The impact of vaccination on the spread of COVID-19: Studying by a mathematical model.

The global spread of COVID-19 has not been effectively controlled, posing a huge threat to public health and the development of the global economy. Currently, a number of vaccines have been approved for use and vaccination campaigns have already started in several countries. This paper designs a mathematical model considering the impact of vaccination to study the spread dynamics of COVID-19. Some basic properties of the model are analyzed. The basic reproductive number ℜ 1 of the model is obtained, and the conditions for the existence of endemic equilibria are provided. There exist two endemic equilibria when ℜ 1 < 1 under certain conditions, which will lead to backward bifurcation. The stability of equilibria are analyzed, and the condition for the backward bifurcation is given. Due to the existence of backward bifurcation, even if ℜ 1 < 1 , COVID-19 may remain prevalent. Sensitivity analysis and simulations show that improving vaccine efficacy can control the spread of COVID-19 faster, while increasing the vaccination rate can reduce and postpone the peak of infection to a greater extent. However, in reality, the improvement of vaccine efficacy cannot be realized in a short time, and relying only on increasing the vaccination rate cannot quickly achieve the control of COVID-19. Therefore, relying only on vaccination may not completely and quickly control COVID-19. Some non-pharmaceutical interventions should continue to be enforced to combat the virus. According to the sensitivity analysis, we improve the model by including some non-pharmaceutical interventions. Combining the sensitivity analysis with the simulation of the improved model, we conclude that together with vaccination, reducing the contact rate of people and increasing the isolation rate of infected individuals will greatly reduce the number of infections and shorten the time of COVID-19 spread. The analysis and simulations in this paper can provide some useful suggestions for the prevention and control of COVID-19.

Introduction

“Coronavirus disease 2019” (COVID-19) [1], a pneumonia epidemic caused by a new type of coronavirus named “Severe Acute Respiratory Syndrome-related Coronavirus type 2” (SARS-CoV-2) [2], remains a serious global issue since its emergence at the end of 2019. For more than a year, COVID-19 has not been effectively controlled, and the numbers of new cases remain at some of the highest levels currently [3]. According to the report by World Health Organization (WHO), as of 21 June, 2021, the cumulative number of confirmed cases worldwide reached 177,108,695, and the cumulative number of deaths stood at 3,840,223 [4]. The COVID-19 pandemic poses a serious threat to public health and economic challenges across the world. Three variants of SARS-CoV-2, Alpha (VOC 202012/01), Beta (501Y.V2) and Gamma (P.1), were identified in the United Kingdom, South Africa and Brazil a few months ago, respectively. Their contagion rate has been confirmed to be higher, aggravating the spread of COVID-19 in multiple countries [5], [6]. Recently, India has suffered from a rapid surge of COVID-19 infection. From April 22 to May 16, there have been more than 300,000 new confirmed cases in a single day in India for 25 consecutive days [4]. The new variant Delta (B.1.617) [5], which has a so-called double mutation, is thought to be fueling India’s deadlier new wave of cases that has made it the world’s second worst-hit country, surpassing Brazil [7].

Since there is no specific medicine, in order to curb the spread of COVID-19, countries mainly adopted non-pharmaceutical intervention measures in the early stage, including contact-tracking, social-distancing, isolating and treating infected persons, lockdown, etc. However, these measures cause a lot of inconvenience to people’s lives and seriously hinder the development of economy. Therefore, in order to develop economy, many countries relaxed these interventions when the COVID-19 epidemic slowed down. As a result, the spread of COVID-19 has not been completely controlled. In order to thoroughly control the spread of COVID-19 and reduce the impact on economic development, people are looking forward to the development and use of effective vaccines. Vaccination is an effective means to control the spread of an epidemic. Through unremitting efforts of all parties, some vaccines have been approved for use, which brings hope to the complete control of the spread of COVID-19.

