Second-wave dynamics of COVID-19: Impact of behavioral changes, immunity loss, new strains, and vaccination

An epidemiological model for COVID-19 developed earlier was extended to determine the effects of behavioral changes, immunity loss, and vaccination on second and subsequent wave dynamics of the pandemic. A model variant that distinguishes four demographic groups with different infection rates and mortality rates was developed to test the hypothesis that behavioral divergence between groups can explain both the larger incidence and lower mortality rate of COVID-19’s second wave. A model version that incorporates immunity loss was developed to test the hypothesis that immunity loss can explain the second wave. Simulations indicate that of the two hypotheses, only the former is consistent with observed trends. Nevertheless, loss of immunity can significantly increase total number of deaths in the long run, particularly in cases where vaccine distribution is barely sufficient to reach herd immunity. The observed trends are illustrated with detailed simulations of the progression of COVID-19 in the United Kingdom, including the appearance of new strains. The U.K. case study indicates the extent to which NPI can be relaxed during the distribution of the vaccine.


Introduction
COVID-19 is a respiratory disease first observed in Wuhan, China, in late 2019. It is caused by the SARS-CoV-2 virus. It is similar to other coronaviruses such as the original SARS virus. The virus spread to Europe and North America in early 2020 and was declared a pandemic by the World Health Organization on March 11, 2020. After an initial peak of daily deaths in April 2020, the death rate declined in the countries first affected but the virus spread to the rest of the world, causing a broad peak of daily deaths in the late summer of 2020.
The widespread use of non-pharmaceutical interventions (NPIs) such as lockdowns, wearing of facemasks, frequent disinfection, and closure of nonessential services led to a retreat of the epidemic in most countries but the easing of these interventions in late summer and early fall 2020 caused a resurgence of the disease in most countries, which persisted to early 2021.
Globally, the "second wave" of fall 2020 was characterized by a vastly increased number of new cases per day, as well as increased number of new deaths per day, in comparison with the first wave of spring 2020, but with a lower case-mortality rate in the second wave in comparison with the first wave. Various factors could be responsible for the latter observation, including a possible decline of the mortality rate of the disease, better clinical procedures, a shift to a younger and less fragile segment of the population as the main carriers of the virus, and increased testing. Because of the significant number of asymptomatic cases, inadequate testing leads to underestimation of the number of cases, and hence higher case-mortality rates. 4 A number of reports point at a gradual loss of immunity over a time span of several months after recovering from COVID-19. Edridge et al. [1] investigated reinfection from four seasonal human coronaviruses and found that reinfection is common after 12 months for all four viruses. Based on this observation, they anticipate that immunity for SARS-CoV-2 is short-lived. Immunity to SARS-CoV-2 is still poorly understood but based on [2,3], a duration of 5 months is a reasonable assumption at this time.
Many mutations of SARS-CoV-2 have been reported [4]. Most of these mutations have not been reported to be physiologically significant.
An early mutation that has been associated with increased infectivity is known as the D614G mutation. Korber et al. [5] observed that mutated virus copies systematically became more prevalent than the pre-mutation variant in a large number of countries and cities. This may explain why, for instance, the initial spread of COVID-19 was more gradual on the North-American West Coast, where the original variant took hold first, than on the North-American East Coast, where the mutated variant took hold from the beginning. There is no evidence that the D614G mutation leads to more severe infections [6].
A mutation of particular concern, VOC 202012/01, was found in southeast England in late 2020.
Modeling the spread of an infectious disease is usually conducted by models based on the SIR (Susceptible-Infected-Recovered) model of Kermack and McKendrick [8]. A review of pre-COVID models is given by Chowell et al. [9]. Early post-COVID models are by Lin et al. [10] and Roosa et al. [11]. De Visscher [12] used an extension of the SIR model to study the effect of NPIs on the expected mortality of the disease and applied the model to project future deaths from 5 past deaths in Italy, France, and Iran. Huang and Qi [13] used a similar model to evaluate the second wave effect of an easing of NPIs. Wang et al. [14] developed a model that accounts for asymptomatic spread of COVID-19 and found that continued control is needed after a lockdown is eased. Kwuimy et al. [15] developed a similar model to evaluate the effect of governmentbased NPI initiatives and the behavioral response of the population. Zlatić et al. [16] developed a model that distinguishes detected and undetected cases in the infected population. Assuming that detected individuals would adjust their behavior to avoid spreading, it was shown that effective early testing can prevent an epidemic from growing explosively. Endo [17] argued, based on modeling results, that an effective way to reduce the spread of the virus is to focus on relatively rare superspreaders or superspreading events.
A much more sophisticated model was developed by Ferguson et al. [18]. This model calculates the spread of the disease within as well as outside family units in a stochastic manner. It also accounts for geographic spreading. The model is based on a similar model for influenza [19].
Based on comparisons with influenza pandemics such as the 1919-1920 "Spanish flu" pandemic, a second wave in the fall of 2020 was anticipated but the severity of the second wave in terms of the number of new cases was unexpectedly high. Factors that may have contributed to the severity include the reopening of schools after the summer vacation as well as NPI "fatigue", where many people relax on the precautions due to a perceived lowering of the risk.
The first objective of this study is to test if the reopening of schools can explain the severity of the second wave. Second, when some degree of "herd immunity" (lowering of the reproduction rate of the disease resulting from partial immunity of the population) is established, a gradual loss of immunity can cause a resurgence of the disease. Our second objective was to test if this process can explain the occurrence of a second wave.
Pfizer developed a vaccine against COVID-19 in late 2020. In most countries, elderly people and other at-risk segments of the population are vaccinated first to minimize the loss of life.
However, this choice may not halt the disease because elderly and at-risk people are not the main spreaders of the disease. An alternative strategy would be to vaccinate the main spreaders first to stop the epidemic in its tracks. Despite the low mortality of COVID among the active segment of the population, it is possible that this strategy would save more lives. The third objective of this paper is to test this hypothesis.
The distribution of vaccinations will alter the dynamic of the disease but this effect may be temporary due to the loss of immunity over time. The fourth objective of this study is to evaluate the long-term effect of vaccination in the case of immunity loss. This paper is concluded with a case study in the United Kingdom. Based on a detailed fitting, a NPI history of the U.K. is reconstructed and followed by projections based on vaccination and NPI relaxing scenarios. The objective of this case study is to evaluate the extent of NPI relaxing that can be afforded by the vaccine, despite the emergence of a more infective strain of the virus.

