Model analysis and data validation of structured prevention and control interruptions of emerging infectious diseases

Abstract

The design of optimized non-pharmaceutical interventions (NPIs) is critical to the effective control of emergent outbreaks of infectious diseases such as SARS, A/H1N1 and COVID-19 and to ensure that numbers of hospitalized cases do not exceed the carrying capacity of medical resources. To address this issue, we formulated a classic SIR model to include a close contact tracing strategy and structured prevention and control interruptions (SPCIs). The impact of the timing of SPCIs on the maximum number of non-isolated infected individuals and on the duration of an infectious disease outside quarantined areas (i.e. implementing a dynamic zero-case policy) were analyzed numerically and theoretically. These analyses revealed that to minimize the maximum number of non-isolated infected individuals, the optimal time to initiate SPCIs is when they can control the peak value of a second rebound of the epidemic to be equal to the first peak value. More individuals may be infected at the peak of the second wave with a stronger intervention during SPCIs. The longer the duration of the intervention and the stronger the contact tracing intensity during SPCIs, the more effective they are in shortening the duration of an infectious disease outside quarantined areas. The dynamic evolution of the number of isolated and non-isolated individuals, including two peaks and long tail patterns, have been confirmed by various real data sets of multiple-wave COVID-19 epidemics in China. Our results provide important theoretical support for the adjustment of NPI strategies in relation to a given carrying capacity of medical resources.

Appendices

Appendix A: Analysis of the maximum number of non-isolated infected individuals with respect to $T_{on}$ and $T_{off}$.

In Theorem 4.1, we summarize the analytical forms of the maximum number of non-isolated infected individuals for cases ($a_{1}$)-($a_{2}$) and ($a_{4}$)-($a_{6}$) with respect to the number of susceptible individuals at the onset $S_{T_{on}}$ and the end $S_{T_{off}}$ of the second intervention. It follows from the first equation of system (4.1) that $S_{T_{on}}$ and $S_{T_{off}}$ are monotonically decreasing with respect to $T_{on}$ and $T_{off}$, respectively, i.e. $\frac{\partial S_{T_{on}}}{\partial T_{on}}<0$ and $\frac{\partial S_{T_{off}}}{\partial T_{off}}<0$. In the following, we analyse the impact of the start and end timings of the second intervention on the maximum number of non-isolated infected individuals in detail.

(i):

The maximum number of non-isolated infected individuals occurs in the region $(a_{1})$, i.e. case $(a_{1})$.

For case ($a_{1}$), the maximum number of non-isolated infected individuals is given as

$$\begin{aligned} I^{c}_{max}= & {} -\frac{\gamma }{\beta _{10}}+\rho _{10} \ln \left( \frac{\gamma }{\beta _{20}}\right) +\left( \frac{\beta _{20}}{\beta _{10}}- \frac{\beta _{2c}}{\beta _{1c}} \right) (S_{T_{off}}-S_{T_{on}}) \nonumber \\ {}{} & {} +(\rho _{1c}-\rho _{10})\ln \left( \frac{S_{T_{off}}}{S_{T_{on}}}\right) +\phi _{10}, \end{aligned}$$

(A.1)

which depends on $S_{T_{on}}$ and $S_{T_{off}}$. It implies that $I^{c}_{max}$ depends not only on $T_{on}$ but also on $T_{off}$. Taking the derivative of equation (A.1) with respect to $T_{off}$, yields

$$\begin{aligned} \frac{\partial I^{c}_{max}}{\partial T_{off}}= & {} \gamma I_{T_{off}}\left[ R_{e2}(T_{off})-1 \right] -\frac{\beta _{1c}\gamma I_{T_{off}}( R_{03}-1) }{\beta _{10}} \nonumber \\ {}{} & {} \le \frac{(\beta _{10}-\beta _{1c}) ( R_{03}-1)\gamma I_{T_{off}}}{\beta _{10}}. \end{aligned}$$

(A.2)

Correspondingly, taking the derivative of Eq. (A.1) with respect to $S_{T_{on}}$, yields

$$\begin{aligned} \begin{aligned} \frac{\partial I^{c}_{max}}{\partial T_{on}}&=\gamma I_{T_{on}} \left[ R_{e1}(T_{on})-1 \right] - \frac{\beta _{10} \gamma I_{T_{on}}( R_{02}-1 )}{\beta _{1c}} \\ {}&- \left[ \frac{\gamma S_{T_{on}} I_{T_{on}}( R_{03}-1)}{S_{T_{off}}} - \frac{\beta _{10} \gamma S_{T_{on}} I_{T_{on}}\left[ R_{e2}(T_{off})-1 \right] }{\beta _{1c}S_{T_{off}}} \right] \frac{\partial S_{T_{off}}}{\partial S_{T_{on}}}, \end{aligned} \end{aligned}$$

(A.3)

where $\frac{\partial S_{T_{off}}}{\partial S_{T_{on}}}=\frac{\beta _{1c}S_{T_{off}}I_{T_{off}}}{\beta _{10}S_{T_{on}}I_{T_{on}}}>0$. It follows from Eq. (A.2) that if $\beta _{1c}>\beta _{10}$, then $\frac{\partial I^{c}_{max}}{\partial T_{off}}<0$. Thus, when $\beta _{1c}>\beta _{10}$, $I^{c}_{max}$ is monotonically decreasing with respect to $T_{off}$. However, it is difficult to calculate the sign of Eq. (A.3), so we cannot determine the monotonicity of $I^{c}_{max}$ with respect to $T_{on}$ for case ($a_{1}$).

