Seroprevalence and attainment of herd immunity against SARS CoV-2: A modelling study

Abstract

Objective:

The present study aims to predict the likelihood of and likely time required to attain herd immunity against COVID-19 in New Delhi due to natural infection.

Method:

An ODE-based mathematical model was constructed by extending the classical SEIR model to predict the seroprevalence rate. We estimated the parameter values for Delhi using available data (reported cases and the seroprevalence rate) and used them for future prediction. Also, changes in the seroprevalence rate with different possibilities of reinfection were predicted.

Results:

Maximum seroprevalence rate obtained through our model is 31.65% and also a reduction in the seroprevalence rate was observed for the upcoming one month (month of January, 2021) due to the reduced transmission rate. After increasing the transmission rate to the value same as the third wave in New Delhi, we obtained a maximum value of 54.96%. This maximum value significantly decreased with the reduction in the reinfection possibilities. Also, a little impact of the duration of persistence of antibodies, 180 vs 105 days, was observed on the maximum seroprevalence.

Conclusion:

This modelling study suggests that natural infection alone, as gauged by serial sero-surveys, may not result in attainment of herd immunity in the state of Delhi.

Introduction

COVID-19, first reported from China on 16^th December 2019, has spread globally affecting more than 91 million people causing 1.9 million deaths. Indian scenario is following the global trends with 10 million cases and 151,918 deaths (as of 16 January 2021).[1] The pandemic is still raging on in different parts of the world showing periodical waxing and waning.

Immune response against SARS CoV-2 is still not completely defined. Antibody test (IgG against SARS CoV-2), being a surrogate method of measuring immune response, has been utilized in a number of studies to detect past infection with COVID-19. Studies have shown that the immune response is not uniform among all the infected individuals and only ~ 80% of the asymptomatic individuals mount a measurable antibody response.[2,3] Also IgG antibody titres start reducing within 2-3 months post COVID-19 infection.[4] Longevity of immune response against COVID-19 is yet to be entirely defined and reinfections are already very well documented.[5]

Oxford dictionary defines herd immunity as “resistance to the spread of an infectious disease within a population that is based on pre-existing immunity of a high proportion of individuals as a result of previous infection or vaccination”. Herd immunity threshold for COVID-19 required is estimated to be 60% to 75% based on the equation (1-1/R₀).[6,7] Few policy makers and groups of scientists around the world proposed allowing unmitigated spread of infection to attain herd immunity by natural infection. Relaxing social restrictions at the peak of pandemic at various places has been suggested to be part of this strategy. However, WHO and other health authorities, on the other hand, have warned against targeting the strategy of attaining herd immunity by natural infection as it entails, potentially significant mortality and morbidity, especially among high-risk individuals.[8]

Seroprevalence surveys are being conducted in various cities to quantify spread of COVID-19 infection as well as to inform various policy decisions. Monthly sero-surveys have been conducted in New Delhi between July and October. However, the trend of seroprevalence in these studies did not increase substantially over several months.[9] With documented rapidly declining antibody response and reported reinfection, will we be ever able to reach the threshold required for herd immunity? To answer this question, we constructed a mathematical model to predict the time required to attain herd immunity against COVID-19 in New Delhi.

Materials and Methods

Data acquisition

The state-wise data were collected from an open-sourced database for COVID-19 stats and patient tracing in India.[10] In this study, we have considered the day-wise confirmed, recovered and deceased cases in Delhi till January 16, 2021. Seroprevalence data between July and October available for New Delhi were used.[9] Ethical clearance for this modeling study was not taken as the model does not involve any individual patient’s data.