Can the COVID-19 epidemic be completely eliminated by vaccination? Here we analyze it by designing a mathematical model. Mathematical models are important tools for predicting and simulating the spread of epidemics, and can provide a theoretical basis for decision-makers to formulate various epidemic prevention measures. There are many mathematical models related to the spread of COVID-19, such as those in [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. There are also some studies on the impact of vaccination on the spread of COVID-19. In [19], the impact of a hypothetical imperfect vaccine on the control of COVID-19 in the United States is studied using a mathematical model. A new SIRV model is proposed to forecast and simulate the COVID-19 epidemic evolution under the effect of vaccination in [20]. In [21], a mathematical model is presented to study the impacts of drugs (vaccination with perfect efficacy) and non-drug prevention measures on the spread of COVID-19 in South Africa. A detailed agent-based model is proposed to study the impact of various prevention and control measures on the spread of COVID-19 in Luxembourg, including testing, contact tracing, lockdown, curfew and vaccination [22]. In [23], utilizing a modified SIR model and using relevant data of COVID-19 in Ontario, Canada, different vaccination strategies are simulated and compared, including without vaccination. The authors believe that non-pharmacological interventions should be continued in the early stage of vaccination and gradually relaxed. In [24], a mathematical model considering vaccination is established to analyze the spread of COVID-19. The authors point out that due to the slow COVID-19 vaccination rate, non-drug prevention and control measures still need to be enforced until a sufficient proportion of the population is vaccinated. There are also some studies discussing vaccination prioritization strategies, such as [25], [26], [27], [28]. In this paper, we present a SVAIRS (susceptible–vaccinated–asymptomatic–symptomatic–removed–susceptible) model to analyze whether the COVID-19 epidemic can be completely eliminated by vaccination alone. By analyzing the basic reproductive number of the model and the existence of the equilibriums, as well as the stability of the equilibria and backward bifurcation, we theoretically prove that relying on vaccination alone may not completely control the COVID-19 epidemic. It is still necessary to properly enforce some non-pharmaceutical intervention measures.

This paper is organized as follows. A mathematical spread model of COVID-19 considering only vaccination is presented in Section 2. The basic reproductive number is obtained and the existence conditions of the endemic equilibria are given in Section 3. In Section 4, the stability of the equilibria is analyzed and the condition for backward bifurcation of the model is given. Numerical simulation and sensitivity analysis are done in Section 5. The last Section concludes this paper.

Modeling

Vaccines are vital to the control of an epidemic. At present, a number of COVID-19 vaccines have been used worldwide. In order to study whether relying solely on vaccination can control the spread of COVID-19, we consider vaccination alone without other prevention and control measures. According to the classic deterministic mathematical spread model SVIR [29], and considering the spread characteristics of COVID-19, we divide the total population into five groups, as shown in Table 1. Here we do not consider the population in the incubation period. Since the incubation period of COVID-19 is short and people in the incubation period are also infectious, we merge the incubation period into the asymptomatic infection period. According to the actual situation, the following assumptions are made:

Table 1

Population classification.

Group	Symbol	Description
susceptible	S	People who do not have antibodies and are easily infected by COVID-19
vaccinated	V	People who have been vaccinated against COVID-19
asymptomatic	A	People who are infected but do not have any symptoms
symptomatic	I	People who have obvious symptoms after being infected
removed	R	People who have recovered from infection or died as a result of infection

Only susceptible individuals are vaccinated.

COVID-19 vaccines are imperfect; that is, some of the vaccinated individuals can become infected and infectious even though they have been vaccinated.

Asymptomatic individuals will experience symptoms or recover after a period of time.

Symptomatic individuals will either recover or die after a period of time.

Individuals who have recovered from COVID-19 infection will produce antibodies; however, after a period of time, the antibodies will weaken or disappear and the recovered people will become susceptible again.

New individuals, including birth and recruitment, are susceptible.

Based on the above assumptions, the transmission relationship of various groups of people is shown in Fig. 1, and the descriptions of parameters are shown in Table 2. The mathematical spread model of COVID-19 is as follows: with the initial condition , where and are the abbreviations of the state variables and , respectively, representing the number of various groups of people at time . is the total population.