Original model description
The model variants used here are extensions of the model of De Visscher [12]. Mathematically, the infection rate (U → I) is given by: (1) where P is the total population. The infection rate r1 is in infections per day. The rate constants k11, … are expressed in day −1 . In the general cases in this paper, P is assumed to be 100 million, whereas in the U.K. case study, P equals 67.8 million. The default value of k11 in this study is 0.4 day −1 for the general cases, and 0.39 day −1 for the U.K. case. The features of new strains will be discussed in Section 2.5. The value of k11 is allowed to vary over time to express NPIs. For details see [12].
The other rate constants in eq. (1) are given by: The model is run with the following initial conditions unless stated otherwise: The same initial conditions are used but for a later date in the case of a second or third strain.
In the case of the U.K. case study, all initial conditions are multiplied with a correction factor as part of the model fit to the reported number of cumulative deaths.

Model extension 1: demographic groups 9
In this model extension, the population is divided into four demographic groups, 1 P, 2 P, 3 P, and 4 P. The properties of each group are shown in Table 1. The numbers were chosen to be consistent with an overall mortality rate of 1.5 % as used in the original model. Onder et al. [20] reported case mortality rates as low as 0.2 % for age groups under forty, whereas the rates ranged from 8 % to 20 % for ages over 70, depending on location and age group. Because underreporting of cases probably inflated these numbers, the lower end, 8 %, was used as the mortality rate for the old/at risk group in this model extension.
The infection rates in this model are split into infection rates for each demographic group, where any demographic group can be infected by any other demographic group as well as its own: To calculate the infection rates of four demographic groups, all 16 possible infection rate constants need to be known for each of k11, k12, etc. They were calculated as follows: (14) for k11, etc.
The model is set up in such a way that the efficiency of each NPI can be different for each demographic group.

Model extension 2: immunity loss
Immunity loss is modeled by assuming an additional transition: 11 with a transition rate: For ki, values of ln(2)/180 and ln(2)/90 day −1 were tested, representing immunity half-lives of 6 months and 3 months.

Model extension 3: vaccination
To represent vaccination, a new state, V, is introduced. Vaccination is represented by the following transition: with a vaccination rate constant kvaccine expressed in vaccinations per day. The vaccination rate rvaccine is given by: In eq. (18), the factor fvaccine is the effectiveness of the vaccine, assumed to be 95 %. The factor U/P expresses the fact that it is not generally known who is immune and who is not, so a fraction of the vaccines is given to people who are already immune.

12
The vaccination rate constant kvaccine as well as the effectiveness factor fvaccine can vary over time.

Model extension 4: multiple COVID strains
Multiple COVID strains need to be accounted for if they have different properties (infection rate, mortality). Approaches of multistrain epidemic dynamics are reviewed by Martcheva [21]. In the current simulation, it is assumed that the strains differ only by their infection rates. For the new strain that emerged in the U.K. in September 2020, an infection rate 70 % higher than the infection rate of the original strain (0.39 day −1 ) was assumed, at the higher end of the confidence interval of Davies et al. [7].
To distinguish infections with two different strains, two different infected states are distinguished: e.g., I1 and I2. Each state progresses in parallel, e.g., I1 to S1 to SS1, etc.
Strain 1 is assumed to be present from the beginning, Strain 2 is introduced in the model on simulation day September 22 nd , 2020. A third strain with infection rate 100 % higher than for Strain 1 was introduced in the model on January 15.