Mathematical model

We propose an extended epidemic model [see Figure 1] which incorporates four infectious classes: pre-symptomatic infectious (P), undetected infectious (U), tested infectious (T), reported infectious (R₁). A large portion of infectious individuals do not show any symptoms or show mild symptoms,[11] thus a portion of such infectious individuals remain undetected.[12,13] Tested infectious (T) individuals are those who went for testing but results are not yet known. After knowing the result, the individuals with COVID-19 positivity will enter the reported infectious (R₁) class. These reported infectious (R₁) individuals usually do not transmit the disease due to home isolation or hospitalization. The susceptible individuals (S) acquire infection from pre-symptomatic infectious (P), undetected infectious (U) and tested infectious (T) individuals and enter into exposed class (E). It is to be mentioned here that an undetected infectious (U) individual may enter into the tested infectious (T) class by rapid antigen or other tests like Reverse transcription polymerase chain reaction (RT-PCR) test. We also incorporated two scaling parameter α_u and α_P for undetected and pre-symptomatic infectious class, respectively, to make the infection spreading rates different from the tested infected ones. Transformation from U and R₁ classes to undetected recovered (U_R) and recorded recovered (R_R) classes occur through the rates and , respectively. The death rates due to this disease are denoted by D _du and D _d for these both U and R₁ classes respectively, and these individuals join into the undetected death (U_D) and recorded death (R_D) classes. Here, the total population (N) is assumed to be constant, which is a reasonable assumption if the epidemic period is not too long. Though re-infections are increasingly being reported in COVID-19, the knowledge about exact frequency, risk factors and period of susceptibility is still evolving. So, we have introduced a parameter (σ) in our model to incorporate the different possibilities of reinfection. The dynamics of these different classes across time (t) are described by the following set of differential equations:

Figure 1.

(Original): Schematic diagram of the proposed model. We extended the classical epidemic model (SEIR) by incorporating ten compartments: susceptible(S), exposed(E), pre-symptomatic infectious(P), undetected infectious(U), tested infectious(T), reported infectious(R₁), undetected recovered(U_R), undetected death(U_D), recorded recovered(R_R) and recorded death(R_D). We have also included the case where the recovered individuals become susceptible again due to loss of antibodies.

with S +E +P +U +T +R +U +U +R +R = N

(total population) and S(O), E(O), P(O), U(O), T(O), R₁(O), U_R(o), U_D, (O), R_R(O), R_D(O)>O. Definitions and values of the parameters with reference are provided in Table 1. Seroprevalence rate at a given time (t) are obtained from the following equation:.

Table 1.

Parameter setting for the model simulation. Parameters for which the references were obtained are provided in the description section and remaining were estimated according to good fit

Parameter	Description	Mar 14-Apr 12	Apr 13-Jun 14	Jun 15-Aug 16	Aug 17-Sept 19	Sept 20-Oct 07	Oct 08-Nov 16	Nov 17-Jan 11
	Transmission rate of tested infectious individuals	0.82	0.242	0.09	0.225	0.12	0.24	0.085
	Ratio of the transmission rate for undetected cases over tested infectious case.[14]	0.75	0.75	0.75	0.75	0.75	0.75	0.75
	Ratio of the transmission rate for pre-symptomatic infectious cases over tested infectious case.[14]	0.5	0.5	0.5	0.5	0.5	0.5	0.5
	Period for presence of antibodies in asymptomatic cases	180	180	180	180	180	180	180
	Period for presence of antibodies in symptomatic cases.[15]	180	180	180	180	180	180	180
	Latent period.[16]	2.9	2.9	2.9	2.9	2.9	2.9	2.9
	Pre-symptomatic infectious period.[16]	2.3	2.3	2.3	2.3	2.3	2.3	2.3
	Transformation rate from pre-symptomatic class to testing class	0.005	0.0085	0.017	0.06	0.06	0.06	0.06
	Transformation rate from undetected class to testing class	0.0001	0.0001	0.0002	0.0004	0.0003	0.0003	0.0003
	Time required for recovery of undetected individual.[14]	10	10	10	10	10	10	10
	Death rate for undetected cases	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001	0.00001
	Time required for sample collection, testing and report generation	2	2	2	2	2	2	2
	Transformation rate from reported infectious class to recorded recovered class	0.0476	0.05	0.1	0.1	0.125	0.143	0.1667
	Death rate for symptomatic cases.	0.009	0.004	0.0021	0.001	0.0012	0.0018	0.0029

Numerical simulation

All the numerical simulation for the mathematical model was performed in MATLAB by using the ODE45 function. Also, the curve fitting toolbox was used to get the R-squared value between the simulated and predicted data. Among the parameters considered in modelling, a few required estimation by trial and error method to best fit the curve [Table 1].