Fig. 1

The state transformation process of individuals, where and .

Table 2

Descriptions of parameters.

Parameter	Description	Value	Source
α	Transmission rate of symptomatic individuals	0.8883	[13]
β	Correction factor for transmission rate of asymptomatic individuals	0.45	[13]
τ	Vaccination rate	0.01/day	Assume
1−ρ	Vaccine efficacy	0.8	Assume
1/δ	Average time of asymptomatic duration	7 days	[14]
θ	Proportion of asymptomatic individuals who develop to symptomatic cases	0.2	[14]
1−θ	Proportion of asymptomatic individuals who recover	0.8	[14]
1/σ	Average removal time for symptomatic individuals	10 days	Assume
γ	Immunization loss rate	0.005/day	Assume
μ	Natural death rate	0.00003349/day	[13]
M	Birth/recruitment rate into the population	1500/day	Assume

Model (1) has the following properties:

For the given initial condition , the solution of system (1) satisfies for all .

We prove the proposition by contradiction. If not, there exists such that at least one of and is non-positive. The continuity of the solution implies that there exists such that at least one of and is equal to . Without loss of generality, we assume that is the minimal time with such property.

(a) If , then the rest of the state variables are non-negative at , and , which implies that there exists such that is strictly monotone increasing in interval . Let . Then , and since , there exists such that by Bolzano’s theorem, which contradicts the assumption of .

(b) If and , then . Similar to (a), we can obtain a contradiction.

(c) If and , then There are two cases:

(1) . Then , which is a contradiction similar to (a).

(2) . Then . It is easy to know that for all is the solution with . This is a contradiction to the uniqueness of the solution since and .

(d) If and , then . Also similar to a), a contradiction can be obtained. Similarly, will lead to a contradiction. Thus, we have and are all positive for all . □

System (1) is bounded.

Adding up all equations of system (1) yields Its solution is . Note that , we have . □

By Proposition 2.1, Proposition 2.2, the set is a positively invariant set of system (1).

Equilibria and basic reproductive number

In this section, we analyze the equilibria (including disease-free equilibrium and endemic equilibria) and the basic regenerative number of system (1). In order to find equilibriums, let where .

The so-called disease-free equilibrium is the equilibrium where the infected person is 0. Let , and it can be obtained by the last two equations of Eqs. (2). Substituting them into the first two equations of (2), we have Thus, the disease-free equilibrium is , and .

We use the next-generation approach [30], [31] to calculate the basic reproductive number. The equations related to infection are the 3rd and the 4th in system (1). Thus, let The Jacobian matrices of and at are where Then we have and the basic reproductive number of vaccination is where is the spectral radius of matrix .

The basic reproductive number of vaccination refers to the number of people infected by an infected individual during the average infection period. Expand as follows and each part has its specific meaning. represents the infected number of susceptible individuals per unit time for asymptomatic individuals. Thus the infected number of susceptible individuals per unit time for one asymptomatic individual is . Note that the proportion of susceptible individuals in the disease-free population is and the average time of asymptomatic duration is . The first term in , , gives the average infected number of susceptible individuals for one asymptomatic individual. Similarly, the second term gives the average infected number of vaccinated individuals for one asymptomatic individual. gives the infected number of susceptible individuals per unit time for symptomatic individuals. Therefore, the infected number of susceptible individuals per unit time for one symptomatic individual is . Since the average duration of symptoms is and asymptomatic individuals develop into symptomatic with proportion per unit time, the third item gives the average infected number of susceptible individuals for one symptomatic individual. Similarly, the last term in gives the average infected number of vaccinated individuals for one symptomatic individual.