Second wave dynamics: effect of changing infection rates
In general, infection rates are calculated with the following equation: 13 where fNPI is the effectiveness of the NPI. In this model, the simulation is run for 450 days, representing March 2020-June 2021 for a typical country. The effectiveness values for the various demographic groups used in the simulation are shown in Table 2. Different scenarios were tested until a combination was found that led to an epidemic progression representative for European countries. The results of the simulation are shown in Figure 1.  Table 2 In the simulation, the first wave peaks at 1400 daily deaths and 200,000 daily new sick cases, for a total population of 100 million. The daily new cases is much higher than actually observed in countries of a similar size because not all cases are identified or reported. Based on simulations for the previous study [12], countries underreport by about a factor 2-10 (data not shown) indicating that the actual reporting can be expected to be on the order of 20,000-100,000 new cases per 100 million, in line with what was actually observed during the first wave.
During the second wave, the daily new deaths peaks at about 1600, similar to the first wave, but the number of new sick cases doubles to about 400,000 (i.e., 40,000-200,000 actually reported).
This means that whereas the incidence of the disease is much higher in the second wave, the mortality rate is about half in the second wave in comparison with the first wave. This is in line with actual observations, although the decline in case mortality rate was generally more pronounced in reality. This may have been due to increased testing during the second wave.
To obtain the trends observed in Figure 1, it was necessary to assume a pronounced change in NPI effectiveness in both Group 1 (active) and Group 4 (school-going). Limiting the change to one of the two groups led to a much more slowly developing second wave with a lower peak.
This indicates that the second wave cannot be attributed to a single demographic group but was the result of broad demographic behavioral changes, representing at least two of the four demographic groups.
To obtain a pronounced drop in the case mortality rate during the second wave, it was necessary to assume very high effectiveness of NPI in the middle-aged and old/at risk groups (99 % and above). These may not be realistic numbers. These high efficiencies were needed to keep cross infection between the more active and less active groups to levels where there is a significant divergence of the number of cases in the groups. This suggests that eq. (14) (for geometric averaging of infection rates) overestimates the actual cross infection rates. In practice, this would mean that there is less mingling between demographic groups than within. This confirms a finding of Klepac et al. [22] that social mixing is fairly stratified by age.
The cumulative number of deaths after 450 days in the model is 216,707.

Second wave dynamics: effect of lost immunity
To evaluate if a second wave can be caused by declining immunity, a base case simulation is run with zero loss of immunity, with a NPI regime chosen so that the daily new deaths remain approximately constant over a prolonged period of time. The parameters are shown in Table 3. The result of the calculations is shown in Figure 3. It is clear from the figure that immunity loss leads to a very gradual increase of the daily deaths over prolonged time, very unlike the dynamics of a second wave, with dramatic increases in death rates within a few weeks. It follows that loss of immunity is not a major contributor to second-wave dynamics of COVID-19.

Vaccination: effect of prioritizing demographic groups
In this section we will evaluate which sequence of vaccination priorities leads to the least number of lives lost. In this section the simulations were run for 800 days to avoid missing slow trends. It is assumed that immunity is not lost. In practice, loss of immunity can be compensated by increased vaccination rates.
In this scenario, it is assumed that vaccination is accompanied by further loosening of NPIs. The sequence of NPI efficiencies is shown in Table 4. It is assumed that vaccination starts on day 300 and occurs in two phases, each lasting 100 days.
During each phase, 200,000 vaccines are administered daily, totalling 40 million vaccines. In each phase, one demographic group is prioritized, receiving 70 % of the vaccines, whereas the other groups receive 10 % each. Every combination is tested and the total number of deaths on day 800 is calculated. The result is shown in Table 5. All prioritizations lead to a total number of deaths ranging from 257,210 to 295,070. This is a surprisingly narrow spread given that the mortality rates of the groups range from 0.2 % to 8 %.
Not vaccinating at all would lead to 467,730 deaths, almost twice as much. Among the scenarios, the most favorable one is vaccinating school-going youth first, followed by vaccinating the active group. It is generally more favorable to switch priorities after 100 days, and the first priority has a greater impact on the overall result than the second priority. However, given the narrow spread of the results, none of these trends should be taken as absolute. It is likely that small changes in the assumptions underlying the calculations will reverse these conclusions. It is more reasonable to conclude that, as long as an adequate level of vaccination is reached, it is of less importance who is prioritized. Figure 3 shows the daily deaths over time in the vaccination scenario where Group 4 is prioritized first and Group 3 next. This scenario is compared with a single vaccination phase, and no vaccination at all. It is clear that vaccination has a profound effect and has the potential to eradicate the epidemic, even with incomplete vaccination. This is because the scenario assumes sufficient NPIs are in place to ensure herd immunity is maintained.

