Modeling the spread dynamics of multiple-variant coronavirus disease under public health interventions: A general framework

Abstract

The COVID-19 pandemic was caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which is a single-stranded positive-stranded RNA virus with a high multi-directional mutation rate. Many new variants even have an immune-evading property, which means that some individuals with antibodies against one variant can be reinfected by other variants. As a result, the realistic is still suffering from new waves of COVID-19 by its new variants. How to control the transmission or even eradicate the COVID-19 pandemic remains a critical issue for the whole world. This work presents an epidemiological framework for mimicking the multi-directional mutation process of SARS-CoV-2 and the epidemic spread of COVID-19 under realistic scenarios considering multiple variants. The proposed framework is used to evaluate single and combined public health interventions, which include non-pharmaceutical interventions, pharmaceutical interventions, and vaccine interventions under the existence of multi-directional mutations of SARS-CoV-2. The results suggest that several combined intervention strategies give optimal results and are feasible, requiring only moderate levels of individual interventions. Furthermore, the results indicate that even if the mutation rate of SARS-CoV-2 decreased 100 times, the pandemic would still not be eradicated without appropriate public health interventions.

1. Introduction

The world is still experiencing the ongoing coronavirus disease 2019 (COVID-19) pandemic, which was caused by the novel human pathogen severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [8], [29]. The outbreak of COVID-19 has become a global crisis that threatens the public health of billions of people, disrupts the economic growth of every country [19], and has progressed with a tremendous impact on social behavior [4], [2], the environment [27], and climate [7]. The continued spread of the COVID-19 pandemic has led to the implementation of non-pharmacological interventions (NPIs), pharmacological interventions (PIs), and vaccine interventions (VIs) to limit its spread over the world. NPIs mean that the interventions can be achieved through limiting human activities to reduce the rate of contact between susceptible and infected individuals [5], [30], while PIs are measures that can improve the health care system to reduce the mortality rate of infected individuals, increase their recovery rate, and enhance the testing capacity to screen out more unconfirmed patients. Unfortunately, there have been no signs of ending at this moment, partially due to the fact that there are no specific anti-SARS-CoV-2 drugs and that multiple dangerous variants have appeared.

When a virus replicates itself to invade cells, some errors may occur and mutations arise due to multiple factors, such as random genetic drift and natural selection [9]. SARS-CoV-2 is a single-stranded positive-strand RNA virus, which has a higher mutation rate than DNA virus and can accrue mutations in a short time [14]. Therefore, the faster and more widely SARS-CoV-2 spreads, the more frequent and diverse the mutations will be and the more new variants will appear. Most viral mutations are trivial and short-lived, with limited harm to humans. However, a few mutations can affect protein generation and have the potential to significantly alter the epidemiological characteristics of SARS-CoV-2 variants, leading to increased infectivity and transmissibility [26], [13], higher clinical severity and mortality rate, and even the development of a neutralization resistance ability [15], namely, immune-evading antibodies induced by either infection or vaccination. Consequently, individuals with antibodies against the “old SARS-CoV-2 variants” will be reinfected by the “new SARS-CoV-2 variants” [24]. As a result, the variants may have different epidemiological characteristics and immune-evading abilities. By April 19, 2022, more than 1,600 SARS-CoV-2 variants had been detected in more than 100 countries. The detailed information of these variants can be found in Section 2.2.

Several dangerous variants with different epidemiological characteristics and immune-evading abilities have been found in different countries [28]. Currently, more than 200 vaccines are in various clinical trial stages, as reported in the website COVID-19 Treatment and Vaccine Tracker (https://covid-19tracker.milkeninstitute.org/). However, most of the vaccines developed to date are based on the reference genome collected on January 5, 2020. Hence, new SARS-CoV-2 variants may evade immunity induced by infection or vaccine against previous variants, generating challenges to vaccine interventions [24]. Additionally, the efficacy of NPIs and PIs, which are designed based on the original SARS-CoV-2 virus, may be limited. The emergence of new dangerous variants highlights the need to re-explore the effectiveness and impact of interventions (including NPIs, PIs, and VIs) to help disentangle the impact of each of them. Furthermore, precise evaluation of particular interventions can help local authorities to find out how to contain or even eliminate these variants and make a good trade-off between fatalities caused by the COVID-19 pandemic and economic costs [1]. To the best of our knowledge, most previous studies developing epidemiological models for investigating the spread of the COVID-19 disease ignore the influence of mutation and variants. The common assumption of these studies is that there is only one virus in the COVID-19 pandemic. Although some have considered transmission dynamics with two variants [34], and few have considered one-directional mutation among multiple variants [36]. However, the real world of the COVID-19 pandemic is more complex than these assumptions. Data-driven models that consider the coexistence of multiple variants of the disease and the quantification of interventions are rare, particularly in the context of the COVID-19 pandemic with the considering multi-directional mutations.

To address the aforementioned challenges posed by the emergence of COVID-19 variants, we used real COVID-19 variants data recorded by researchers and vaccination data released by the governments in different countries to determine important factors that are utilized to describe the mutation rate, vaccine development and supply. Then, we propose a novel epidemiological framework, named the Susceptible-Exposed-Unreported-Asymptomatic-Confirmed-Recovered-Deceased with Vaccine and Mutation (SEUACRD-VM) framework, for simulating the emergence of realistic COVID-19 virus mutations and the transmission of multiple variants with human interventions. In this framework, mutations accumulate as infections increase. Significant mutations can alter the epidemiological features of the virus, including its infectivity, transmissibility, clinical severity, the mortality rate, and even its immune-evading ability. The SEUACRD-VM framework contains three main components: (1) a proposed Susceptible-Exposed-Unreported-Asymptomatic-Confirmed-Recovered-Deceased with Vaccination Population (SEUACRD-VP) model, which is an extension of our previous work [31], [32], for modeling the transmission of each variant in the population; (2) a mechanism for mimicking multi-directional mutation and generating new variants with different epidemiological features; (3) a mechanism for simulating vaccine development, supplies and distribution, which is based on real-world data. Overall, the main feature of our proposed model is its ability to capture the dynamic processes of virus mutation, the transmission of multiple variants, and the impact of interventions.

Then, based on the developed framework, we investigated the cost-effectiveness of a wide spectrum of single or combined intervention strategies, including non-pharmaceutical interventions (restricted social contact, mandatory mask-wearing and quarantine of confirmed cases), pharmaceutical interventions (enhanced medical resources to increase the recovery rate, reduce the mortality rate and enhance the testing capacity), and vaccination interventions (accelerating vaccine development by enhancing virus monitoring and shortening the development period, increasing the vaccine supply and enhancing the vaccine acceptance rate). This comprehensive analysis has allowed us to determine the most effective strategies applicable to containing the COVID-19 pandemic under the scenarios with new variants that have different epidemiological features. To ensure the reliability and validity of the experiments, we collected a large amount of real SARS-CoV-2 mutation data from December 24, 2019 to February 22, 2022 and vaccine data from December 3, 2019 to May 25, 2022 to support the simulation of the COVID-19 virus mutation mechanism and vaccination process. Ultimately, we found and validated the following results in our experiments:

• •In scenarios where single interventions are implemented, experiments show that strictly restricting social contact and enhancing the vaccine acceptance rate are the optimal interventions to contain and eradicate epidemics and pandemics when considering the occurrence of multi-directional virus mutations and the coexistence of multiple variants. Certain interventions such as reducing the mortality rate, accelerating vaccine development, and enhancing virus monitoring had almost no impact on containing the pandemic. In contrast, other interventions such as quarantining of confirmed cases, increased testing capacity, increased vaccine supply and increased recovery rate could contain the transmission rate of the virus effectively;
• •By combining several intervention strategies together, our experimental results show that the combined interventions are highly effective in containing or even eradicating the COVID-19 pandemic. The implementation of combined intervention strategies is always more feasible and effective under a more relaxed scenario, as the control intensity of each intervention is looser than that adopting only one intervention;
• •Under the scenarios with no (or loosening) intervention, our experimental results reveal that the number of daily infections would reach its first peak in the middle of the second year after the first COVID-19 patient was detected. Then, the number of daily infections decreases and reaches the first valley in the early third year. After that, as multiple variants emerge, the pandemic will spread rapidly with an unstoppable trend; that is, there are almost no valleys anymore.

It is worth noting that this research is based on previous studies incorporating and evaluating some of the most common intervention strategies adopted by most local authorities around the world. Moreover, the developed framework can effectively capture the dynamic processes of virus mutations and fully evaluate the impact of interventions on the spread of COVID-19. It is our hope that this work can help local authorities to adopt suitable intervention strategies to contain the spread of future pandemics.

The reminder of this paper is organized as follows. Section 2 describes and discusses our proposed framework. Section 3 introduces intervention measures under the established framework. Section 4 presents experimental results, and Section 5 concludes the paper with some discussions.

2. A general epidemic spreading framework

This paper proposes a general epidemiological framework, i.e., the aforementioned SEUACRD-VM, by considering multi-directional mutations, multiple variants, and massive vaccinations. Specifically, this framework includes three main parts: (1) a single-virus dynamics model (Sec. 2.1); (2) a mechanism for mimicking variant mutation (Sec. 2.2); (3) a mechanism for simulating vaccination (Sec. 2.3). A detailed description of the SEUACRD-VP model is presented in the Appendix A.

