Infection-Immunity Competition: A Simple Model for Illustrating the Background of Individual Response on Herd Immunity
Round 1
Reviewer 1 Report
I have to note that the proposed model contains a couple of assumptions, which have to improve in the next steps. For instance, the individual susceptibility (parameter s in equation 2) can be changed after antibodies production. Also, there are reports that demonstrated SARS-CoV-2 infection resulted in strong autoimmune disease means the immune system begins attacking healthy tissue instead of infected cells. For instance, data for concern autoimmune disease in rheumatoid arthritis can be found https://doi.org/10.1016/j.autrev.2020.102523 So the analyzed here infection – immunity correlation in case of SARS-CoV-2 not so simple and straightforward.
Some minor points in the manuscript have to be corrected.
- the equation of the competition model not provided. In the provided file the line 51 on page 2 is empty.
- I have no idea why the ticks in the x-axis (time interval) in Figures 2, 3 and 8 are rotated by 90o.
Author Response
Additional literature on immunity induction, quantitative effects and herd immunity is referenced.
Additional literature on immunity induction, quantitative effects and herd immunity is referenced.
Equation of competition model, line 51, p2:
The related equations for the model are provided as eq (3) and (4):
dv/dt = k*v(0)*[1-v(0)/v(max)]-r*v(0)*a(0) (3)
da/dt = sqrt{s*v(0)*[1-s*a(0)/a(max)]}-r*v(0)*a(0) (4)
For simulation, the iterative model of eqs. (1) and (2) was used with application of arbitrary chosen parameters k, r, s, v(max) and a(max) and a time step dt=1.
Fig 2,3,8 (rotated ticks)
The figures are corrected.
Reviewer 2 Report
The review manuscripts describes an important and currently relevant topic on how the interplay between virus infection and induced immune responses effect the establishment of herd-immunity. Unfortunately the review does not present any conceptual or novel ideas to the field and simplifies "immunity" as a response to viruses based on antibody induction that comes and goes. The way the manuscript is presented does not show much knowledge of the complexity of immune induction or virus/immune interplay, and therefore reads as very unconvincing.
Some comments:
- What is exactly meant by yes/no decision between susceptibility and immunity? This requires further explanation. As well as terms such as "individual immunity".
- What is meant with "low level impact of infecting object such as viruses"? Line 39
- What is meant with ratio of virus concentration and "individual immune situation"? Line 74
- Fig. 1: the dominance of virus over immunity on an individual scale is usually caused by a compromised immune system and not by the initial viral dose of infection.
- Fig.2: This shows a model where existing antibody titers are present before initial infection and the Abs disappear over time. How is this very unlikely scenario explained?
Author Response
The discussion on induction of immunity with respect to literature was improved. Therefore, the last part of introduction was extended:
The individual response on viruses is important for the formation of herd immunity, but herd immunity is a complex principle. In case of influenza, a mixing between social distancing, vaccination of children [9] and pre-existing antibodies from vaccination or earlier infection in older age people can contribute to her immunity [10]. An earlier study on HIV vaccines speaks for the possibility that herd immunity can be achieved even in case of imperfect vaccines [11].
In the following, some simple simulations will be presented which illustrate the effect of a step-wise low-level increase of immunity of an individual. The approach is related to the possible response on the level of individuals, but will be discussed in their consequences for the spreading of infection in a population and for supporting the development of herd immunity, too.
The meaning of the term “Yes/No decision” was clearified in the introduction:
Population models for herd immunization are based, mainly, on the concept of a clear presence or absence of immunity (“yes/no decision”).
The term “individual immunity was substituted by “immunity”
- What is meant with "low level impact of infecting object such as viruses"? Line 39
The related paragraph was modified:
The intention behind the term „low level impact“ is the assumption, that not only the strength of antibody production can vary in case of a massive infection, but there might also be an induction of the production of a lower number of antibodies or of less specific antibodies in case of an exposition against a lower concentration of antigens.
- What is meant with ratio of virus concentration and "individual immune situation"? Line 74
The term "individual immune situation" was substituted by “strength of the individual immune system”:
For the consequences of an impact of viruses the ratio of virus concentration and the strength of the individual immune system are important.
- 1: the dominance of virus over immunity on an individual scale is usually caused by a compromised immune system and not by the initial viral dose of infection.
The mentioned fact is very important. But, the figures should illustrate the consequences of a thinkable gradual response. For explanation, following text block was added:
The study presented here is based on two assumptions : It is assumed that the velocity of initial antibody production might depend on the strength of antigen exposure (viral dose). And it is assumed that immunity can drift from a safe immune state into a sensitive state by lowering of the antibody concentration. Vice versa, it is taken in mind that the manifestation of an infection can be dependent on the virus load and the virus doses in case of multiple exposure situations. It is well clear, that a certain level of antibodies can mean a safe protection over decades or the whole life, on the one hand. On the other hand, it should be respected, the immunity can got lost over time in some cases.
- 2: This shows a model where existing antibody titers are present before initial infection and the Abs disappear over time. How is this very unlikely scenario explained?