Prediction of the number of confirmed, active, recovered and deceased cases for the upcoming one month were done by considering the parameter set obtained for the period of 17 November 2020–16 January 2021 range. This allowed us to maintain the changes in the number of confirmed, active, recovered and deceased cases with the same rate obtained at the end of the study period.

Results

Model analysis

The aim of this study was to capture and predict the future seroprevalence rate in Delhi through mathematical modelling. We extended the classical SEIR model by including pre-symptomatic infectious (P), undetected infectious (U) and reported infectious (R₁) classes [see methods and Figure 1] and considered the daily data of confirmed, active, recovered and deceased cases in Delhi till January 16, 2021.[10] The model estimated time-series curves for confirmed, active, recovered and deceased cases are provided in Figure 2 along with their reported data. The obtained R-squared value of >0.97 indicates a good fit of the model results. The parameter values corresponding to time-series curves are presented in Table 1, which was further used for predictions.

Figure 2.

(Original): Comparison between the reported data (blue colour curve) and the model simulation data (red colour curve) of Delhi. The data contains day-wise information for the period March 14, 2020 to January 16, 2021 for Delhi and the parameter values are given in Table 1. The R-squared values are also mentioned here in the inset

Model prediction

We used the parameter set obtained for the period of 17 November 2020 to 16 January 2021 range [see Table 1] to predict the number of confirmed, active, recovered and deceased cases for the upcoming 1 month (month of January 2021) [Figure 3]. We also plotted the variation in the seroprevalence rate with respect to time [Figure 4] for the whole study period. Here, we were able to capture the reported seroprevalence rate of New Delhi for the month of July, September and October but not for August.[9] We observed that from around the last week of November this value reached above 30% and from the 2^nd week of December, the seroprevalence rate started to fall. During our study period, we obtained a maximum seroprevalence rate around 31.65% on December, 8. Also, a reduction in seroprevalence rate was observed for the upcoming one month (month of January, 2021) due to the reduced transmission rate in the period of 17 November 2020–16 January 2021.

Figure 3.

(Original): Number of confirmed, active, recovered and deceased cases after one month from January 16, 2021. We have used the parameter set of the period of Nov 17, 2020- Jan 16, 2021 range [see Table 1] for this prediction

Figure 4.

(Original): Variation in estimated seroprevalence rate with time. The reported seroprevalence rates for July 1^st week, August 1^st week, September 1^st week and October 3^rd week are also provided as diamond shapes. In the last week of November this value is expected to reach above 30%. Thereafter, it is predicted to fall and reach around 24.4% at the end of the study period

We tried various simulations to understand the possibility of getting herd immunity by achieving a >65% seroprevalence rate, as calculated by the formula = 1-1/R_0.[17] Thus, we increased the value of transmission rate to 0.24, a value same as that during the third wave (8 October 2020–16 November 2020). We observed that the seroprevalence rate was projected to go beyond 50% after 156 days before falling, after reaching the maximum value of 54.96% [see Table 2]. In this case, we have considered 100% reinfection possibility of the individuals who lose their anti-SARS CoV-2 antibodies. We also captured the maximum seroprevalence rate for different possibilities of re-infection and for the case where antibodies persist for only 105 days [see Table 2 for details].

Table 2.