2.1. Brief introduction of the SEUACRD-VP model

Assume that at time t, there exist j variants $\{v^{(1)},v^{(2)},\cdots ,v^{(j)}\}$. According to the known epidemiological features of COVID-19, individuals in a population can be roughly classified into eleven classes considering variant-i [33]: Susceptible ($S^{(i)}$), Exposed (or presymptomatic, $E^{(i)}$), Unreported cases with clinical features ($U^{(i)}$), Asymptomatic patients without clinical features ($A^{(i)}$), Confirmed cases with clinical features ($C^{(i)}$), Recovered asymptomatic infectious ($R_a^{(i)}$), Recovered unreported infectious ($R_u^{(i)}$), Recovered confirmed cases ($R_c^{(i)}$), Deceased unreported cases ($D_u^{(i)}$), Deceased confirmed cases ($D_c^{(i)}$), and Vaccinated individuals ($V^{(i)}$). These eleven classes can transit from one to another (shown in Fig. 1 ).

Fig. 1.

Flow chart of the SEUACRD-VP model. Each rectangle represents a unique individual state, the solid black line represents the direction of state transition, and the dashed line represents that S individuals can be infected by 4 different classes of infected individuals who have infective properties and can spread the SARS-CoV-2.

Additionally, we make some fundamental assumptions as follows:

• •One individual can only be infected by one specific variant at a time. This assumption is reasonable as most of the infected individuals may be quarantined or hospitalized, resulting in a low chance of contact with other infected individuals and infected by other variants;
• •Individuals with antibodies against variant-i would not be infected by variant-i but have a possibility to be infected by another variant-j, where $i\ne j$.

Given the existence of variant-i, for variant-j, we develop the SEUACRD-VP model $f(\cdot )$ for describing the dynamic properties of the number of each class at time $t+1$ as follows:

$$X^{(i)}(t+1)=f(X^{(i)}(t)|{\theta }^{(i)}),$$ (1)

where ${\theta }^{(i)}=\{{\theta }_I^{(i)},{\theta }_R^{(i)},{\theta }_M^{(i)},{\theta }_C^{(i)},{\theta }_O^{(i)}\}$ and $X^{(i)}=\{S^{(i)},E^{(i)},U^{(i)},A^{(i)},C^{(i)},R_a^{(i)},R_u^{(i)},R_c^{(i)},D_u^{(i)},D_c^{(i)},V^{(i)}\}$. The parameter sets of the SEUACRD-VP model determining the epidemiological features of variant-i are ${\theta }_I^{(i)}=\{{\alpha }_e^{(i)},{\alpha }_a^{(i)},{\alpha }_u^{(i)},{\alpha }_c^{(i)}\}$ (determining the infectivity), ${\theta }_R^{(i)}=\{{\gamma }_{ur}^{(i)},{\gamma }_{cr}^{(i)},{\gamma }_{ar}^{(i)}\}$ (determining the recovery rates), ${\theta }_M^{(i)}=\{{\gamma }_{ud}^{(i)},{\gamma }_{cd}^{(i)}\}$ (determining the mortality rates), ${\theta }_C^{(i)}=\{{\beta }_{eu}^{(i)},{\beta }_{ea}^{(i)}\}$ (determining the incubation period) and other important parameters ${\theta }_O^{(i)}$. Note that mutations can alter an epidemiological feature by altering several parameters (including ${\theta }_I^{(i)}$, ${\theta }_R^{(i)}$, ${\theta }_M^{(i)}$, ${\theta }_C^{(i)}$, and $k_{ev}^{(i)}$) of the SEUACRD-VP model. Here, $k_{ev}^{(i)}$ is a parameter that determines the immune-evading property of variant-i. Meanwhile, there exist several parameters that cannot be altered by mutations. Fig. 5 shows which parameters can or cannot be altered. The detailed description of these parameters is shown in Table 1, Table 2 . More details of the SEUACRD-VP model are provided in the Appendix A.

Fig. 5.

Factors can be varied by intervention measures, including NPIs, PIs and VIs.

Table 1.

Parameters of the SEUACRD-VP model.

Variable	Epidemiological denotation of parameters	Initial value
$N_p^{(1)}$	The number of population in a region before the start of the pandemic	2,000,000,000
kA	The proportion of asymptomatic cases among the infectious cases	0.4
${\alpha }_e^{(1)}$	The infection rate of transmission per contact from the exposed cases	$\frac{1}{22}$
${\alpha }_a^{(1)}$	The infection rate of transmission per contact from the asymptomatic cases	$\frac{9}{10}{\alpha }_e^{(1)}$
${\alpha }_u^{(1)}$	The infection rate of transmission per contact from the unreported cases	$\frac{4}{5}{\alpha }_e^{(1)}$
${\alpha }_c^{(1)}$	The infection rate of transmission per contact from the confirmed cases	$\frac{1}{5}{\alpha }_u^{(1)}$
${\beta }_{ea}^{(1)}$	The transition rate from the exposed cases to the asymptomatic cases	$k_A\frac{1}{10}$
${\beta }_{eu}^{(1)}$	The transition rate from the exposed cases to the unreported cases	$(1-k_A)\frac{1}{10}$
${\beta }_{uc}^{(1)}$	The transition rate from the unreported cases to the confirmed cases	$\frac{1}{10}$
${\beta }_{ac}^{(1)}$	The transition rate from the asymptomatic cases to the confirmed cases	$\frac{1}{10000}$
${\gamma }_{cr}^{(1)}$	The transition rate of the confirmed cases to the recovered cases	$\frac{1}{50}$
${\gamma }_{ar}^{(1)}$	The transition rate of the asymptomatic cases to the recovered cases	$\frac{6}{5}{\gamma }_{cr}^{(1)}$
${\gamma }_{ur}^{(1)}$	The transition rate of the unreported infectious cases to the recovered cases	$\frac{1}{2}{\gamma }_{cr}^{(1)}$
${\gamma }_{cd}^{(1)}$	The transition rate of the confirmed cases to the deceased cases	$\frac{1}{2000}$
${\gamma }_{ud}^{(1)}$	The transition rate of the unreported cases to the deceased cases	2γcd
${\gamma }_{rs}^{(1)}$	The transition rate of the recovered cases to the susceptible cases	$\frac{1}{200}$
${\gamma }_{vs}^{(1)}$	The transition rate of the vaccinated cases to the susceptible cases	$\frac{1}{200}$

Table 2.

Parameters of mutation and vaccination.

Variable	Parameter	Initial value
$k_{ev}^{(1)}$	Immune-evading rate of a variant	0.3
NM	Threshold of a mutation happening	5,000,000
δv	Vaccine acceptation rate	0.75
kv	Daily increase in production capacity	169,354
nV,Thr	Maximum productivity of vaccine	0.006Np
IT	Threshold for starting vaccine development for a variant	200,000
TV,i	Time for developing the vaccine against variant-i	150 days
γp	Rate of natural population increase	$\frac{0.02}{365}$

2.2. Mutation and variants of COVID-19

The PANGO Lineages database (https://cov-lineages.org/lineage_list.html) records detailed information about SARS-CoV-2 variants, such as lineage name, most common countries, earliest date, first reported area, WHO names, etc. [21], [18]. By April 19, 2022, more than 1,600 SARS-CoV-2 variants had been found in more than 100 countries. Fig. 2 (a) shows the number of variants found in different countries around the world. The statistical results indicate that about 85%, 9%, 5%, and 1% of total variants were found in high-income, upper-middle-income, lower-middle-income, and lower-income countries, respectively. Obviously, most variants are first reported by high-income countries that have extensive sequencing resources, as the detection of genome sequence variants requires sufficient research ability [16]. Obviously, there is a high probability that a large group of unconfirmed mutations and variants in low-income countries have not been detected. Hence, the actual number of variants may be much larger than the reported number. Fig. 2(b) shows the cumulative number of global COVID-19 confirmed cases $C_w(t)$ and the number of reported variants $v_w(t)$ from the end of 2019 to mid-2022. Here, we define a rate, $m_U^{(i)}(t)=\frac{C_w(t)}{v_w(t)}$, to represent how many confirmed cases against one variant. The statistical results indicate that we can screen out one variant per 310,000 confirmed cases by mid-2022. Note that, since the actual number of variants may be much larger than the number of recorded variants, the actual $m_U^{(i)}(t)$ should be smaller than 310,000.

Fig. 2.

(a) The number of variants founded in different countries around the world by April 19, 2022, and the proportion of variants found in high-income, upper-middle-income, lower-middle income, and lower-income countries; (b) the cumulative number of confirmed cases and recorded variants around the world.

2.2.1. A framework for multi-directional mutation

Based on the above observation, we propose a framework to mimic multi-directional mutation for generating new variants with different epidemiological features. Here, we assume that all but the first virus evolve from one of the previous viruses. At time $t_j$, there exist j variants $\{v^{(1)},v^{(2)},\cdots ,v^{(j)}\}$. Then, assume that, at time $t_{j+1}$, the number of cumulative infected individuals reaches $k{N}_M$ ($k=1,2,\cdots$, and $N_M>0$ is a constant). One of the j variants, still denoted variant-j, has significant mutations (shown in Fig. 3 (a)), resulting in a new variant-$(j+1)$ and the total number of variants increases to $j+1$. Actually, a mutation happens randomly, which means that $N_M$ is a random number. However, in this study, we assume that $N_M$ is constant for simplicity.

Fig. 3.

(a) The time when the cumulative number of infected case becomes larger than the threshold kN_M. At time T_i, one new variant appears; (b) The flowchar shows multi-directional mutation.

