left(t\right)}$, ${\beta }_2\left(t\right)=\frac{{\beta }_0(m_{\text{ac}}{I}_{\text{ac}}\left(t\right)+m_{\text{sc}}{I}_{\text{sc}}\left(t\right)+u_2\left(t\right)m_{\mathrm{h}}{H}_{\text{ICU}}\left(t\right))}{N\left(t\right)-Q\left(t\right)-(1-u_2\left(t\right))H_{\text{ICU}}\left(t\right)}$, ${\beta }_0=0.3531$$(1/\text{day})$, where $m_{\mathrm{q}}=0.3\times 7/3$, $m_{\mathrm{a}}=0.425\times 7/3$, $m_{\mathrm{s}}=0.425\times 7/3$, $m_{\text{ac}}=0.225\times 7/4$, $m_{\text{sc}}=0.3\times 7/3$, and $m_{\mathrm{h}}=0.57\times 7/4$.

Fig. 10

Model 7: Mathematical model of COVID-19 that accounts for the influence of public health education, quarantine and hospitalization. $S\left(0\right)=12\times {10}^6$, $E\left(0\right)=1565$, $Q\left(0\right)=800$, $I\left(0\right)=695$, $H\left(0\right)=326$, and $R\left(0\right)=200$, $\lambda =0.03521$$(1/\text{day})$, ${\lambda }_{\mathrm{h}}=0.04255$$(1/\text{day})$, $\mu =0.0079$$(1/\text{day})$, ${\mu }_{\mathrm{h}}=0.0068$$(1/\text{day})$, ${\mu }^{\prime }=0.000034$$(1/\text{day})$, $\Gamma =600$$(1/\text{day})$, ${\delta }^{\prime }=0.07143$$(1/\text{day})$, $\psi =0.1327$$(1/\text{day})$, ${\psi }_{\mathrm{q}}=0.1259$$(1/\text{day})$, $R_0=1.51$ when the exposed population is not quarantined, while $R_0=0.76$ when all of them are on quarantine, and $R_0=2.5$ when the infected individuals are not quarantined or isolated. Control intervention $u\left(t\right)$ is assumed to alter $\beta \left(t\right)$, where $\beta \left(t\right)=(1-u\left(t\right))\frac{{\beta }_0(I\left(t\right)+m_{\mathrm{e}}E\left(t\right)+m_{\mathrm{q}}Q\left(t\right)+m_{\mathrm{h}}H\left(t\right))}{N\left(t\right)}$, ${\beta }_0=0.25$$(1/\text{day})$, $u\left(t\right)=[0,1]$, $m_{\mathrm{e}}=0.3$, and $m_{\mathrm{h}}=m_{\mathrm{q}}=0.1$, .

Fig. 11

Model 8: Mathematical model of COVID-19 that accounts for age-wise interaction rates. The parameters are estimated for 9 age groups such as 0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70–79, and $\ge 80$, for $i=1,2\dots ,9$, respectively, ${\beta }^i\left(t\right)=\varphi {\sum }_{j=1}^{9}A(i,j)\frac{I_{\mathrm{s}}^j\left(t\right)}{N^j}+\varphi {\sum }_{j=1}^{9}A(i,j)\frac{I_{\mathrm{a}}^j\left(t\right)}{N^j}$, ${\tau }_{\text{in}}=\frac{1}{\gamma }=5$, ${\tau }_{\mathrm{a}}=\frac{1}{{\lambda }_{\mathrm{a}}}=4$, ${\tau }_{\mathrm{s}}=\frac{1}{{\lambda }_{\mathrm{s}}}=10$, ${\tau }_{\mathrm{h}}=\frac{1}{{\lambda }_{\mathrm{h}}}=10.4$ ($\text{days}$). $N^1=12\%$, $N^i=13\%$, $i=2$, $\dots ,5$, $N^6=14\%$, $N^7=12\%$, $N^8=7\%$, $N^9=4\%$, $a^1=11.1\%$, $a^i=12.1\%$, $i=2$, $\dots ,5$, $a^6=17.5\%$, $a^i=28.7\%$, $i=7,8,9$, ${\psi }^1=18.2\%$, ${\psi }^2=5.5\%$, ${\psi }^3=6.8\%$, ${\psi }^i=13.9\%$, $i=4,5$, ${\psi }^6=25.1\%$, ${\psi }^i=51.2\%$, $i=7,8$, ${\psi }^9=61.7\%$, ${\mu }_{\mathrm{h}}^i=0.2\%$, $i=1,3$, ${\mu }_{\mathrm{h}}^2=0\%$, ${\mu }_{\mathrm{h}}^i=0.9\%$, $i=4,5$, ${\mu }_{\mathrm{h}}^6=3.6\%$, ${\mu }_{\mathrm{h}}^i=14.9\%$, $i=7,8$, ${\mu }_{\mathrm{h}}^9=32.8\%$, ${\mu }^i=0.1\%$, $i=1,3$, ${\mu }^2=0\%$, ${\mu }^i=0.4\%$, $i=4,5$, ${\mu }^6=1.4\%$, ${\mu }^i=5.9\%$, $i=7,8$, and ${\mu }^9=12.9\%$.