Predicted maximum seroprevalence rates for different reinfection possibilities. Here, we considered an increased value of transmission rate with the same obtained for the third wave for each case

Reinfection possibility	Persistence of antibodies (in days)	Maximum seroprevalence rate	Time point
100%	180	54.96%	196 days after January 16, 2021
	105	51.86%	165 days after January 16, 2021
95%	180	51.58%	203 days after January 16, 2021
	105	49.47%	164 days after January 16, 2021
75%	180	40.3%	209 days after January 16, 2021
	105	35%	173 days after January 16, 2021
50%	180	31.77%	201 days after January 16, 2021
	105	32.4%	December 3, 2020
25%	180	31.43%	December 9, 2020
	105	32.65%	December 4, 2020

Discussion

In our study, we saw that the maximum seroprevalence achievable in Delhi was less than 65%, suggesting that herd immunity, as predicted by serial sero-surveys, may not be achieved by natural infection alone. The maximum seroprevalence achievable was 54.96%, assuming that the re-infection rate and transmission rate was very high. Maximum seroprevalence achievable when parameters were unaltered was much lower (only 31.65%). The duration of persistence of antibodies, 180 vs 105 days, had minimal impact on the seroprevalence rate according to the prediction by our model. In both the scenarios, the required threshold of herd immunity i.e. 65% was not achieved.

Nevertheless, some limitations noted in our model are,

• 1) We have considered data of only one state (New Delhi) which may not be extrapolated to other states or regions. Since average seroprevalence for the whole state was taken, population heterogeneity in the different districts or zones was not accounted for.

• 2) In this model, antibody response, as measured by sero-survey, was taken as surrogate for immunity. Role of CMI (cell-mediated immunity), which may have persisted after waning of IgG antibodies, has not been studied in sero-surveys, and was not considered in the above model.

• 3) The 29% mark of seroprevalence of August 2020, could not be accounted for in our study.

• 4) Though our findings were validated internally, the availability of sero-surveillance data for only 4 months made external validation improbable.

• 5) Intervention with vaccines will have an impact on transmission rate, which was not considered in the prediction model.

Various factors leading to failed herd immunity could be,

1. Antibody response to COVID-19 though robust starts fading off in the early convalescent phase.[2] Immunity against SARS CoV-2, measured as antibody response, lasts for 3–6 months rendering the individuals susceptible to reinfection, which though rare is well documented.[5]

2. Secondly, among the asymptomatic individuals, only ~80% mount a measurable antibody response. While those that do not develop antibodies, potentially are susceptible to reinfection.

To the best of our knowledge, this is the first study that has used modelling for predicting future trajectories of sero-surveillance and its implications on herd immunity. Interestingly, a study had suggested the presence of possible cross-reacting antibodies for explaining the trend of sero-survey result in New Delhi and had predicted the seroprevalence rate in the city for the month of September.[18] However, unlike our ODE-based dynamical model it was an entirely statistical model. A good internal validation score makes our study a near-accurate prediction model. Also various levels of re-infections and duration of persistence of antibodies were considered to gauge the impact on seroprevalence.

Notably, sero-survey of Mumbai showed decrease in seroprevalence from 40% to 33.4% in the second survey, similar to that of New Delhi,[19] which suggests that a similar trend in Mumbai may be expected, as predicted for New Delhi. Results of serial sero-survey in other cities are not available and hence difficult to extrapolate.

The maximum seroprevalence predicted in our study was 54.96%, which is very close to the last sero-survey of New Delhi, conducted in January 2021, which showed a seroprevalence of 56%.[20] Also supporting our prediction, a devastating second wave of COVID-19 was witnessed during April and May 2021.

Primary care physicians being the first contact of patients should reinforce COVID appropriate behaviour for protection of vulnerable population. Also they should address vaccine hesitancy and motivate their patients for vaccination whenever appropriate.

Key points of our study are

1. If we consider seroprevalence to be surrogate of immunity against SARS CoV-2 in the community, natural infection alone cannot lead to attainment of herd immunity.

2. Repeated sero-surveys may not be much informative, at least till the time a sizable portion of the population is immunized.

Conclusion

This modelling study suggests that natural infection alone, as determined by serial sero-surveys, will not result in attainment of herd immunity in the state of Delhi, highlighting the importance of continuing COVID appropriate behaviour and role of widespread implementation of vaccination against COVID-19.

Financial support and sponsorship

Nil.

Conflicts of interest

There are no conflicts of interest.