If not vaccinated, i.e., , the basic reproductive number is which is called the basic reproductive number in the absence of vaccination. It is not difficult to see that is a decreasing function of , which means the larger the vaccination rate , the smaller the basic reproductive number of vaccination . This also indicates that vaccination plays an important role in controlling the spread of COVID-19. On the other hand, we have It can be seen that if the efficacy of the COVID-19 vaccine is not high ( is large), even if everyone is vaccinated, it may not be possible to make the basic reproductive number of vaccination to be less than 1, which means that COVID-19 cannot be eliminated. Therefore, the efficacy of the vaccines is also critical.

When or the efficacy of the vaccine , the COVID-19 epidemic might be controlled by vaccination. In order to make , the vaccination rate must satisfy

The remainder of this section discusses the existence of endemic equilibria. Adding up all equations of (2) yields which implies that when the system is in equilibrium. Assume . Divide all equations in (2) by , and let Then in equilibrium, (2) becomes From the last two equations of (4), we obtain and where By the first two equations of (4), we have Substituting (5), (6) into the third equation of (4), and eliminating , we obtain Expanding the above formula and dividing both sides by , we get a quadratic equation about . where and

Noticing that , we have which implies that

The discriminant of quadratic Eq. (7) is By and Eq. (12), we obtain or

It is clear that by Eq. (3), (12). According to the discriminant , Eq. (7) has two different real roots if or , has two same real roots if or , and has no real root if .

and have the following properties:

(1) , and the equal holds if and only if .

(2) if .

(1) Since and we have which is equivalent to

(2) If , then that is, . Similarly, . □

Regarding the existence of the positive roots of Eq. (7), we have the following result:

(1) Eq. (7) has only one positive real root if , or , or .

(2) Eq. (7) has two positive real roots if .

(3) Eq. (7) has no positive real root on the other cases.

It is easy to see that by Eqs. (8), (11). Combining with Eqs. (9), (10) and Lemma 3.1, according to Veda’s theorem and Remark 3, it is easy to verify the conclusions. □

Correspondingly, we have the following result about the existence of the equilibria of system (1).

(1) System (1) has a disease-free equilibrium and an endemic equilibrium if , or , or .

(2) System (1) has only a disease-free equilibrium and no endemic equilibrium if or .

(3) System (1) has a disease-free equilibrium and two different endemic equilibria if .

Analysis of stability and backward bifurcation

Stability of the disease-free equilibrium

Disease-free equilibrium is locally asymptotically stable if and unstable if .

The Jacobian matrix of system (1) at the disease-free equilibrium is where Its characteristic equation is It can be seen that has at least three negative real eigenvalues and . Thus, we only need to consider the following equation. If , or which implies , we have In this case, we also have . By the Veda theorem, the real parts of the two roots of Eq. (15) are all negative. Thus, all eigenvalues of have negative real parts if , which implies that is locally asymptotically stable.

If , we have , which implies that Eq. (15) has a positive real root, or has a positive eigenvalue. Thus is unstable if . □

Bifurcation analysis

By Theorem 3.3, when , in addition to a disease-free equilibrium, there may exist two endemic equilibria in system (1), and backward bifurcation may occur. That is, there may exist a locally stable endemic equilibrium. In this case, the disease-free equilibrium is not globally stable when , which means that the spread of COVID-19 may not be completely controlled even if . Therefore, it is necessary to analyze the backward bifurcation phenomenon of system (1). Here we apply Castillo-Chavez and Song Bifurcation Theorem [32] to analyze the bifurcation phenomenon of system (1).

(1) System (1) undergoes backward bifurcation at if .

(2) System (1) undergoes forward bifurcation at if .