Fig. 12

Model 9: Mathematical model of COVID-19 that model how school opening delay will affect the pandemic, ${\beta }_i\left(t\right)={\sum }_{j=1,2}{\beta }_{0ij}\frac{I_{\mathrm{j}\left(\mathrm{t}\right)}}{N_i\left(t\right)}$, where ${\beta }_{0ij}=\left(\genfrac{}{}{0pt}{}{7.9910.7172}{0.71724.7531}\right)$, ${\beta }_{0ij}=\left(\genfrac{}{}{0pt}{}{7.81920.8544}{0.85444.6452}\right)$, ${\beta }_{ij}^0=\left(\genfrac{}{}{0pt}{}{8.55311.0558}{1.05585.0616}\right)$ represent transmission matrices for Data 1, Data 2, and Data 3, respectively, ${\zeta }_i\left(t\right)={\zeta }_{0i}(1-{\mathrm{e}}^{-e{Q}_i\left(t\right)})$, $e=0.001$, ${\zeta }_{01}=1.8138$$(1/\text{day})$, ${\zeta }_{02}=1.69$$(1/\text{day})$, and ${\zeta }_{03}=2.121$$(1/\text{day})$, for Data 1, Data 2, and Data 3, respectively, $m=0.02$, $\lambda =1/14$$(1/\text{day})$, $\delta =1/4$$(1/\text{day})$, and $\gamma =1/4.1$$(1/\text{day})$.

Fig. 13

Model 10: Mathematical model of COVID-19 that accounts for undetected, detected, symptomatic, asymptomatic, and severely ill cases, where, ${\gamma }_{\text{au}}={\gamma }_{\text{au}0}{I}_{\text{au}}\left(t\right)$, ${\gamma }_{\text{ad}}={\gamma }_{\text{ad}0}{I}_{\text{ad}}\left(t\right)$, ${\gamma }_{\mathrm{u}}={\gamma }_{\mathrm{u}0}{I}_{\mathrm{u}}\left(t\right)$, ${\gamma }_{\mathrm{d}}={\gamma }_{\mathrm{d}0}{I}_{\mathrm{s}}\left(t\right)$, $S\left(0\right)=1-I_{\text{au}}\left(0\right)-I_{\text{ad}}\left(0\right)-I_{\mathrm{u}}\left(0\right)-I_{\mathrm{d}}\left(0\right)-I_{\mathrm{s}}\left(0\right)-R\left(0\right)-D\left(0\right)$, $I_{\text{au}}\left(0\right)=200/(60\times {10}^6)$, $I_{\text{ad}}\left(0\right)=20/(60\times {10}^6)$, $I_{\mathrm{u}}\left(0\right)=1/(60\times {10}^6)$, $I_{\mathrm{d}}\left(0\right)=2/(60\times {10}^6)$, $I_{\mathrm{s}}\left(0\right)=R\left(0\right)=D\left(0\right)=0$. Parameter values are estimated for 6 different stages of interventions. On Day 1, ${\gamma }_{\text{au}0}=0.57$, ${\gamma }_{\text{ad}0}={\gamma }_{\mathrm{d}0}=0.011$, ${\gamma }_{\mathrm{u}0}=0.456$, ${\Delta }_{\text{ad}}=0.171$, ${\Delta }_{\text{uu}}={\Delta }_{\text{dd}}=0.125$, ${\lambda }_{\text{au}}={\lambda }_{\mathrm{a}}=0.034$, ${\Delta }_{\mathrm{d}}=0.371$, ${\Delta }_{\mathrm{s}}={\Delta }_{\mathrm{u}}={\lambda }_{\mathrm{s}}={\lambda }_{\mathrm{u}}=0.017$, ${\Delta }_{\text{ds}}=0.027$, $\mu =0.01$, and $R_0=2.38$. After Day 4, with hand washing and social distancing measures, ${\gamma }_{\text{au}0}=0.422$, ${\gamma }_{\text{ad}0}={\gamma }_{\mathrm{d}0}=0.0057$, ${\gamma }_{\mathrm{u}0}=0.285$, and $R_{\mathrm{e}}=1.66$. After Day 12, with screening limited to individuals with symptoms, ${\Delta }_{\text{ad}}=0.143$ and $R_{\mathrm{e}}=1.8$. After Day 22, with incomplete lockdown, ${\gamma }_{\text{au}0}=0.36$, ${\gamma }_{\text{ad}0}={\gamma }_{\mathrm{d}0}=0.005$, ${\gamma }_{\mathrm{u}0}=0.2$, ${\Delta }_{\text{uu}}={\Delta }_{\text{dd}}=0.034$, ${\lambda }_{\text{au}}=0.08$, ${\Delta }_{\mathrm{s}}=0.008$, ${\Delta }_{\text{ds}}=0.015$, ${\lambda }_{\mathrm{a}}=0.017$, and $R_{\mathrm{e}}=1.6$. After Day 28, with complete lockdown ${\lambda }_{\text{au}0}=0.21$, ${\gamma }_{\mathrm{u}0}=0.11$, and $R_{\mathrm{e}}=0.99$. After Day 38, with mass testing campaigns, ${\Delta }_{\text{ad}}=0.2$, ${\Delta }_{\text{uu}}={\Delta }_{\text{dd}}=0.025$, ${\lambda }_{\mathrm{a}}={\lambda }_{\mathrm{s}}={\Delta }_{\mathrm{u}}=0.02$, $\lambda =0.01$, and $R_{\mathrm{e}}=0.85$.