(ii):

The maximum number of non-isolated infected individuals occurs in the region ($a_{2}$), i.e. case $(a_{2})$.

For case ($a_{2}$), the maximum number of non-isolated infected individuals is given as

$$\begin{aligned} I^{c}_{max}= & {} -\frac{\gamma }{\beta _{1c}}+\rho _{11} \ln \left( \frac{\gamma }{\beta _{2c}}\right) +\left( \frac{\beta _{2c}}{\beta _{1c}}-\frac{\beta _{20}}{\beta _{10}} \right) S_{T_{on}} \nonumber \\ {}{} & {} -(\rho _{1c}-\rho _{10})\ln (S_{T_{on}})+\phi _{10} \end{aligned}$$

(A.4)

which only depends on $S_{T_{on}}$. This implies that $I^{c}_{max}$ only depends on $T_{on}$ for case ($a_{2}$). Taking the derivative of equation (A.4) with respect to $T_{on}$, yields

$$\begin{aligned} \frac{\partial I^{c}_{max}}{\partial T_{on}}= & {} \left[ \gamma I_{T_{on}} \left[ R_{e1}(T_{on})-1 \right] - \frac{ \beta _{10} \gamma I_{T_{on}}( R_{02}-1 )}{\beta _{1c}} \right] \nonumber \\ {}{} & {} \ge \frac{(\beta _{1c}-\beta _{10})( R_{02}-1 ) \gamma I_{T_{on}}}{\beta _{1c}}. \end{aligned}$$

(A.5)

Thus, if $\beta _{1c}>\beta _{10}$, then $\frac{\partial I^{c}_{max}}{\partial T_{on}}>0$. Therefore, when $\beta _{1c}>\beta _{10}$, $I^{c}_{max}$ is monotonically increasing with respect to $T_{on}$ for case ($a_{2}$).

(iii):

The maximum number of non-isolated infected individuals occurs in the region $(a_{4})$, i.e. case $(a_{4})$.

For case ($a_{4}$), the maximum number of non-isolated infected individuals is given as

$$\begin{aligned} I^{c}_{max}=-\frac{\beta _{20}}{\beta _{10}}S_{T_{on}}+\rho _{10} \ln (S_{T_{on}})+\phi _{10} \end{aligned}$$

(A.6)

which only depends on $S_{T_{on}}$. This implies that $I^{c}_{max}$ also only depends on $T_{on}$. Taking the derivative of the above equation (A.6) with respect to $T_{on}$, we have

$$\begin{aligned} \frac{\partial I^{c}_{max}}{\partial T_{on}} =\gamma I_{T_{on}} \left[ R_{e1}(T_{on})-1 \right] >0. \end{aligned}$$

(A.7)

Thus, $I^{c}_{max}$ is monotonically increasing with respect to $T_{on}$ for case ($a_{4}$).

(iv):

The maximum number of non-isolated infected individuals occurs in the region $(a_{5})$, i.e. case $(a_{5})$.

For case ($a_{5}$), the maximum number of non-isolated infected individuals is given as

$$\begin{aligned} I^{c}_{max}=\{ I_{1}, I_{2}\}. \end{aligned}$$

(A.8)

If $I^{c}_{max}=I_{2}$, then $\frac{\partial I^{c}_{max}}{\partial T_{off}}=0$, and when $\beta _{1c}>\beta _{10}$, $\frac{\partial I^{c}_{max}}{\partial T_{on}}<0$. Thus, when $\beta _{1c}>\beta _{10}$, $I^{c}_{max}$ is monotonically increasing with respect to $T_{on}$. If $I^{c}_{max}=I_{1}$, it follows from Eq. (A.2) that $\frac{\partial I^{c}_{max}}{\partial T_{off}}<0$. Thus, in this case, $I^{c}_{max}$ is monotonically decreasing with respect to $T_{off}$. Therefore, for case ($a_{5}$), $\frac{\partial I^{c}_{max}}{\partial T_{off}}\le 0$ and $I^{c}_{max}$ is quasi-monotonically decreasing with respect to $T_{off}$. However, it is difficult to calculate the value of Eq. (A.3), and we cannot determine the monotonicity of $I^{c}_{max}$ with respect to $T_{on}$ for case ($a_{5}$).