Let and . System (1) is rewritten as Choose as a bifurcation parameter. According to , the critical value of is Substituting it into the Jacobian matrix (14), we have By (15), it is not difficult to know that only one eigenvalue of is , and the other four eigenvalues are all negative. After a simple calculation, a right eigenvector and a left eigenvector corresponding to eigenvalue of are obtained respectively, where and After tedious calculations, we obtain and Since , the sign of is determined by . By Castillo-Chavez and Song Bifurcation Theorem, if , equivalent to , system (1) undergoes backward bifurcation at . If , equivalent to , system (1) undergoes forward bifurcation at . □

According to Lemma 3.1, it follows that if . Meanwhile, combining with Theorem 3.3, Theorem 4.1 and Theorem 4.2, it can be seen that system (1) has not only a locally asymptotically stable disease-free equilibrium but also a locally asymptotically stable endemic equilibrium when , and has only a locally asymptotically stable disease-free equilibrium when .

Theorem 4.2 shows that relying only on imperfect COVID-19 vaccine may lead to backward bifurcation. Even if the basic regenerative number is less than 1, there may exist a locally asymptotically stable endemic equilibrium. In this case, it cannot completely eliminate the spread of COVID-19. In order to completely eliminate COVID-19, it is necessary to reduce the basic regenerative number to no more than , a positive number smaller than 1, which will bring greater difficulties to the prevention and control of COVID-19. The reason for the backward bifurcation is that an incompletely effective COVID-19 vaccine leads to two types of susceptible individuals, naive susceptible and the vaccinated susceptible.

In order to show the bifurcation phenomenon of system (1), the coefficients of Eq. (7) are expressed by . Divide both sides of Eq. (7) by and substitute . Noticing that when system (1) is in equilibrium, we rewrite Eq. (7) as where Choose appropriate parameters and consider Eq. (17) to define a curve in the positive quadrant, see Fig. 2. In Fig. 2(a), we set the parameter values and . We obtain and system (1) undergoes forward bifurcation at . In this case, a stable endemic equilibrium exists if and no endemic equilibrium exists if . In Fig. 2(b), the parameters are taken as and . Then we have and system (1) undergoes backward bifurcation at . Furthermore, through calculation, we obtain . In this case, there are two endemic equilibria if (the upper is locally stable and the lower is unstable) and only one stable endemic equilibrium exists if .

Fig. 2

Bifurcation of the endemic equilibria.

Simulation

In this section, we simulate the effect of vaccination on the prevention and control of COVID-19 to verify the previous analysis. The values of some parameters in system (1) are derived from some literature, and the others are assumptions made based on actual conditions. According to reports from relevant departments, currently, the efficacies of the most widely used types of vaccines recognized by WHO are 95% for COVID-19 mRNA vaccine BNT162b2 (Pfizer), 94.1% for mRNA-1273 vaccine (Moderna), 70.4% for ChAdOx1 nCoV-19 vaccine/AZD1222 (AstraZeneca) vaccine and 78% for sinovac respectively [33]. Combining the efficacies of these vaccines, we take . People who have recovered from COVID-19 will develop antibodies, but over time, the antibodies will gradually weaken and disappear. In [34], a follow-up study was conducted on 37 asymptomatic infections, and it was found that after two or three months, their antibodies to COVID-19 were significantly weakened. A screening study of 30082 survivors infected with COVID-19 in New York showed that the antibodies of infected individuals will remain stable for several months [35]. Based on these studies, we assume that the antibodies in the recovered patient can exist for an average of 200 days, that is, . The average time spent for removing (recovery or death) from symptomatic infections ranges from 7 days [15] to 14 days [19]. Here we choose an intermediate value 10 days. The values of the rest parameters are derived from [13], [14]. The all values of parameters are shown in Table 2. Based on these parameters, the basic reproductive number without vaccination is calculated as , which indicates that a person infected with COVID-19 can infect an average of 4 to 5 susceptible persons during the infection period. In this case, the spread of COVID-19 is very fast. After vaccination, the basic reproductive number is , which is already less than 1. Although it is very close to 1, it is much smaller than . This shows that vaccination is effective in controlling the spread of COVID-19. Taking the initial state values as shown in Table 3, we compare the trends of the number of different populations within 350 days in the two cases of with and without vaccination, as shown in Fig. 3.

Table 3

Initial values of states.