Vaccination: effect of loss of immunity
In this section, the same scenarios will be repeated, but this time it is assumed that immunity is only temporary, with immunity half-lives of 6 months or 3 months. It is assumed that the rate of immunity loss is the same regardless of whether immunity was obtained by the disease or the vaccine. It is assumed that health authorities anticipate the loss of immunity by continuing the vaccination. During days 500-800 it is assumed that all demographic groups are prioritized according to their size. The results are shown in In the absence of loss of immunity, the total number of deaths is 257,210 in 800 days. When the half-life of immunity is 6 months, this number is 821,830, with a half-life of 3 months, the death toll is as high as 1,594,800. This profound effect is because 200,000 vaccinations per day, 0.2 % of the total population per day, is barely enough to achieve herd immunity.
The loss of immunity can be compensated by administering more vaccines. This is demonstrated in Figure 5, with 200,000, 600,000, and 1,200,000 vaccinations administered per day and an immunity half-life of 3 months.   Table 6, along with the dates they take effect. Because detailed disaggregation of deaths per day by age group is not publicly available, the entire population was modeled as a single demographic group, as in the original version of the model [12]. The model predicts that 2.7 million people in the U.K. had contracted COVID-19 by the end of June 2020. This corresponds well with an antibody study by Ward et al. [23] which led to an estimate of 3.36 million. Given that the actual COVID-19 death number is probably underreported, leading to an underestimated number of cases in the model, this is an excellent agreement.
As indicated above, a second strain with infection rate 70 % higher than the original strain and a third strain with an infection rate 100 % higher than the original strain (worst-case scenario) was introduced on September 22 nd and January 15 th , respectively. In the base case scenario, the third strain does not gain a hold because of the high effectiveness of the NPI used in the model at that time (97.5 %). This high value was needed to predict recent deaths occurring with the new strain and may indicate that the infectiveness of the second strain (at 70% higher than the first) is overestimated in our model.
In the base case scenario, it is assumed that vaccination starts on January 1 st , 2021 with 250,000 vaccinations per day, which is ramped up to 400,000 per day on February 1 st , 2021, and thereafter. These numbers are conservative estimates. The vaccinations started on December 8 th , 2020 in the U.K. but immunity takes three weeks to take effect. In the model, it is assumed that the effectiveness of the vaccines administered during the first month is 70 %, and that the effectiveness of the vaccines administered after the first month is 95 %. This allows for possible optimizations of the vaccination, partly owing to the variety of vaccines that are and will be available in the U.K..
The result of this scenario is shown in Figure 6 as the solid line. In the base scenario, the cumulative number of deaths after 800 days of simulation (i.e., on April 10 th , 2022) is 128,436.
In scenario 2, the same situation is simulated in the absence of vaccinations. The cumulative number of deaths is only slightly higher than in the base case, at 133,712. The small difference is due to the fact that the assumed effectiveness is very high. To calculate the effect of loosening the NPIs after the start of the vaccination program, a decrease of the NPI effectiveness by 15 % was introduced on March 7 th , 2021, from 97.5 % to 82.5 %. This date was chosen because it has been suggested as a potential reopening day for schools by the U.K. Government. This is scenario 3, which is also shown in Figure 6. This leads to a cumulative deaths number of 134,082, about the same as in the absence of a vaccine, but maintaining the high NPI effectiveness after March 7.

Conclusions
An epidemiological model developed recently was extended in a number of ways to evaluate factors contributing to second-wave dynamics of COVID-19. It was shown that changes in the effectiveness of NPIs for younger age groups (labelled "active" and "school-going") comprising 50 % of the population can explain both the second wave itself and the reduced mortality rate during the second wave. Loss of immunity affects the dynamics of COVID-19 too slowly to be a major factor in second-wave dynamics. However, loss of immunity has a pronounced long-term effect on the number of deaths resulting from COVID. The effect of prioritizing demographic groups for vaccination has a surprisingly small effect on the overall death number from the virus.
In the scenarios run in this paper, vaccinating the most active segments of the population had the most favorable outcome, but this may be specific to the assumptions used in the model and may not bear out in actual practice.
A case study of the United Kingdom shows that the model can accurately describe death numbers due to COVID-19. This calibrated model predicts that vaccination in the U.K. at the current and planned rates will be adequate if maintained, and will enable a loosening of the NPIs, in this case on March 7th 2021 when it is thought U.K. schools might reopen, allowing the effectiveness of the NPIs to fall by 15 %.