Here, we simply assume that the more individuals a virus has infected in a time interval $\mathrm{\Delta }{t}_j=t_{j+1}-t_j$, the greater the probability that it will mutate. For the existing j variants, the cumulative infected number of the variant-i from time $t_j$ to $t_{j+1}$ is $\mathrm{\Delta }{P}_I^{(i)}(t)$, where $i=1,2,3,\cdots ,j$ and $N_M={\sum }_{i=1}^{i=j}\mathrm{\Delta }{P}_I^{(i)}$. Then, the mutation probability of variant-i is proportional to $\mathrm{\Delta }{P}_I^{(i)}$; that is, the probability is $\frac{\mathrm{\Delta }{P}_I^{(i)}}{N_M}$. Obviously, ${\sum }_{i=0}^{i=j}\frac{\mathrm{\Delta }{P}_I^{(i)}}{N_M}=1$. Then, according to this probability of mutation, one variant will be randomly chosen from these j variants for mutation and one variant will likely mutate multiple times to generate different new variants, which mimics multi-directional variant mutations. Based on the above, a framework for simulating multi-directional mutation is now proposed (shown in Fig. 3).

2.2.2. Epidemiological characteristics of new variant-$(j+1)$

The epidemiological features of the new variant-$(j+1)$, including the viral infectivity (${\theta }_I^{(j+1)}$), recovery rate (${\theta }_R^{(j+1)}$), mortality rate (${\theta }_M^{(j+1)}$), and incubation period (${\theta }_C^{(j+1)}$), are highly related with and varied from the epidemiological features of variant-i (shown in Fig. 3(b)):

$$\{{\theta }_I^{(i)},{\theta }_R^{(i)},{\theta }_M^{(i)},{\theta }_C^{(i)}\}\stackrel{M}{\to }\{{\theta }_I^{(j+1)},{\theta }_R^{(j+1)},{\theta }_M^{(j+1)},{\theta }_C^{(j+1)}\},$$ (2)

where M represents a mapping relationship.

Next, we show how to alter the epidemiological features. We assume that the epidemiological features of variant-$(j+1)$ differ from variant-i as follows:

• •
  Infectivity:

  $${\alpha }_e^{(j+1)}=(1+{\delta }^{(I)})\cdot {\alpha }_e^{(i)},{\alpha }_a^{(j+1)}=(1+{\delta }^{(I)})\cdot {\alpha }_a^{(i)},{\alpha }_u^{(j+1)}=(1+{\delta }^{(I)})\cdot {\alpha }_u^{(i)},{\alpha }_c^{(j+1)}=(1+{\delta }^{(I)})\cdot {\alpha }_c^{(i)},$$ (3)

  where ${\delta }^{(I)}$ is a single uniformly distributed random number in the interval $[{\delta }_l^{(I)},{\delta }_u^{(I)}]$.
• •
  Mortality and recovery rate:

  $${\gamma }_{ur}^{(j+1)}=(1+{\delta }^{(R)})\cdot {\gamma }_{ur}^{(i)},{\gamma }_{cr}^{(j+1)}=(1+{\delta }^{(R)})\cdot {\gamma }_{cr}^{(i)},{\gamma }_{ar}^{(j+1)}=(1+{\delta }^{(R)})\cdot {\gamma }_{ar}^{(i)},{\gamma }_{ud}^{(j+1)}=(1+{\delta }^{(M)})\cdot {\gamma }_{ud}^{(i)},{\gamma }_{cd}^{(j+1)}=(1+{\delta }^{(M)})\cdot {\gamma }_{cd}^{(i)},$$ (4)

  where ${\delta }^{(R)}$ and ${\delta }^{(M)}$ are all uniformly distributed random numbers in the interval $[{\delta }_l^{(R)},{\delta }_u^{(R)}]$ and $[{\delta }_l^{(M)},{\delta }_u^{(M)}]$, respectively.
• •
  Incubation time:

  $${\beta }_{eu}^{(j+1)}=(1+{\delta }^{(C)})\cdot {\beta }_{eu}^{(i)},{\beta }_{ea}^{(j+1)}=(1+{\delta }^{(C)})\cdot {\beta }_{ea}^{(i)},$$ (5)

  where ${\delta }^{(C)}$ is a uniformly distributed random number in the interval $[{\delta }_l^{(C)},{\delta }_u^{(C)}]$. Note that the longer the incubation period, the longer the time an infected individual can spread the virus. Hence, the variant will be more dangerous.

Note that there is no clear evidence or study to explain the direction of mutation of the variant. Hence, mutations have an equal chance of increasing or decreasing infectivity, mortality, recovery rate, and incubation time in the experiment. Then, we set the interval range for infectivity, mortality, recovery rate, and incubation time as $[-0.2,+0.2]$. Now, we are ready to develop an epidemiological framework considering multi-directional mutations. The detailed framework is shown in the Appendix A.

2.2.3. Immune-evading properties of new variant

Based on the assumption in Sec. 2.1, for variant-i, the population ($P_a^{(i)}(t)$) with antibodies against variant-i is as follows:

$$P_a^{(i)}(t)=R_a^{(i)}(t)+R_u^{(i)}(t)+R_c^{(i)}(t)+V^{(i)}(t),$$ (6)

where $R_a^{(i)}(t)$, $R_u^{(i)}(t)$, and $R_c^{(i)}(t)$ represent the numbers of recovered individuals who are infected by variant-i and $V^{(i)}(t)$ represents the number of individuals who were vaccinated against variant-i at time t.

The new variant-$(j+1)$ may have immune-evading properties, which means that some individuals with antibodies against other variants $\{v^{(1)},v^{(2)},\cdots ,v^{(j)}\}$ may not be fully protected from being infected by variant-$(j+1)$. Therefore, the number of susceptible individuals and the population for each variant are different. Here, we assume that $0<k_{ev}^{(j+1)}<1$ population with antibodies against other variants would be infected by variant-$(j+1)$. Then, the number of the population with antibodies against variant-i but would be infected by the new lineage variant is $k_{ev}^{(j+1)}{P}_a^{(i)}(t)$, where $i\ne j+1$ and $k_{ev}^{(j+1)}$ is a uniformly distributed random number in the interval $[{\delta }_l^{(E)},{\delta }_u^{(E)}]$. In SEUACRD-VM, we assume that the total population in day t is $N_p(t)$ and that it changes daily with a natural population increase rate of ${\gamma }_p$. Then, the number of susceptible individuals for the variant-$(j+1)$ is

$$S^{(j+1)}(t)=N_p(t)-[(1-k_{ev}^{(j+1)})(\sum _{i=1}^n{P}_a^{(i)}(t)-P_a^{(j+1)}(t))+P_a^{(j+1)}(t)],$$ (7)

where n is the number of variants in day t.

The number of all the vaccinated populations in day t is:

$$V_{all}(t)=\sum _{i=1}^{(j+1)}{V}^{(i)}(t).$$ (8)

Similarly, we define $E_{all}(t)$, $A_{all}(t)$, $C_{all}(t)$ and $U_{all}(t)$ as the numbers of all the exposed, asymptomatic infections, confirmed infections, and unreported infections, respectively.

Then, the individuals who would be infected by variant-$(j+1)$ include $E^{(j+1)}$, $A^{(j+1)}$, $C^{(j+1)}$ and $U^{(j+1)}$. Therefore, the population of variant-$(j+1)$ in day t is:

$$N_p^{(j+1)}(t)=N_p(t)-((E_{all}(t)+A_{all}(t)+C_{all}(t)+U_{all}(t))-(E^{(j+1)}(t)+A^{(j+1)}(t)+C^{(j+1)}(t)+U^{(j+1)}(t))),$$ (9)

where $i\ne j+1$. To this end, we can derive the number of variables in day $t+1$, such as $E^{(j+1)}(t+1)$, $A^{(j+1)}(t+1)$, etc., from the previous ones.

2.3. Vaccine development, supplies and distribution

In this subsection, we introduce the mechanism for simulating vaccine development, supplies and distribution.

2.3.1. Vaccine development

Note that not every variant can cause an outbreak and spread all over the world, as most mutations are deleterious to the virus itself and only a few mutations would enhance the virus. Hence, most variants will not be widely transmitted and may even become extinct in the near future, while only a few variants, such as the Alpha, Beta, Lambda, and Omicron variants, can spread and become the main strains. Scientific research takes time to discover which variant is dangerous and medical resources are used to develop a vaccine against a dangerous variant. Here, we make a simple assumption in the SEUACRD-VM that, if a variant-i infected more than $I_T$ individuals at time $t_{i,d}$, it will be confirmed as a “dangerous variant” and a vaccine will be developed against it (shown in Fig. 4 (a)). The time for developing a vaccine against variant-i is $T_{V,i}$. Then, at time $t_{v,0}^{(i)}=t_{i,d}+T_{V,i}$, a new vaccine is ready to distribute to people (shown in Fig. 4).

Fig. 4.

(a) The condition for confirming a variant as a “dangerous variant” and the time for developing a new vaccine; (b) The illustrative chart for vaccine development time and the increase of vaccine supplies.

2.3.2. Vaccine supplies

Under the real scenario, in the early stages of the production of a new vaccine, the vaccine supplies are always insufficient. The number of daily dosed individuals will gradually increase and eventually reach a maximum value (shown in Fig. 4(b)). Hence, it is reasonable to define the number of daily dosed individuals by

$$n_V(t)=\min \{n_{V,M},k_v(t-t_{v,0}^{(1)})\},t_{v,0}^{(1)}\le t,$$ (10)

where $n_{V,M}$ is the maximum number of daily dosed individuals. $k_v=169,354$ is a constant equal to the slope of the approximate line fitting the real numbers of daily dosed individuals in the whole world (shown in Fig. 4(b)). In a real scenario, not everyone wants to take the vaccine. For instance, studies indicate that a considerable proportion of the population refuse vaccines [3]. The majority of survey studies among the general public, stratified by country, have revealed a level of acceptance of COVID-19 vaccination of 70% [22]; that is, 70% of the population is willing to be vaccinated. Hence, it is reasonable to define a rate of acceptance of vaccines $0<{\delta }_v<1$. Then, human society will stop vaccination when $V_{all}(t-1)>{\delta }_v{N}_p(t)$, and set $n_V(t)=0$.