Initial state	Value
S(0)	50000000
V(0)	0
A(0)	1000
I(0)	100
R(0)	50

Fig. 3

The spread dynamics of COVID-19 about different .

Fig. 3(a) is the change trend of the number of asymptomatic individuals, . Without vaccination, reaches its peak on the 38th day, and there are about asymptomatic individuals. In the case of vaccination (), the peak of appeared on the 44th day, and there are about asymptomatic individuals. It can be seen that vaccination delays and reduces the peak of . Fig. 3(b) is the change curve of symptomatic individuals . Symptomatic individuals are the focus of the prevention and control of COVID-19. Without vaccination, peaks on about the 45th day, and there are about symptomatic individuals. In the case of vaccination (), the peak of appears on about the 52nd day, and there are about asymptomatic individuals. The peak of is postponed by 7 days, and also reduced by about 1 million, which will greatly reduce the burden of medical resources due to COVID-19, allowing more people to receive timely medical treatment, thereby reducing mortality. After about 130 days, the number of symptomatic individuals is always smaller in the case of vaccination ().

Figs. 3 (a) and (b) show that vaccination () is effective in reducing the number of infected individuals and has a positive effect on the prevention and control of COVID-19. However, it can also be seen that even if the COVID-19 vaccines have an efficacy of 80%, the epidemic could not been eliminated within 350 days. There are two possible reasons: one is the occurrence of backward bifurcation; and both converge to be non-zero constants. In this case, COVID-19 eventually develops into an endemic disease and coexists with humans for a long time. The other is that and both converge to 0, but the convergence speed is very slow because that the basic regenerative number is close to 1 and not small enough. In order to find out the reason, we need to calculate and . It is not difficult to obtain that and (the second column of Table 4). Since , backward bifurcation appears in system (1), as shown in Fig. 4(a) (the curve corresponding to ). However, implies that system (1) has only one disease-free equilibrium and no endemic equilibrium in this case, which means that COVID-19 will eventually be eliminated. Since is not small enough, the final elimination will take a long time.

Table 4

Values of and in different cases.

Symbol	ρ=0.2(τ=0.01)	ρ=0.1	ρ=0.03	τ=0.015	τ=0.025
ℜ1∗	0.6305	0.3240	0.1099	0.8372	1.2516
r_1	0.9429	0.7725	0.5012	0.9907	0.9828
ℜ_1	0.9268	0.4710	0.1520	0.9228	0.9195

Fig. 4

Bifurcation of different and .

In order to quickly and completely control the spread of COVID-19, in addition to avoiding backward bifurcation, reducing the basic reproductive number to make it sufficiently small is more important. Therefore, it is necessary to study how robust and are with respect to the changes of parameters and which parameters are the key factors that affect and respectively. Sensitivity analysis can provide valuable insights.

Sensitivity analysis

We adopt the sensitivity analysis method in [36]. Suppose that function is differentiable to parameter . Then, the sensitivity index of for is defined as Sensitivity index reflects the robustness of function to variable . Specifically, when the values of other variables (or parameters) remain unchanged, if , increases (or decreases) by when increases (or decreases) by 1% ; if , decreases (or increases) by when increases (or decreases) by 1%. Sensitivity indices of and to each parameter are shown in Table 5. It can be seen that has the greatest influence on . The sensitivity index is 1; that is, when decreases (increases) by 10%, will also decrease (increase) by 10%. Since does not contain , has no effect on it. has similar impacts on and . When decreases (increases) by 10%, and decrease (increase) by 9.8353% and 9.7236%, respectively. The effects of on and are opposite. When increases by 10%, decreases by 0.13131%, while increases by 6.5424%. Similarly, both and have little effect on , but have relatively greater effect on .

Table 5

Sensitivity index.