Fig. 14

Model 11: Mathematical model of COVID-19 that accounts for protection of a subpopulation by vaccination, $\beta \left(t\right)={\beta }_{0}I\left(t\right)$, $\Omega \left(t\right)={\Omega }_0{\beta }_{0}I\left(t\right)$, $\phi \left(t\right)={\phi }_0+{\phi }_{{\Lambda }_M}\left(t\right)$, ${\phi }_{{\Lambda }_M}\left(t\right)=(1-{\phi }_0-{\phi }_1)\frac{M{\Lambda }_{\mathrm{M}}\left(t\right)}{1+M{\Lambda }_{\mathrm{M}}\left(t\right)}$, ${\dot{\Lambda }}_{\mathrm{M}}\left(t\right)={\Lambda }_{\mathrm{t}}({\Lambda }_{\mathrm{c}}I\left(t\right)-{\Lambda }_{\mathrm{M}}\left(t\right))$, $\eta \left(t\right)={\eta }_0{\beta }_{0}I\left(t\right)$, ${\beta }_0=1.07$$(1/\text{person}/\text{days})$$\lambda =0.278$$(1/\text{days})$, ${\Lambda }_t=(0.00833,\infty )$, $M=500$, $1-{\phi }_1=0.99$, ${\eta }_0=0.15$, and ${\Omega }_0=0.2$. Control intervention $u\left(t\right)=\phi \left(t\right)$, where $\phi \left(t\right)$ is the vaccination rate.

Fig. 15

Model 12: Mathematical model of COVID-19 that accounts for virus dynamics in a host. $U_{\text{cells}}\left(0\right)=4\times {10}^8$$\left(\text{cells}\right)$, $I_{\text{cells}}\left(0\right)=0$$\left(\text{cells}\right)$, and $C\left(0\right)=0.31$ ($\text{copies}/\text{ml}$), $\beta \left(t\right)={\beta }_{\mathrm{v}}C\left(t\right)$, ${\beta }_{\mathrm{v}}=4.71\times {10}^8$ ($\text{ml}/(\text{copies}.\text{days})$), $r_{\mathrm{v}}=3.07$$\left(\right((\text{copies}/\text{ml})/\text{days})/\text{cells}))$, $c_1=1.07$$(1/\text{days})$, and $c_2=2.4$$(1/\text{days})$.