2.3.3. Vaccine distribution

Here, we consider an ideal scenario in which vaccines can be accurately distributed to individuals without antibodies; that is, no individual with an antibody against variant-j will take a booster dose against variant-j. Additionally, the vaccine development and distribution focus primarily on combating currently severely infected variants in the population. Then, at time t, the daily count of individuals who took vaccine against variant-i is proportional to the daily infections by variant-i ($\mathrm{\Delta }{I}^{(i)}(t)$):

$$n_V^{(i)}(t)=\frac{\mathrm{\Delta }{I}^{(i)}(t-1)}{{\sum }_{j=0}^{M(t)}(\mathrm{\Delta }{I}^{(j)}(t-1))}{n}_V(t),$$ (11)

where $M(t)$ is the number of variants at time t and $n_V^{(i)}(t)$ is the number of daily dosed individuals who have developed immunity against variant-i. Note that since SEUACRD-VM considers multiple variants, its computational formula for $n_V^{(i)}(t)$ is slightly different from that of the SEUACRD-VP model.

At time t, there are $V^{(i)}(t)$ individuals who still have immunity through vaccination against variant-i. Note that a vaccinated individual will lose immunity to a variant over a period of time and has the potential to be reinfected [20]. We set a rate ${\gamma }_{VS}^{(i)}$ for the vaccinated group to lose immunity and transit into the susceptible group. Then, the total number of vaccinated individuals who still have antibodies against variant-i is

$$V^{(i)}(t+1)=V^{(i)}(t)+n_V^{(i)}(t+1)-{\gamma }_{VS}^{(i)}{V}^{(i)}(t).$$ (12)

3. Intervention measures

In this work, we want to investigate the effects of intervention measures (NPIs, PIs, and VIs) to determine whether these interventions can contain or even eradicate the pandemic under the multi-directional mutation scenario. For instance, the government can restrict social contact to reduce the transmission rate of the virus. To simulate the effect of this intervention, we can reduce transmission rate parameters (${\alpha }_e^{(\cdot )}$, ${\alpha }_u^{(\cdot )}$ and ${\alpha }_a^{(\cdot )}$) to ${\delta }_S{\alpha }_e^{(\cdot )}$, ${\delta }_S{\alpha }_u^{(\cdot )}$, and ${\delta }_S{\alpha }_a^{(\cdot )}$, respectively, where $0\le {\delta }_S\le 1$ stands for the strength of restricting social contact, and $(\cdot )$ represents all variants. Here, ${\delta }_S=1$ means no restriction of social contact so that people can interact freely, while ${\delta }_S=0$ means implementing the strictest restriction so that no one can have any contact with others. For other interventions, we have similar definitions. In this work, we consider the following NPIs:

• •Restricted social contact with mandatory: These intervention measures include limiting mobility, travels, contacts, and even lockdown cities, etc., which have been shown in various countries to be an effective non-pharmacological intervention (NPIs) to control the COVID-19 pandemic [5], [30], as these measures can reduce the risk of transmission from exposed, unconfirmed, and asymptomatic cases by reducing the transmission rates ${\alpha }_e^{(\cdot )}$, ${\alpha }_u^{(\cdot )}$ and ${\alpha }_a^{(\cdot )}$ to ${\delta }_S{\alpha }_e^{(\cdot )}$, ${\delta }_S{\alpha }_u^{(\cdot )}$, and ${\delta }_S{\alpha }_a^{(\cdot )}$, respectively;
• •Quarantine of confirmed patients: This intervention measure can help control an outbreak by reducing the risk of transmission from confirmed infected individuals [23] by reducing the transmission rate ${\alpha }_c^{(\cdot )}$ to ${\delta }_Q{\alpha }_c^{(\cdot )}$, where $0\le {\delta }_Q\le 1$.

On the other hand, PIs include:

• •Enhancing health care resources: The mortality rate can be reduced and the recovery rate increased by enhancing availability of the health care resources [12]; that is, the recovery rate ${\gamma }_{cr}^{(\cdot )}$ increases to ${\delta }_R{\gamma }_{cr}^{(\cdot )}$ and the mortality ${\gamma }_{cr}^{(\cdot )}$ reduces to ${\delta }_D{\gamma }_{cd}^{(\cdot )}$, where $1\le {\delta }_R$ and $0\le {\delta }_D\le 1$. Note that if we assume that unconfirmed and asymptomatic individuals would not receive additional medical help, then the mortality rate of unconfirmed and asymptomatic cases, including ${\gamma }_{ur}$, ${\gamma }_{ar}$ and ${\gamma }_{ud}$, would not be influenced by any NPIs or PIs;
• •Enhancing testing capacity: This can greatly influence the rate of detected asymptomatic cases and unconfirmed cases with clinical features (${\beta }_{ac}^{(\cdot )}$ and ${\beta }_{uc}^{(\cdot )}$). Countries with sufficient testing capacities can screen out asymptomatic cases and unreported cases quickly [31]. Then, enhancing the testing capacity means that ${\beta }_{ac}^{(\cdot )}$ and ${\beta }_{uc}^{(\cdot )}$ increase to ${\delta }_T{\beta }_{ac}^{(\cdot )}$ and ${\delta }_T{\beta }_{uc}^{(\cdot )}$, where $1\le {\delta }_T$. The confirmed cases can be isolated, and the spread of COVID-19 can be effectively contained.

Enhancing the ability to monitor new variants will shorten the time gap between the appearance of a new variant and the time it has been confirmed as a dangerous variant, which can help public health policymakers develop new vaccines in time [11]. In addition, a shortage of vaccine has become a difficult issue in fighting against COVID-19 [25]. Hence, it is believed that a sufficient vaccine supply can help contain the spread of COVID-19 [6]. Additionally, the acceptance of COVID-19 vaccines is critical for achieving adequate immunization coverage to end the COVID-19 pandemic [10]. These interventions for vaccine development and distribution are also considered in this work. Specifically, VIs include:

• •Enhancing the virus monitoring ability for developing vaccine: In our model, $I_T$ represents the monitoring ability. For instance, $I_T=20,000$ means that pharmaceutical factories could produce a new vaccine against a new variant when it has only infected 20,000 individuals. Then, we can reduce $I_T$ to simulate a strong virus monitoring ability;
• •Accelerating vaccine development: In our model, the parameter $T_{v,i}$ represents the vaccine development time. Accelerating vaccine development by shortening the development time $T_{v,i}$ can help to quickly distribute new vaccines against new dangerous variants. Then, we can reduce $T_{v,i}$ to simulate the effect of shortening the vaccine development time;
• •Acceptance of COVID-19 vaccines: Different areas have different acceptance rates ${\delta }_v$. Here, we analyze how the acceptance rate ${\delta }_v$ influences the spread of COVID-19.

The mutation rate is a vital factor that determines the number of variants. In this work, parameter $N_M$ is adopted to imitate the mutation rate. A small $N_M$ means a high mutation rate, while a large $N_M$ represents a low mutation rate. The more variants, the more difficult it is to contain the spread of the pandemic. The mutation rate is difficult for human beings to intervene in, as it is the natural property of a virus. In this work, however, we attempt to investigate optimistic scenarios that the mutation rate will become low and fewer variants will appear in the future.

Note that there are other factors (parameters) that are difficult for human beings to intervene in. So we keep these factors fixed. All the factors are classified and described in Fig. 5 . More detailed descriptions of interventions are given in the Appendix B.

In this work, a criterion for evaluating the harm of the pandemic is established. Here, we utilize the cumulative infected number between the 3551-st and 3650-th days with intervention (${\sum }_{t=3551}^{3650}{I}_{all}^{(it)}(t)$) over the cumulative infected number without intervention (${\sum }_{t=3551}^{3650}{I}_{all}^{(0)}(t)$) as a criterion to quantify the effectiveness of the intervention, in the form of

$$p_{it}=\frac{{\sum }_{t=3551}^{3650}{I}_{all}^{(it)}(t)}{{\sum }_{t=3551}^{3650}{I}_{all}^{(0)}(t)}.$$ (13)

Based on this criterion, we can separate the performances of these interventions into four classes: (1) interventions that have almost no effect on containing the spreading of the pandemic, with $0.9\le p_{it}$; (2) interventions that can only contain the pandemic with $0.4\le p_{it}\le 0.9$, that is, only slightly reducing the disease prevalence; (3) interventions that can greatly contain the pandemic, with $0.001\le p_{it}\le 0.4$, including the reduction of disease prevalence, morbidity or mortality to a locally acceptable level; (4) interventions that can eradicate the COVID-19 pandemic, with $p_{it}\approx 0$ $(p_{it}\le 0.001)$, that is, a permanent reduction to zero of the worldwide incidence of infection caused by a specific agent as a result of deliberate efforts. Intervention measures are no longer needed and the active infections finally become zero. In addition, we utilize the cumulative death toll (${\sum }_{t=1}^{3650}{D}_{all}^{(it)}(t)$) with intervention over the cumulative death toll (${\sum }_{t=1}^{3650}{D}_{all}^{(0)}(t)$) without intervention to evaluate the overall effectiveness of the intervention, in the following form:

$$p_{it}^D=\frac{{\sum }_{t=1}^{3650}{D}_{all}^{(it)}(t)}{{\sum }_{t=1}^{3650}{D}_{all}^{(0)}(t)}.$$ (14)

4. Experimental results

Considering the multi-directional mutation of SARS-CoV-2, vaccine is of great importance for understanding the dynamics of the spread of the pandemic and for implementing COVID-19 control and intervention. It is vital to precisely evaluate particular interventions to find out how to contain or even eliminate the virus. In the following, we investigate the effectiveness of NPIs, PIs and VIs.