Parameter	Value	Sensitivity index of ℜ_1	Sensitivity index of ℜ1∗
α	0.8883	1	0
ρ	0.2	0.98353	0.97236
τ	0.01	−0.013131	0.65424
δ	1/7	−0.61149	−0.022714
σ	0.1	−0.38814	−0.0064874

When non-pharmaceutical interventions are not adopted, in order to control the spread of COVID-19 as soon as possible, we can only consider two aspects: improving the vaccine efficacy (reducing ) and increasing the vaccination rate (increasing ). From the sensitivity analysis results (Table 5), has a great effect on both and . When decreases by 10%, and decrease by 9.8353% and 9.7236%, respectively. It can be seen that as long as and other conditions remain unchanged, improving the vaccine efficacy (reducing ) will greatly reduce , such that COVID-19 can be controlled as soon as possible. If other parameters remain unchanged, the trends of and are shown as in Fig. 3 for and 0.03, respectively. The corresponding , and are shown in Table 4, where when and , and 0.1099, respectively, both of which are less than 1. These two cases both mean that backward bifurcation occurs in system (1), and the bifurcation diagrams are shown in Fig. 4(a). Regardless of or 0.03, is less than , which means system (1) has only one disease-free equilibrium and no endemic equilibrium, and COVID-19 will eventually be effectively controlled. It can also be seen from Table 4 that the smaller the , the smaller the corresponding , and the faster the numbers of people infected with COVID-19, and , decrease, which is also manifested in Fig. 3.

Although improving vaccine efficacy has a very positive effect on the prevention and control of COVID-19, it cannot be achieved in a short term. For this reason, another measure that is easier to implement needs to be considered — increasing the vaccination rate (increasing ). From the sensitivity index (Table 5), has a small impact on , but has a large impact on . If increases by 10%, will only decrease by 0.13131%, while will increase by 6.5424%. Let the remaining parameter values be unchanged (where ). Take and 0.025 to simulate the trends of and , and the results are shown in Fig. 5. The corresponding values of and are shown in Table 4, and the bifurcation diagrams are shown in Fig. 4(b). From Table 4 and Fig. 4(b), we can see that when and 0.015, the corresponding is all less than 1, and backward bifurcation occurs in system (1). When , , and forward bifurcation (transcritical bifurcation) occurs in system (1). From Table 4, it can be seen that although decreases as increases, the magnitude of the decrease is very small, which is consistent with the sensitivity analysis results. However, as shown in Fig. 5, increasing the vaccination rate (increasing ) has a significant effect on reducing and delaying the peak of the number of people infected with COVID-19, which is very helpful to avoid a serious run on medical resources.

Fig. 5

The spread dynamics of COVID-19 about different .

Combining non-pharmaceutical interventions

Comparing Fig. 3, Fig. 5, we can see that improving vaccine efficiency can control the spread of COVID-19 faster, while increasing the vaccination rate can reduce and postpone the peak of infection to a greater extent. However, in reality, the improvement of vaccine efficacy cannot be realized in a short time, and only relying on increasing the vaccination rate cannot quickly achieve the control of COVID-19. Therefore, in addition to vaccination, some non-pharmaceutical interventions need to be taken. In addition to and , according to the sensitivity analysis, the basic regenerative number can also be reduced by measures such as reducing and increasing and . The details are as follows:

Reduce . The essence is to reduce the contact rate between people, such as wearing a mask, maintaining social distance, imposing curfews and lockdown.

Increase and . These can be achieved through quarantine of asymptomatic cases, hospital isolation and treatment of symptomatic cases.

Based on the above two aspects of non-pharmacological intervention, model (1) is improved as follows: where measures the control over the contact rate of people. The smaller means the stricter the prevention and control measures, and the lower the contact rate of people. represents infected individuals treated in isolation, including symptomatic and asymptomatic. We assume that once an infected individual is diagnosed, he/she will be isolated and treated, and the isolation measures are strict, in other words, the isolated individuals no longer have the opportunity to infect others. Using the next-generation approach, the basic reproductive number of model (18) is obtained as where .

