Estimating the impact of lock-down, quarantine and sensitization in a COVID-19 outbreak: lessons from the COVID-19 outbreak in China

Model development

Some of the recommended control measures to reduce the spread of COVID-19 include: sensitization of the public on how to reduce transmission, isolation/quarantine of infectious individuals, prompt treatment of infectious individuals etc. Here, we formulate a mathematical model that takes these control measures into consideration. From the mode of transmission there is some distance at which the virus cannot be transmitted. Consider a location (country, community, etc.) and let N(t) be the total population of individuals in that location whose probability of contacting the disease is greater than zero. Next, we partition N(t) into five classes namely: Susceptible class (S(t)), Exposed class (E(t)), Infectious class (I(t)), Quarantine class (Q(t)) and Recovered/removed/death class (R(t)). Individuals enter the susceptible class through migration/movement or birth at a rate Λ. Susceptible individuals become exposed by direct person-to-person contact at a rate β_I. Susceptible individuals can also be exposed through contact with virus in the environment (shed by persons or animals) at a rate β_W. These exposed individuals move to the infectious class at a rate σ. Infectious individual recover at a rate γ and die from the disease at a rate δ. There is also a natural death rate μ.W(t) represent the density of pathogens in the environment that cause the disease. Infectious individuals shed pathogen into the environment at a rate ν and these decay at a rate ξ.

Several control measures are introduced. Sensitization on methods that reduce an individuals chance of being infected (regular hand washing etc.) is assumed to reduce the number of susceptible individuals at a rate ϕ. Lock-down reduces the contact rates between the susceptible individuals and infectious individuals at a rate ε. Individuals found to be infected are quarantined at a rate ρ. Based on these assumptions, the mathematical model that describes a COVID-19 outbreak is given by

Model development

The meaning of variables and parameters are given in Tables 1 and 2.

Since humans and pathogens have different space and time scales, it was necessary to non-dimensionalized model (1). To non-dimensionalize the model, we set the following: $\Lambda =\mu N,s=\frac{S}{N},e=\frac{E}{N}$, $i=\frac{I}{N},q=\frac{Q}{N},w=\frac{\xi W}{\nu N},\beta =b_{I}N$, $\alpha =\frac{b_W\nu N}{\xi }$. Based on these, the dimensionless model is given by (2)${\displaystyle \frac{ds}{dt}=\mu -\left(1-\epsilon \right)\beta i\left(t\right)s\left(t\right)-\alpha w\left(t\right)s\left(t\right)-\left(\mu +\varphi \right)s\left(t\right),\frac{de}{dt}=\left(1-\epsilon \right)\beta i\left(t\right)s\left(t\right)+\alpha w\left(t\right)s\left(t\right)-\left(\mu +\sigma \right)e\left(t\right),\frac{di}{dt}=\sigma e\left(t\right)-\left(\mu +\rho +\delta +\gamma \right)i\left(t\right),\frac{dq}{dt}=\rho i\left(t\right)-\left(\mu +\delta +\gamma \right)q\left(t\right),\frac{dr}{dt}=\varphi s\left(t\right)+\left(\delta +\gamma \right)\left(i\left(t\right)+q\left(t\right)\right)-\mu r\left(t\right),\frac{dw}{dt}=\xi \left(i\left(t\right)-w\left(t\right)\right).}$

Model analyses

The model is used to investigate the dynamics of a COVID-19 outbreak. First, the basic reproduction number, denoted by ${\mathcal{R}}_0$, is calculated as epidemiologically it represents the expected number of secondary infections that result from introducing a single infectious individual into an otherwise susceptible population (Van den Driessche & Watmough, 2002). If ${\mathcal{R}}_0<1$, the outbreak is likely to eventually end. On the other hand, if ${\mathcal{R}}_0>1$ the outbreak will continue. Thus, the value of the basic reproduction number gives an indication on the overall rate at which the disease spreads.

The basic reproduction number (${\mathcal{R}}_0$) of model (2) computed using the next generation matrix approach (Van den Driessche & Watmough, 2002) is (3)${\displaystyle {\mathcal{R}}_0=\frac{\mu \sigma \left(\alpha +\left(1-\epsilon \right)\beta \right)}{\left(\mu +\varphi \right)\left(\mu +\sigma \right)\left(\mu +\rho +\delta +\gamma \right)}.}$ The equilibrium point of model (2) when there is no disease in the population (disease free equilibrium (DFE)) is given by (4)${\displaystyle \left(s^0,e^0,i^0,q^0,r^0,w^0\right)=\left(\frac{\mu }{\mu +\varphi },0,0,0,0,\frac{\varphi }{\mu +\varphi },0\right).}$

Second, the model is fit to the real data (active cases of COVID-19 Worldometer (2020)). In fitting the model, the parameters γ, μ, δ, ξ were fixed with values taken from Chen et al. (2020) except δ which we estimated based on a high estimate of the death rate due to Covid-19 (i.e., Italy at the peak of their epidemic). The values are: γ = 0.172,  μ = 0.002,  δ = 0.113,  ξ = 0.100. To estimate the remaining parameters the data is also non-dimensionalized to allow for a best fit to the model. The resulting parameter values are: β = 1.097,  α = 0.098,  ϕ = 0.006,  σ = 0.947,  ρ = 0.242,  ε = 0.211. The results are then converted back to the original dimensions (Fig. 2). In this way the actual data is a benchmark for the ensuring visual comparisons. In terms of the overall trend, there is a reasonable fit of the data for active cases of the COVID-19 outbreak in China from 22 January 2020 to 30 March 2020. Thus, the model can be used further to explore the dynamics of COVID-19 outbreaks.

Basic reproduction number

Using the parameter values, the basic reproduction number for the COVID-19 outbreak in China as at 30 March 2020 is ${\mathcal{R}}_0=0.454<1$. If we exclude the control measures the basic reproduction number becomes ${\mathcal{R}}_0=4.155>1$. Also, using the model to estimate the basic reproduction number for the COVID-19 outbreak in China before 20 February 2020 gives a basic reproduction number ${\mathcal{R}}_0>1$.

Numerical simulations

Numerical simulations enable us to explore the impact of each of the control measures used in the COVID-19 outbreak in China.

Figure 3 compares the results of using all controls with the case where there are no controls. With all controls this result agrees with the case of the COVID-19 outbreak in China where the outbreak is coming to an end. On the other hand, if no control measures are used, Fig. 3 shows that many more individuals could become infectious within a short period.

Next we explore the impact of each of the various control measures. Figure 4 is a comparison of the impact of each separate control as compared to implementing all or no controls. From the figure, each control on its own reduces the number of infectious individuals but not to the extent of using all controls. Quarantine is the best individual control. When all controls are used the number quarantined is compared to those people removed from the system in Fig. 5. From the figure, the quarantining effort is greatest in the early part of the outbreak and is associated with a steady rise in individuals no longer susceptible.

The effect of implementing any two of the controls is presented in Fig. 6. Still quarantine is the essential control and together with lock-down the result more closely matches that of all controls. However, the case of sensitization together with lock-down is also significantly more effective. Further simulations, not presented here, showed that increasing any one of the separate controls can improve their effectiveness.