4.1. Scenarios with single intervention

Non-pharmaceutical interventions (NPIs): The impact of non-pharmaceutical interventions, including restricting social contact, mandatory mask-wearing and quarantining confirmed cases, in reducing the burden of the COVID-19 pandemic is explored first. These measures generally include reducing on the transmission rate within the general community through reductions in the number of contacts with infectious individuals. In particular, we consider the possibility that restricting social contact is implemented only while allowing the reduction levels in the transmission rate to vary from 70 to 100%. Similarly, we analyze the case in which quarantine of confirmed cases is implemented. Our results show that restricting social contact is more effective than quarantining confirmed cases.

• •We investigate the role of restricting social distancing and mandatory mask-wearing for several pandemic scenarios. Our experimental results indicate that by implementing strict social restrictions for reducing ${\alpha }_e^{(\cdot )}$, ${\alpha }_u^{(\cdot )}$, and ${\alpha }_a^{(\cdot )}$ to $0.72{\alpha }_e^{(\cdot )}$, $0.72{\alpha }_u^{(\cdot )}$, and $0.72{\alpha }_a^{(\cdot )}$ can almost totally eradicate COVID-19 with $p_{it}=0.00036$, resulting in up to 94.53% reduction of deaths ($p_{it}^D=0.05662$) in comparison with the baseline case (shown in Fig. 6 (a) and Table 3 ). Our results also indicate that the 28% reduction in contact with infectious individuals led to a more than 90% decrease in mortality and could almost eradicate the pandemic. Additionally, only a 10% reduction in ${\alpha }_e^{(\cdot )}$, ${\alpha }_u^{(\cdot )}$, and ${\alpha }_a^{(\cdot )}$ results in a dramatic decrease in infections of the pandemic. Hence, even under the scenario with multiple variants, strict social restriction is an effective measure for containing the spread of the virus spreading and even eradicating the pandemic;
• •Enhanced quarantine measures can influence the contact rate between susceptible and confirmed patients ${\alpha }_c^{(\cdot )}$. Note that ${\alpha }_c^{(\cdot )}=0$ means implementing the strictest quarantine measure so that no confirmed case can infect other individuals. Experimental results indicate that a 100% reduction in contacts between confirmed cases and susceptible individuals (${\alpha }_c^{(\cdot )}=0$) results in up to about 85% reduction of daily cases during days 3551 to 3650 ($p_{it}=0.38394$). However, there are still about 0.0016 million new cases every day. Hence, only strictly quarantining confirmed cases cannot eradicate the COVID-19 pandemic but can greatly contain the spread (shown in Fig. 6(b)). In this scenario, the infected cases and deaths are reduced dramatically, resulting in up to 71.937% reduction of deaths, shown in Table 3). Note that, in our experiment, the total population is 2 billion, which means that only an average of 0.0459% of the population is quarantined per day from the 3551-st to the 3650-th days.

The results summarized in Table 3 are further illustrated graphically in Fig. 6(b), which clearly shows that by only reducing contact between confirmed cases and susceptible, almost 100% reduction is needed to reduce the morbidity and mortality significantly.

Fig. 6.

The effect of implementing NPIs in 500 days after the detection of the first COVID-19 case. Here, the intervention time is marked by a red vertical dashed line, and the mean (solid line) and 95% confidence interval (shaded) of daily infections are given: (a) Enhancing social restriction to reduce the transmissibility to 90%, 80% and 70% of the original transmissibility; (b) Improving quarantine intensity to reduce ${\alpha }_c^j$ to $0.8{\alpha }_c^j$, $0.4{\alpha }_c^j$ and 0.

Table 3.

The influence of factors.

Single intervention	Intervention scenario	Almost no effect	Contain	Greatly contain	Eradicate	Criterion pit	Criterion $p_{it}^D$
Non-pharmaceutical interventions
(1) Restrict social contact	$0.72{\alpha }_e^{(\cdot )}$,$0.72{\alpha }_u^{(\cdot )}$,$0.72{\alpha }_a^{(\cdot )}$				✓	0.00036	0.05662
(2) Strict Quarantine of confirmed cases	$0{\alpha }_c^{(\cdot )}$			✓		0.38394	0.29073
Pharmaceutical interventions
(3) Increased recovery rate	$9{\gamma }_{cr}^{(\cdot )}$		✓			0.61740	0.15102
(4) Reduce mortality rate	$0.1{\gamma }_{cd}^{(\cdot )}$	✓				0.95451	0.29633
(5) Enhance testing capacity	$9{\beta }_{ac}^{(\cdot )},9{\beta }_{uc}^{(\cdot )}$		✓			0.51148	0.32556
Vaccines
(6) Enhance vaccine acceptance rate	γv = 1				✓	0.00067	0.08809
(7) Accelerating vaccine development	TV,i = 30 days	✓				1.00000	1.00000
(8) Enhance virus monitoring	IT = 10,000	✓				1.00000	0.98548
Natural process
(9) Slowed virus mutation rate	100NM			✓		0.22690	0.26870
Combination of intervention strategies
(10) Combination strategy 1: (1) + (2) + (5)	$0.8{\alpha }_e^{(\cdot )}$,$0.8{\alpha }_u^{(\cdot )}$,$0.8{\alpha }_a^{(\cdot )}$;				✓	0.00006	0.05035
	$0.7{\alpha }_c^{(\cdot )};3{\beta }_{ac}^{(\cdot )},3{\beta }_{uc}^{(\cdot )}$
(11) Combination strategy 2: (2) + (3) + (5)	$0{\alpha }_c^{(\cdot )};9{\gamma }_{cr}^{(\cdot )}$;			✓		0.00243	0.02343
	$9{\beta }_{ac}^{(\cdot )},9{\beta }_{uc}^{(\cdot )}$
(12) Combination strategy 3: (5) + (6)	$3{\beta }_{ac}^{(\cdot )},3{\beta }_{uc}^{(\cdot )};{\gamma }_v=0.95$				✓	0.00061	0.07008

Note: Almost no effect: $0.9\le p_{it}$; Contain: $0.4\le p_{it}\le 0.9$; Greatly contain: $0.001\le p_{it}\le 0.4$; Eradicate: $p_{it}\approx 0$$(p_{it}\le 0.001)$.

Pharmaceutical/medical interventions (PIs): Here, we assume that medical resources can be enhanced for fighting against the COVID-19 pandemic, such as massive PCR tests, and antivirals can be administered therapeutically and prophylactically. Our study assumes several scenarios with different antiviral efficacy and testing capacities.

• •First, we assume that the recovery rate of confirmed individuals ${\gamma }_{cr}^{(\cdot )}$ can be increased by improving the availability of local health care resources. We also assume that the recovery rate of unconfirmed cases is not influenced by the health care system, as this group of patients has less possibility of seeking medical help from health professionals. Here, several pandemic scenarios (corresponding to $\delta =1$, 3, 6 and 9) are considered (shown in Fig. 7 (a)). Our experimental results show that improving the health care system by increasing the recovery rate nine times can contain the spread, resulting in up to 38.260% reduction of daily new cases ($p_{it}=0.61740$) during the 3551-st to 3650-th days in comparison with the baseline scenario. Additionally, the number of deaths is reduced by 84.908% (corresponding to $p_{it}^D=0.15102$, shown in Table 3).
• •With the improvement of local health care resources, the mortality rate of confirmed individuals (${\gamma }_{cd}^{(\cdot )}$) can also be reduced. Here, we consider several scenarios with δ=1, 0.9, 0.5 and 0.1. Fig. 7(b) shows the experimental results. The reduction of the mortality rate has almost no effect in containing the spread of COVID-19 with $p_{it}=0.95451$; that is, the number of infected cases only reduces by less than 5% from days 3551 to 3650 (as given in Table 3). However, the number of deaths decreases significantly to about 70% of the baseline scenario ($p_{it}^D=0.29633$).
• •Similarly, if COVID-19 PCR kits are sufficient to cover a large percentage of the population, the testing capacity can be enhanced, resulting in quickly screening out more unconfirmed patients. Hence, the rate at which unconfirmed cases transit into confirmed cases (${\beta }_{ac}^{(\cdot )}$ and ${\beta }_{uc}^{(\cdot )}$) increases. Here, we investigate several scenarios, corresponding to increasing the testing capacity 3, 6, and 9 times (shown in Fig. 7(c)). Our experimental results indicate that the pandemic can be contained with sufficient testing capacity by increasing ${\beta }_{ac}^{(\cdot )}$ and ${\beta }_{uc}^{(\cdot )}$. Additionally, we found that the pandemic situation under the scenarios with 6 and 9 times testing capacity is similar. This phenomenon may indicate that there exists a threshold transmission rate, above which the pandemic situation will not be changed by increasing the transmission rate. Our experimental results also indicate that it is not possible to totally eradicate the pandemic by only increasing the testing capacity. However, enhancing the testing capacity can contain the spread of the COVID-19 pandemic, with $p_{it}=0.51148$ and $p_{it}^D=0.32556$.