(v):

The maximum number of non-isolated infected individuals occurs in the region $(a_{6})$, i.e. case $(a_{6})$.

For case ($a_{6}$), the maximum number of non-isolated infected individuals is given as

$$\begin{aligned} I^{c}_{max}=\{ I_{T_{on}}, I_{1}\}. \end{aligned}$$

(A.9)

If $I^{c}_{max}=I_{T_{on}}$, then $\frac{\partial I^{c}_{max}}{\partial T_{on}}>0$ and $\frac{\partial I^{c}_{max}}{\partial T_{off}}=0$. Thus, $I^{c}_{max}$ is monotonically increasing with respect to $T_{on}$. If $I^{c}_{max}=I_{1}$, it follows from Eq. (A.2) that $\frac{\partial I^{c}_{max}}{\partial T_{off}}<0$. Thus, in this case, $I^{c}_{max}$ is monotonically decreasing with respect to $T_{off}$. Therefore, $\frac{\partial I^{c}_{max}}{\partial T_{off}}\le 0$ and $I^{c}_{max}$ is quasi-monotonically decreasing with respect to $T_{off}$ for case ($a_{6}$). However, it is difficult to calculate the sign of Eq. (A.3), so we cannot determine the monotonicity of $I^{c}_{max}$ with respect to $T_{on}$ for case ($a_{6}$)

Appendix B: Analysis of the special case $T_{off}=T^{c}_{end}$ in model (2.1) with (2.2)

Fig. 12

The impact of the quarantine rate $q_{c}$, contact rate $c_{c}$ and the start time $T_{on}$ of the second intervention on the maximum number of non-isolated infected individuals and the time needed to realize the dynamic zero-case aim, where a and b $c_{c}=10$, c and d $q_{c}=0.01526$, e and f $T_{on}=3$. Other parameters are as follows: $N=763$, $S_{0}=762$, $I_{0}=1$, $\beta =0.155$, $c_{0}=10$, $q_{0}=0.01526$, $\gamma _{I}=0.1$ and $\delta _{I}=0.2504$

In the following, we carry out the numerical analyses to understand the effect of the timing of the second intervention and close contact tracing strategy on the maximum number of non-isolated infected individuals and the time needed to realize the dynamic zero-case aim. Here we let $N=763$, $S_{0}=762$, $I_{0}=1$, $\beta =0.155$, $c_{0}=10$, $q_{0}=0.01526$, $\gamma _{I}=0.1$ and $\delta _{I}=0.2504$, and produce the contour plots of the maximum number of non-isolated infected individuals, and the time needed to realize the dynamic zero-case with respect to $T_{on}$ and $\Delta q$, $T_{on}$ and $\Delta c$, $\Delta q$ and $\Delta c$, respectively, as shown in Fig. 12a–f. In Fig. 12a and b, we let $c_{c}=10$, i.e. the contact rate during the second intervention is the same as that without the second intervention. It follows from Fig. 12a that the earlier the second intervention, the stronger the contact tracking intensity during the second intervention, the smaller the maximum number of non-isolated infected individuals. In Fig. 12b, it is worth mentioning that $T^{c}_{end}$ is non-monotonically dependent on $\Delta q$ for a fixed small $T_{on}$, which implies that even if a second intervention is implemented early, it may take longer to realize the dynamic zero-case policy due to the contact tracing intensity not being strong enough. Therefore, in this case, to minimize the maximum number of non-isolated infected individuals and shorten the duration, the best strategy is to intervene early and fully strengthen the contact tracing intensity. As shown in Fig. 12a and b, it is clear that when $(T_{on}, \Delta q)$ nears the point (1, 0.85), both $I^{c}_{max}$ and $T^{c}_{end}$ are minimized.

Comparing this with the results shown in Fig. 12a, we can conclude that the contact rate should be reduced to a very low level to get the same $I^{c}_{max}$ in Fig. 12c. This implies that strengthening the contact tracing intensity is more effective than reducing the contact rate in minimizing the maximum number of infected individuals. It follows from Fig. 12b and d that strengthening the contact tracing intensity is also more effective than reducing the contact rate in shortening the time needed to realize the dynamic zero-case aim.

The results from Fig. 12e with $T_{on}=3$ reveal that the stronger the contact tracing intensity, the smaller the contact rate during the second intervention, the more effective they are to minimize the maximum number of non-isolated infected individuals. In terms of the time needed to realize the dynamic zero-case aim, it follows from Fig. 12f that $T^{c}_{end}$ is monotonically decreasing with respect to $\Delta q$ for any fixed $\Delta c$. It is worth mentioning that for small and intermediate $\Delta q$, $T^{c}_{end}$ is an upward function of $\Delta c$. This means that when the quarantine rate ($q_{c}$) is not large enough, in order to shorten the time needed to realize the dynamic zero-case aim, it is necessary to reduce the contact rate to a very low level (i.e. $\Delta c$ is large enough).