From the sensitivity index in Table 5, it follows that has the greatest impact on the basic reproductive number , followed by and . Corresponding to are and , respectively. We focus on the impact of measures to reduce the contact rate (reducing ) on the spread of COVID-19 firstly. Since the asymptomatic individuals are difficult to be diagnosed, the isolation rate is not high, so set . The diagnosis rate of symptomatic individuals is higher as they generally seek medical treatment. Their isolation rate is taken as . The average days for isolated individuals to be removed are shorter than that for non-isolated infected individuals, which is taken as days, i.e., . Other parameters and state initial values remain unchanged, see Table 2, Table 3. Let and 0.6. Then the corresponding basic reproductive numbers and 0.1518, respectively. The simulation results are shown in Fig. 6. It can be seen that the smaller the , the smaller the basic reproductive number , and the faster the spread of COVID-19 is controlled. When , COVID-19 can be completely controlled in about 40 days, as shown in Fig. 6.

Fig. 6

The spread dynamics of COVID-19 about different .

Secondly, we study the impact of the isolation rate of asymptomatic infections on the spread of COVID-19. Let the remaining parameters be unchanged, where and . Let and 0.25. Then the corresponding basic reproductive numbers and 0.1766, respectively. The spread dynamics of COVID-19 are shown in Fig. 7. It can be seen that the larger , the smaller , and the faster COVID-19 is controlled. As shown in Fig. 7, when , the spread of COVID-19 is completely controlled in about 50 days.

Fig. 7

The spread dynamics of COVID-19 about different .

Finally, we focus on the impact of the isolation rate () on the spread of COVID-19. Let the remaining parameters be unchanged, where and . Let and 0.4, then the corresponding basic reproductive numbers and 0.2130, respectively. The spread dynamics of COVID-19 are shown in Fig. 8. Compared with and , we can see that has a much smaller impact on the basic reproductive number . As shown in Fig. 7, the larger , the more helpful it is to control the spread of COVID-19. However, the time spent to eliminate the epidemic cannot be significantly reduced in the three cases when and 0.4, which all need about 80 days.

Fig. 8

The spread dynamics of COVID-19 about different .

Based on the above simulations and analysis, it can be seen that when non-pharmaceutical intervention is not implemented, whether improving the vaccine efficiency or increasing the vaccination rate, the spread of COVID-19 cannot be effectively and quickly controlled. Therefore, some non-pharmaceutical intervention measures need to be implemented. Among non-pharmaceutical interventions, reducing the contact rate of people (reducing ) and increasing the isolation rate of asymptomatic individuals (increasing ) are very effective, which can greatly reduce the peak of infection and quickly control the spread of COVID-19.

Conclusion

COVID-19 has been spreading globally for more than a year, and it has not yet been effectively controlled. In addition, multiple variants of SARS-CoV-2 have been identified, which are more infectious, posing greater challenges to the global COVID-19 epidemic prevention. Especially in the recent outbreak of COVID-19 in India, the number of confirmed cases in a single day has repeatedly reached new highs. The number of confirmed cases in a single day has exceeded 300,000 for 25 consecutive days, and the number of deaths per day has also remained high. The prevention and control of COVID-19 remains a long-term and arduous task. Although a number of vaccines have been put in use, relying only on vaccines to control the spread of COVID-19 will lead to backward bifurcation. In this case, when the basic reproductive number , there exists a locally asymptotically stable endemic equilibrium in addition to a locally asymptotically stable disease-free equilibrium, which means that even if the basic reproduction number is less than 1, COVID-19 cannot be completely eliminated. In order to achieve a complete and quick control of COVID-19, in addition to population vaccination, it is still necessary to appropriately enforce some non-pharmaceutical intervention measures, such as reducing the contact rate of people and increasing the isolation rate of asymptomatic individuals.

CRediT authorship contribution statement

Bo Yang: Investigation, Methodology, Writing – original draft, Writing – review & editing. Zhenhua Yu: Formal analysis, Investigation, Writing – review & editing. Yuanli Cai: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Supervision, Writing – review & editing.