In conclusion, our experimental results suggest that enhancing the testing capacity and increasing the recovery rate is always more effective than reducing the mortality rate in mitigating a future pandemic. However, a sufficient number of COVID-19 PCR kits and medical resources must be available and well distributed. Note that we assume that the recovery rate will increase by 9 times. This condition may be difficult to satisfy as the probability of developing effective antiviral therapies for COVID-19 in a short time is low. However, enhancing the testing capacity can achieve better performance than increasing the recovery rate. Hence, our results suggest that enhancing the testing capacity may be a pragmatic optimum since it is easy to provide a sufficient number of COVID-19 PCR kits to the community.

Fig. 7.

The effect of implementing PIs in 500 days after detecting the first COVID-19 case. Here, the intervention time is marked by a red vertical dashed line, and the mean (solid line) and 95% confidence interval (shaded) of daily infections are given: (a) The effect of improving the health care system to increase the recovery rate of confirmed cases to 3 times, 6 times and 9 times; (b) The effect of improving the health care system to reduce the mortality rate of confirmed cases by 10%, 50% and 90%; (c) Enhance the testing capacity by 3 times, 6 times, and 9 times; (d) represents the relationships between variants. Each node represents a unique variant, and the lines between the two nodes represent that one variant is muted from the other one. The presence of several nodes with large degrees suggests that this variant mutes many times and has a lot of “offsprings”.

Vaccine-only intervention: Here, we investigate vaccine-only intervention by assuming that a suitable vaccine is available at some time after the appearance of a new variant. We evaluate three kinds of scenarios, including scenarios with different vaccine development periods $T_{V,i}$, scenarios with different virus monitoring ability $I_T$ and scenarios with different acceptance rates ${\gamma }_v$.

• •We can enhance the virus monitoring ability and start to develop a vaccine for the new variant even when the number of confirmed cases $I_T$ is still low. Here, we consider scenarios with $I_T$=150,000, 100,000, and 50,000; that is, a vaccine starts to be developed for a new variant when the variant has only infected 150,000, 100,000, and 50,000 individuals (shown in Fig. 8 (a)). However, our experimental results indicate that this strategy had almost no effect on containing the spread even the threshold is set to 10,000 individuals (shown in Table 3). Hence, these results indicate that even if new viruses can be identified quickly and new vaccines can be developed quickly, the spread of the COVID-19 pandemic still cannot be contained. The reason is that the speed of vaccine development cannot keep up with the speed of virus mutation.
• •We assume that the vaccine development time $T_{V,i}$ can be shortened from 120 days to 90, 60, and 30 days against a new variant. Our experimental results indicate that even shortening the vaccine development time to 30 days has almost no effect on containing the pandemic, with $p_{it}\approx 1$ and $p_{it}^D\approx 1$ (shown in Fig. 8(b) and Table 3).
• •We consider the rate of acceptance of vaccine ${\delta }_v$ as a factor influencing the spread of COVID-19 and investigate scenarios with ${\delta }_v=75$, 80, 90 and 100%, respectively. Our experimental results indicate that the spread of COVID-19 can be greatly suppressed with an increase of the vaccine acceptance rate (shown in Fig. 8(c)). Additionally, in the scenario with a 100% acceptance of the vaccine (100% of the population is willing to be vaccinated), the pandemic can be eradicated, with $p_{it}=0.00066$, resulting in up to 91.191% reduction of deaths (given in Table 3).

Overall, we have explored the potential role of vaccine-only intervention, and our experimental results indicate that increasing the vaccine acceptance rate can be highly effective. However, enhancing the virus monitoring ability and shortening the vaccine development time have almost no effect on containing the pandemic, which is a counterintuitive phenomenon. The reason is that the vaccination acceptance rate has a limit. Thus, if the vaccine acceptance rate is low, a large number of individuals are unwilling to be vaccinated. Therefore, vaccination coverage is still low even if the vaccine supply is timely and sufficient. Hence, the acceptance rate of vaccines is a vital factor in controlling the pandemic.

Fig. 8.

The effect of implementing vaccine interventions in 500 days after the detection of the first COVID-19 case. Here, the intervention time is marked by a red vertical dashed line, and the mean (solid line) and 95% confidence interval (shaded) of daily infections are given: (a) The effect of enhancing the virus monitoring capacity and human society's beginning to develop a new vaccine for a new variant if the total infections are larger than 0.75I_T, 0.5I_T, and 0.25I_T; (b) The effect of shortening the vaccine development time is demonstrated by reducing it to 90 days and 30 days; (c) The effect of changing the vaccine acceptance rate to δ_v = 0.8, 0.9, and 1.

Additionally, we have investigated the impact of the mutation rate by considering several scenarios with $N_M=$5, 50, and 500 millions, respectively (shown in Fig. 9 (a)). Our experimental results indicate that, even if the mutation threshold increases to 500,000,000; that is, if the virus has to infect more than 500,000,000 individuals to trigger mutations to generate a new variant, the COVID-19 pandemic can not be eradicated although it can be greatly contained, with $p_{it}=0.22690$ and $p_{it}^D=0.26870$ (given in Table 3). Simulation results indicate that in 10 years, on average, there are 1708 variants. Hence, even if the mutation rate has slowed down, the pandemic can still be a great threat to human communities (shown in Fig. 9(a)). Fig. 7(d) shows the relationships between variants in one experiment, in which each node represents one variant. Fig. 9(b) shows the number of variants and daily infected cases in the baseline situation (without any interventions). Obviously, the number of daily infections rises sharply between the 200-th and 500-th day and reaches its peak around the middle of the second year. After that, there is a rapid decline and a trough period around the beginning of the third year. However, on the 1000th day, the number of daily infected cases rises slowly again and the rate of growth gradually increase, showing an unstoppable upward trend. On the other hand, the number of variants begins to increase gradually after the 300-th day, and the rate increases slowly. After the 1500-th day, the number of variants shows a sharp increase. The experimental results indicate that under the scenario considering multi-directional mutation and the existence of multiple variants, there would be a large-scale rapid outbreak at the beginning of transmission. Then the spread of the pandemic would quickly slow down. However, due to the mutation of the virus, the epidemic will repeatedly outbreak, causing multiple waves. Without effective interventions, new waves will erupt again and again, resulting in an unstoppable transmission.

Fig. 9.

(a) The effect of reducing mutation rate. Here, we assume the mutation rate is slowing down in 500 days after the detection of the first COVID-19 case (marked by a red vertical dashed line). The mean (solid line) and 95% confidence interval (shaded) of daily infections are also given. Here, the mutation threshold increases from N_M = 5,000,000 to 50N_M, 75N_M and 100N_M; (b) represents the number of mutations and the number of daily infection cases in the baseline situation.

In conclusion, the experimental results for single interventions show that strictly restricting social contact and enhancing the vaccine acceptance rate yield the most optimal intervention for eradicating the COVID-19 pandemic. Several intervention strategies for reducing the mortality rate, accelerating vaccine development, and enhancing virus monitoring have almost no influence on containing the spread of COVID-19. The other interventions can contain or greatly suppress the spread of the pandemic. The results also indicate that the human society should not slacken when the outbreak declines, but be more vigilant to prevent another resurgence.

4.2. Scenarios with combined interventions

Local authorities always adopt multiple interventions at the same time to contain the spread of the pandemic. One of the reasons is that it is difficult to achieve the optimal effect by a single intervention. For instance, enhancing the vaccine acceptance rate to 100% can eradicate the pandemic. However, in the real world, this is impossible to achieve. Another example is that strictly restricting social contact can also eradicate the pandemic. However, strictly restricting social contact has certain economic and social impacts, so almost no country can restrict social contact for a long time. Some studies have found that a combined intervention is more effective than a single intervention in containing the spread of flu [17]. Additionally, human society cannot always maintain a single intervention at a high intensity when fighting a pandemic due to social and economic factors. Hence, an appropriate combination of multiple intervention strategies is one of the effective means to reduce pandemic outbreaks while minimizing social disruption. Here, we investigate the effect of combined interventions.

• •Combined intervention strategy 1: We begin our study by assessing the potential impact of the combined use of strictly restricted social contact, increasing the recovery rate and enhancing the testing capacity. We evaluate several scenarios with different levels of these three interventions (shown in Fig. 10 (a)). In this scenario, we only have to (1) reduce the transmission rate to 80% of the original rate, that is, ${\alpha }_e^{(\cdot )}=0.8{\alpha }_e^{(\cdot )}$, ${\alpha }_a^{(\cdot )}=0.8{\alpha }_a^{(\cdot )}$ and ${\alpha }_u^{(\cdot )}=0.8{\alpha }_u^{(\cdot )}$; (2) impose strict quarantine on confirmed cases with ${\alpha }_c^{(\cdot )}=0.7{\alpha }_c^{(\cdot )}$; (3) enhance the testing capacity to increase ${\beta }_{uc}^{(\cdot )}$ and ${\beta }_{ac}^{(\cdot )}$ to $3{\beta }_{uc}^{(\cdot )}$ and $3{\beta }_{ac}^{(\cdot )}$. The results, shown in Table 3, indicate that the pandemic can be eradicated with $p_{it}=0.00006$ and a 94.965% reduction in deaths. Hence, the combined intervention strategy is more effective than the scenario with a single intervention strategy.
• •Combined intervention strategy 2: This strategy includes strict quarantine of confirmed cases, increasing the recovery rate, and enhancing the testing capacity. In this scenario, we only (1) increase the strict quarantine of confirmed cases to reduce ${\alpha }_c^{(\cdot )}$ to $0{\alpha }_c^{(\cdot )}$; (2) increase the recovery rate from ${\gamma }_{cr}^{(\cdot )}$ to $9{\gamma }_{cr}^{(\cdot )}$; (3) enhance the testing capacity to increase ${\beta }_{uc}^{(\cdot )}$ and ${\beta }_{ac}^{(\cdot )}$ to $9{\beta }_{uc}^{(\cdot )}$ and $9{\beta }_{ac}^{(\cdot )}$. Then, the pandemic can be greatly contained, with $p_{it}=0.00243$ and $p_{it}^D=0.02343$. Although this combined strategy cannot eradicate the pandemic, the pandemic can be greatly contained;
• •Combined intervention strategy 3: We further consider a combined intervention strategy with an enhanced testing capacity and vaccine acceptance rate. Several scenarios with different levels of these two interventions (shown in Fig. 10(c)) are evaluated. The results show that enhancing the testing capacity by 3 times and increasing the vaccine acceptance rate to 95% can eradicate the pandemic, with $p_{it}=0.00061$.