The scenario of Fig. 2 is a result of the above mentioned model and calculated by the same parameters as in case of the other simulations. For interpretation, an additional text block was added:
The scenario of Fig. 2 was obtained by the assumption of a certain concentration of antibodies which could bind to the infecting viruses. These antibodies can either be a residual of an earlier specific immune response with a low remaining concentration of specific antibodies or it can be regarded as a less specific immune response which was formerly induced by an exposure to a related class of viruses. The initial decrease in the antibody concentration can be interpreted by a fast consumption of antibodies by reactions between antibodies and viruses, the later increase by the stimulating effect of the virus exposure on the antibody production. The simple model presented, here, does not distinguish between a higher concentration of antibodies with lower specificity and a lower concentration of antibodies with a higher specificity.
Reviewer 3 Report
This work presents interesting hypothetical models that can be used in coronavirus infection, particularly in the acquisition of herd immunity. Although the coronavirus COVID-19 pandemic is recent, there are many references to refer to, the introduction is insufficient. Beyond this revision, the following issues should be addressed in preparing a revised version of the manuscript.
L36: "population dynamic(s) model"?
L52: "enhancement" could be "increase"?
L55-57: the role of v(max) and a(max) is not clear: what are these? max virus and antibody concentrations in the considered time interval? If so, during a phase of increasing concentration, it could be that v(max) = v(0) and the second term of the equation would be equal to 0. Or am I missing something? Additionally, it is not clear whether equations (1) and (2) refer to a hypothetical individual, a pool of individual/population, or something else. Please clarify.
Author Response
References on Covid19 for introduction:
The discussion of herd immunity plays a very important role for decisions of the management of the recent SARS Covid19 pandemic [1].
…
Model simulations have been also applied in order to predict the evolution of the SARS-CoV-2-pandemic [4].
L36: "population dynamic(s) model"?
Was corrected: … population dynamics models …
L52: "enhancement" could be "increase"?
Was corrected: The increase of virus concentration …
L55-57: the role of v(max) and a(max) is not clear: what are these? max virus and antibody concentrations in the considered time interval? If so, during a phase of increasing concentration, it could be that v(max) = v(0) and the second term of the equation would be equal to 0. Or am I missing something?
The values v(max) and a(max) have been introduced arbitrarily in order to limit the concentrations of viruses and antibodies to a maximum value. Their introduction leads to a reduction of growth rates in the simulation when the values are narrowing to the maximum values. Thus, v is always smaller than v(max):
v(max) > v(t) and a(max) > a(t)
These conditions are added into the manuscript, for clearify:
for v(max) > v(t) and a(max) > a(t) (5)
Additionally, it is not clear whether equations (1) and (2) refer to a hypothetical individual, a pool of individual/population, or something else. Please clarify.
The simple model describes possible responses on the level of individual. But, the discussed quantitative approach could have large importance of the development of herd immunity. For clarify, the end of introdution is modified:
In the following, some simple simulations will be presented which illustrate the effect of a step-wise low-level increase of immunity of individua. The approach is related to the possible response on the level of individuals, but will be discussed in their consequences for the spreading of infection in a population and for supporting the development of herd immunity, too.
Extended reference list:
[1] Tian, S.J.; Hu, N.; Lou, J. et al. Characteristics of COVID-19 infection in Beijing. J. Infection 2020, 80, 401-406.
[2] Fine, P.; Eames, K.; Heymann, D. L. "'Herd immunity': A rough guide”. Clinical Infectious Diseases. 2011, 52, 911-916.
[3] Daley, D. J.; Gani, J. Epidemic Modeling: An Introduction, NY, Cambridge University Press 2005.
[4] Richard, A.N.; Robert, D.; Valentin, D.; Emma, B.H.; Jan, A. Potential impact of seasonal forcing on a SARS-CoV-2 pandemic. Swiss Medical Weekly 2020, 150, w20224.
[5] Brauer, F.; Castillo-Chávez, C. Mathematical Models in Population Biology and Epidemiology. Springer 2001.
[6] Reyes-Silveyra, J.; Mikler, A.R. Modeling immune response and its effect on infectious disease outbreak dynamics. Theor. Bio. Med. Model. 2016, 13,10.
[7] Huisman, W.; Martina, B.E.E.; Rimmelzwaan, G.F.; Gruters, W; Osterhaus, A.D.M.E. Vaccine-induced enhancement of viral infections. Vaccine 2009, 27, 505-512.
[8] Hughes, L.J.; Twonsendl M.B.; Gallardo-Romerol, N. et al. Magnitude and diversity of immune response to vaccinia virus is dependent on route of administration. Virology 2020, 544, 55-63.
[9] Reichert, T.A.; Sugaya, N.; Fedson, D.S.; Glezen, W.P., Tashiro, M. The Japanese experience with vaccinating schoolchildren against influenza. New England Journal of Medicine 2001, 344, 889-896.
[10] Miller, E.; Hoschler, K.; Hardelid, P.; Stanford, E.; Andrews, N.; Zambon, N. Incidence of 2009 pandemic influenca A H1N1 infection in England: a cross-sectional serological study. Lancet 2010, 375, 1100-1108.
[11] McLean, A.R.; Blower, S.M.: Imperfect vaccines and herd-immunity to HIV. Proceed. Royal Soc. B 1993, 253, 9-13.
Round 2
Reviewer 2 Report
The authors have replied to the comments and answered the questions asked. Thereby they have improved the quality of the manuscript. However, this does not change the low impact of this study, which does not provide any conceptual novelties for the field.