To sum up, the combined interventions are highly effective in containing or even eradicating the COVID-19 pandemic. However, several requirements are still difficult to meet. Nevertheless, we may encourage more than 75% of the population to accept vaccines. Our study shows that restricting social contact to reduce transmission rates is effective and should be explored as a realistic and cost-effective alternative. In fact, we find that the effective implementation of transmission control alone may significantly contain the spread of the pandemic. Combined intervention strategies are always more effective under a more relaxed scenario.

Fig. 10.

The effect of implementing combination intervention strategies in 500 days after the detection of the first COVID-19 case. A red vertical dashed line marks the beginning time, and the mean (solid line) and 95% confidence interval (shaded) of daily infections are given. (a) combined intervention strategy including social restriction, quarantine of confirmed cases, and enhancing testing capacity; (b) combined intervention strategy including quarantine of confirmed cases, increase recovery rate and enhancing testing capacity; (c) combined intervention strategy including enhancing testing capacity and enhancing vaccine acceptance rate.

5. Discussion and conclusion

Effective and benign coordination among government organizations, public health agencies, and other stakeholders is necessary to respond to the COVID-19 pandemic. At the same time, there are many PIs, NPIs, and VIs to reduce the risk of epidemic transmission as well as to achieve universal herd immunity. In most previous computational epidemic modeling studies, only one strain is assumed to be present in the environment when modeling the epidemiological dynamics of disease transmission, with no mutations occurring in that strain. However, SARS-CoV-2 has shown mutational properties and there were more than 1,600 variants detected worldwide by April 19, 2022. Therefore, the traditional epidemiological transmission framework of a single strain is no longer adequate to describe the dynamic transmission of COVID-19, resulting in unreliable results for evaluating the effectiveness of interventions in COVID-19 transmission. In this paper, to address this issue, we propose an epidemic dynamics transmission framework by considering the co-existence of multiple variants with multi-directional mutations for describing epidemic transmission taking into consideration of different single and combined intervention strategies.

Seven main interventions were considered: restricting social contact, quarantining confirmed cases, enhancing medical resources, enhancing the testing capacity, increasing the vaccine acceptance rate, accelerating vaccine development, and enhancing virus monitoring. Furthermore, the impact of the mutation rate on transmission was re-explored. The experimental results show that in scenarios where a single intervention is used, restricting social contact and increasing the vaccine acceptance rates are the most effective among the seven interventions. However, shortening the vaccine development time and enhancing virus monitoring, which accelerates the development of new vaccines, have almost no impact on curbing the spread of COVID-19. In terms of public health resources, not all countries have a well-established system of medical resources to implement PIs and VIs. Based on this observation, our experimental results suggest that restricting social contact should be an optimal intervention measure in areas where medical resources and vaccines are limited. Additionally, even if a region has a well-established health care system, it may face problems in balancing social and economic factors with the effectiveness of epidemic prevention, resulting in a single intervention that cannot cope with the spread of epidemics influenced by complex factors.

We show that combined intervention strategies always give the most desirable results and achieve better performance than single interventions. Our results indicate that the vaccine acceptance rate and restricted social contact are key components of a combined intervention strategy. In particular, combined intervention strategies can enhance the effectiveness of interventions while reducing the burden of pandemic events on public health resources, thus improving the feasibility and effectiveness of epidemic intervention strategy implementation. In addition, we find that not all combined interventions can eliminate pandemics.

The limitations of our study lie in the following aspects:

• •We assume that each vaccine has 100% effectiveness against one variant. However, practically this is not 100% in the real world. Additionally, there exist many vaccines against one variant and these vaccines have different impacts;
• •We assume that in different regions, one virus would have the same transmission characteristics. However, each region has its own specific social, economic and environmental conditions, which may influence the transmission characteristics of the virus [35]. Hence, even one virus may have different transmission characteristics in different regions;
• •We assume that the intensity of intervention is constant after the onset of implementation. However, the local authorities would dynamically adapt the intensity of interventions to the real pandemic situation;
• •We assume that all people are homogeneous and do not consider the impact of social relationships between people on the spread of epidemics [38]. When making group decisions in communities, considering networking factors can help curb the transmission of epidemics [37]. Therefore, epidemiological models should be modeled with network factors in the future.

In the future, it would be interesting to extend the proposed framework so as to address these limitations more effectively.

CRediT authorship contribution statement

Choujun Zhan and Haijun Zhang conceived of the presented idea and planned the experiments. Choujun Zhan carried out the experiments and took the lead in writing the manuscript with support from Yufan Zheng, Lujiao Shao, Guanrong Chen and Haijun Zhang. All authors discussed the results and helped shape the research, analysis and manuscript by providing critical feedback.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Susceptible-Exposed-Unreported-Asymptomatic-Confirmed-Recovered-Deceased with Vaccination Population (SEUACRD-VP) model

The epidemiological characteristics of COVID-19 are far from the normal epidemic diseases and have some typical features. There exists a larger group of unregistered COVID-19 patients. Pre-symptomatic, asymptomatic and undiagnosed COVID-19 patients form a large group of unconfirmed infected cases, who can travel from one area to another and spread the virus freely. The model should therefore take into account the relevant presence of pre-symptomatic, asymptomatic, and unreported COVID-19 patients. According to the known epidemiological characteristics of SARS-CoV-2, individuals in a population can be roughly classified into eleven classes.

• •Susceptible ($S^{(i)}$): The susceptible individual is vulnerable to SARS-CoV-2 but has not been infected by the disease;
• •Exposed (or pre-asymptomatic, $E^{(i)}$): The mean incubation period of the original COVID-19 was about one week, with the 95th percentile of the distribution being about two weeks. In the incubation period, the exposed (or pre-asymptomatic) individuals have no typical clinical symptoms, such as fever, but have infective properties and can spread the SARS-CoV-2, which is different from many other epidemic diseases;
• •Asymptomatic COVID-19 infectious ($A^{(i)}$): After the incubation period, some exposed individuals become asymptomatic COVID-19 patients, who have no visible clinical signs or symptoms, and even no apparent abnormalities in lung computed tomography (CT), just like exposed individuals. Hence, there is a low probability of an asymptomatic patient taking reverse transcriptase-polymerase chain reaction (RT-PCR) tests and being documented by the authorities. However, asymptomatic patients are still infectious. For simplicity, we assume that the detection rate of asymptomatic patients is close to zero;
• •Unreported infectious with clinical symptoms ($U^{(i)}$): After the incubation period, some COVID-19 patients begin to develop clinical symptoms and can also infect other susceptible individuals. These infectious individuals with clinical symptoms have a high probability of taking PCR tests but have not yet taken PCR tests. However, due to limitations of the testing capacity and other factors, only some but not all of the infectious individuals with clinical symptoms would be screened out before they recover or pass away.
• •Confirmed cases with clinical symptoms ($C^{(i)}$): After the incubation period, some of the infectious individuals with clinical symptoms would be laboratory confirmed to be positive for SARS-CoV-2 through PCR tests, who are confirmed as COVID-19 patients and reported by local authorities. The confirmed cases would be in self-quarantine at home, or be centrally quarantined, or be hospitalized. However, these isolated infectious individuals still have a possibility to infect other susceptible individuals and the level of quarantine measures influences the infection rate of confirmed cases;
• •Recovered asymptomatic infectious ($R_a^{(i)}$): As asymptomatic patients with no visible symptoms, we assume that most of the asymptomatic patients finally recover and no one dies due to COVID-19. Hence, for the sake of simplicity, all the asymptomatic patients are assumed recover and return to normal;
• •Recovered unreported infectious ($R_u^{(i)}$): Some unreported COVID-19 infectious individuals with symptoms would recover;
• •Deceased unreported infectious ($D_u$): Some unreported COVID-19 patients with symptoms would not survive the disease;
• •Recovered confirmed cases ($R_c^{(i)}$): Most confirmed cases would take medical treatment and then recover;
• •Deceased confirmed cases ($D_c^{(i)}$): A few confirmed cases could not survive the disease and passed away.
• •Vaccinated individuals ($V^{(i)}$): Individuals who have been vaccinated.

We represent the numbers of individuals at time t in the above eleven classes by $S^{(i)}(t)$, $E^{(i)}(t)$, $A^{(i)}(t)$, $U^{(i)}(t)$, $C^{(i)}(t)$, $R_A^{(i)}(t)$, $R_U^{(i)}(t)$, $D_U^{(i)}(t)$, $R_C^{(i)}(t)$, $D_C^{(i)}(t)$ and $V^{(i)}(t)$, respectively.

An individual can cycle through the eleven classes based on different state transition probabilities (shown in Fig. 1). Then, a generalized SEUACRD-VP model considering the vaccinated population is proposed, denoted as the Susceptible-Exposed-Unreported-Asymptomatic-Confirmed-Recovered-Deceased Vaccination-Population (SEUACRD-VP) model (shown in Eq. (A.16a), (A.16b), (A.16c), (A.16d), (A.16e), (A.16f), (A.16g), (A.16h), (A.16i), (A.16j), (A.16k), (A.16l), and more detailed information of this model can be found in [33]). In this model, $t_{v,0}$ is the initial time when the first vaccine dose is given. Here, $t_{v,0}>t_0$, where $t_0$ represents the detection time of the first infected individual.

$$S^{(i)}(t+1)=S^{(i)}(t)-\frac{({\alpha }_e^{(i)}{E}^{(i)}(t)+{\alpha }_u^{(i)}{U}^{(i)}(t)+{\alpha }_c^{(i)}{C}^{(i)}(t)+{\alpha }_a^{(i)}{A}^{(i)}(t))S^{(i)}(t)}{N_p^{(i)}(t)}+{\gamma }_{rs}^{(i)}(R_c^{(i)}(t)+R_u^{(i)}(t)+R_a^{(i)}(t))+{\gamma }_{vs}^{(i)}{V}^{(i)}(t)-n_V^{(i)}(t),$$ (A.16a)

$$E^{(i)}(t+1)=E^{(i)}(t)+\frac{({\alpha }_e^{(i)}{E}^{(i)}(t)+{\alpha }_u^{(i)}{U}^{(i)}(t)+{\alpha }_c^{(i)}{C}^{(i)}(t)+{\alpha }_a{A}^{(i)}(t))S^{(i)}(t)}{N_p^{(i)}(t)}-{\beta }_e^{(i)}{E}^{(i)}(t),$$ (A.16b)

$$A^{(i)}(t+1)=A^{(i)}(t)+k_A{\beta }_e^{(i)}{E}^{(i)}(t)-{\beta }_{ac}^{(i)}{A}^{(i)}(t)-{\gamma }_{ar}^{(i)}{A}^{(i)}(t),$$ (A.16c)

$$U^{(i)}(t+1)=U^{(i)}(t)+(1-k_A){\beta }_e^{(i)}{E}^{(i)}(t)-{\beta }_{uc}^{(i)}{U}^{(i)}(t)-{\gamma }_{ur}^{(i)}{U}^{(i)}(t)-{\gamma }_{ud}^{(i)}{U}^{(i)}(t),$$ (A.16d)

$$C^{(i)}(t+1)=C^{(i)}(t)+{\beta }_{uc}^{(i)}{U}^{(i)}(t)+{\beta }_{ac}^{(i)}{A}^{(i)}(t)-{\gamma }_{cr}^{(i)}{C}^{(i)}(t)-{\gamma }_{cd}^{(i)}{C}^{(i)}(t),$$ (A.16e)

$$R_a^{(i)}(t+1)=R_a^{(i)}(t)+{\gamma }_{ar}^{(i)}{A}^{(i)}(t)-{\gamma }_{rs}^{(i)}{R}_a^{(i)}(t),$$ (A.16f)

$$R_u^{(i)}(t+1)=R_u^{(i)}(t)+{\gamma }_{ur}^{(i)}{U}^{(i)}(t)-{\gamma }_{rs}^{(i)}{R}_u^{(i)}(t),$$ (A.16g)

$$D_u^{(i)}(t+1)=D_u^{(i)}(t)+{\gamma }_{ud}^{(i)}{U}^{(i)}(t),$$ (A.16h)

$$R_c^{(i)}(t+1)=R_c^{(i)}(t)+{\gamma }_{cr}^{(i)}{C}^{(i)}(t)-{\gamma }_{rs}^{(i)}{R}_c^{(i)}(t),$$ (A.16i)

$$D_c^{(i)}(t+1)=D_c^{(i)}(t)+{\gamma }_{cd}^{(i)}{C}^{(i)}(t),$$ (A.16j)

$$n_V^{(i)}(t)=\min \{n_{V,Thr},f_v^s(t)\},$$ (A.16k)

$$N_p^{(i)}(t+1)=(1+{\gamma }_p)N_p^{(i)}(t)-{\gamma }_{ud}^{(i)}{U}^{(i)}(t)-{\gamma }_{cd}^{(i)}{C}^{(i)}(t).$$ (A.16l)

Note that a recovered individual would acquire immunity against a particular virus. However, this immunity would be lost as time elapsed. Here, we assume that recovered individuals would transit into susceptible individuals with a rate of ${\gamma }_{vs}$. Hence, by considering vaccination, the number of susceptible individuals is represented by Eq. (A.16a). Note that recovered individuals would lose their immunity to a virus over a period of time. Here, we assume that the transition rate from recovered cases to susceptible cases is ${\gamma }_{rs}$. Then, at time t, there would be ${\gamma }_{rs}(R_c^{(i)}(t)+R_u^{(i)}(t)+R_a^{(i)}(t))$ recovered individuals who return to the susceptible state (shown in Eq. (A.16a), (A.16f), (A.16g), and (A.16i)).

The term ${\beta }_e{E}^{(i)}(t)$ stands for the rate at which an exposed individual finishes the incubation period and transits into the next states (in Eq. (A.16b)), including asymptomatic (A) and unreported (U), respectively. Some exposed individuals ($k_A{E}^{(i)}(t)$) would transit into asymptomatic cases without obvious clinical symptoms (in Eq. (A.16c)), while the others ($(1-k_A{E}^{(i)}(t))$) would transit into unreported cases with obvious clinical symptoms (in Eq. (A.16d)). Note that in the real scenario, asymptomatic infections do not have obvious clinical symptoms, resulting in a low probability of accessing medical help for COVID-19 and being difficult to diagnose. Hence, it is reasonable to assume that the number of confirmed asymptomatic cases (${\beta }_{ac}{A}_a^{(i)}$) is close to zero (in Eq. (A.16c) and (A.16e)), that is, ${\beta }_{ac}{A}_a^{(i)}\approx 0$ and ${\beta }_{ac}\approx 0$. Here, the incidence rate ${\beta }_{uc}{U}^{(i)}(t)$ stands for the transmission from unreported cases to confirmed cases (in Eq. (A.16d) and (A.16e)). The incidence ${\gamma }_{rs}(R_c^{(i)}(t)+R_u^{(i)}(t)+R_a^{(i)}(t))$ represents the rate of recovered cases that have lost the immunity and can be infected again by the same disease (in Eq. (A.16a), (A.16f), (A.16g), and (A.16i)). Here, we also consider the influence of the vaccine. We assume that on day t, $n_V^{(i)}(t)$ susceptible individuals would be vaccinated, where $f_v^s(t)$ denotes the vaccination power, and it cannot exceed the maximum threshold for daily vaccine production $n_{V,Thr}$ (in Eq. (A.16k)). Additionally, we consider the increasing rate of the population ${\gamma }_p$ (in Eq. (A.16l)).

Appendix B. Interventions of the COVID-19 based on SEUACRD-VM

We define some interventions in the mathematical model. We set $M(t)$ as the number of variants at time t and $T^{(c)}$ as the intervention or changing time, which means that we begin implementing the intervention at $T^{(c)}$. Moreover, we set ${\delta }_S$, ${\delta }_Q$, ${\delta }_R$, ${\delta }_M$, ${\delta }_T$ and ${\delta }_I$ as the intensity of different interventions, and set $t_0^{(i)}$ as the detection time of the first infected individual of variant-i. We assume that each intervention will affect all existing variants. When the intervention is performed at time $T^{(c)}$, it affects $M(t)$ variants present at that time, $0⩽i<M(t)$. Then, we consider some interventions as follows.

B.1. Restricting social contact

$${\beta }_{(\cdot )}^{(i)}(t)=\{\begin{array}{cc}{\delta }_S{\beta }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & t⩾T^{(c)}\hfill \\ {\beta }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & 0<t<T^{(c)}\hfill \end{array}$$ (B.17)

where $(\cdot )=\{e,a,u\}$.

B.2. Quarantine of confirmed cases

$${\beta }_c^{(i)}(t)=\{\begin{array}{cc}{\delta }_Q{\beta }_c^{(i)}\left(t_0^{(i)}\right),\hfill & t⩾T^{(c)}\hfill \\ {\beta }_c^{(i)}\left(t_0^{(i)}\right),\hfill & 0<t<T^{(c)}\hfill \end{array}$$ (B.18)

B.3. Enhanced medical resources for increasing recovery rate

$${\gamma }_{(\cdot )}^{(i)}(t)=\{\begin{array}{cc}{\delta }_R{\gamma }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & t⩾T^{(c)}\hfill \\ {\gamma }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & 0<t<T^{(c)}\hfill \end{array}$$ (B.19)

where $(\cdot )=\{cr,ar,ur\}$.

B.4. Enhanced medical resources for reducing mortality rate

$${\gamma }_{(\cdot )}^{(i)}(t)=\{\begin{array}{cc}{\delta }_M{\gamma }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & t⩾T^{(c)}\hfill \\ {\gamma }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & 0<t<T^{(c)}\hfill \end{array}$$ (B.20)

where $(\cdot )=\{cd,ud\}$.

B.5. Enhancing the testing capacity

$${\beta }_{(\cdot )}^{(i)}(t)=\{\begin{array}{cc}{\delta }_T{\beta }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & t⩾T^{(c)}\hfill \\ {\beta }_{(\cdot )}^{(i)}\left(t_0^{(i)}\right),\hfill & 0<t<T^{(c)}\hfill \end{array}$$ (B.21)

where $(\cdot )=\{ac,uc\}$.

Data availability

The authors do not have permission to share data.