Super-Spreaders Out, Super-Spreading In: The Effects of Infectiousness Heterogeneity and Lockdowns on Herd Immunity
SSuper-Spreaders Out, Super-Spreading In: The Effects ofInfectiousness Heterogeneity and Lockdowns on Herd Immunity
Jhonatan Tavori ∗ Hanoch Levy † January 25, 2021
Abstract
Recently, [8] has proposed that heterogeneity of infectiousness (and susceptibility) acrossindividuals in infectious diseases, plays a major role in affecting the Herd Immunity Thresh-old (HIT). Such heterogeneity has been observed in COVID-19 and is recognized as overdis-persion (or ”super-spreading”). The model of [8] suggests that super-spreaders contributesignificantly to the effective reproduction factor, R , and that they are likely to get infectedand immune early in the process. Consequently, under R ≈ Personal-Trait- , and (2)
Event-Based-
Infectiousness (Susceptibility). The former is a personal trait of specific indi-viduals ( super-spreaders ) and is nullified once those individuals are immune (as in [8]). Thelatter is event-based (e.g cultural super-spreading events) and remains effective throughoutthe process, even after the super-spreaders immune. We extend [8]’s model to account forthese two factors, analyze it and conclude that the HIT is very sensitive to the mix between(1) and (2), and under R ≈ adverse effects and reduce the herd immunity. ∗ Blavatnik School of Computer Science, Tel Aviv University, Israel. E-mail: [email protected] † Blavatnik School of Computer Science, Tel Aviv University, Israel. E-mail: [email protected] a r X i v : . [ q - b i o . P E ] J a n Introduction
In 1923 Topley and Wilson described experimental epidemics in which the rising prevalenceof immune individuals would end an epidemic. They named this phenomenon as ”Herd-Immunity” [14]. Once the Herd Immunity Threshold (HIT, measured in fractions of thepopulation that got immune) is surpassed, then the effective reproduction number, R , reducesbelow 1 and the number of infection cases decreases. The exact value of this threshold is animportant measure used in infectious disease control and immunization, and its estimationfor the COVID-19 disease are used by governments worldwide in determining policies to fightagainst the current pandemic.A recent study of Oz, Rubinstein and Safra [8] proposed a new model for the spreading ofinfectious diseases, such as COVID-19. Under that model and the assumption that the basicreproduction number, R , of COVID-19 is approximately 3, the HIT is approximately 5%,namely, when 5% of the population is infected herd immunity is reached. This estimation wasin contrast to the allegedly “axiomatic” cutpoint of HIT ≈
67% assumed for COVID-19 [10].Preliminary data from COVID-19 suggests that herd immunity is not reached at the ap-proximate 5% fraction expected by [8]. For Example, the US states of North Dakota, SouthDakota, Iowa, Utah and Tennessee with at least 12%, 11.5%, 9.5%, 9.5% and 9% infected,respectively. Four of them have current
R > overdispersion or super-spreading ( super-spreaders are a class of individuals whose secondary infection rate is very high [5]). The estimates forthe COVID-19 pandemic fits this property and asserts that between 5% to 10% of the infectedindividuals cause 80% of the secondary infections [3, 7].Furthermore, a correlation between the infectiousness and susceptibility of each individualhave a drastic effect on the over-time reduction of the effective reproduction number, R , underthe spreading model of [8]. The heterogeneity and correlation of these parameters yields thatthe ”super-spreaders” are extremely likely to get infected and develop immunity in an earlystage of the pandemic process. [8]’s 5% estimation of the percentage of the population thatcontract the disease before herd immunity is reached was based on these properties.In this work, we follow-up on [8] regarding the effect of infectiousness heterogeneity on theHIT. However, we propose that infectiousness (and susceptibility) should be classified into twoinherently different types: (1) Personal-Trait Infectiousness (Susceptibility) , and (2)
Event-Based Infectiousness (Susceptibility) . We will refer to the combination of the infectiousnessand the susceptibility as
Spreading . 2he first type stems from traits of an individual. The second type relates to social events inwhich every individual may participate, regardless of its personal traits. To demonstrate thesetwo types, consider, for example, a package-delivery person and compare it with an academicresearcher. During a single day, the delivery person has interactions with tens or hundreds ofpeople, and therefore has high personal-trait infectiousness. On the other hand, the researchermay work most of the time in his/her office or interact with a small research group and as aresult has a lower personal-trait infectiousness. Yet, both of them may participate in a social-gathering event (such as a concert, a wedding, or ”just” a family birthday party). Duringsuch an event, both have approximately the same amount of interaction (which may be quitelarge), and therefore have the same event-based spreading degree, regardless of their personaltraits. It is important to note that personal-trait infectiousness relate not only to the socialbehaviour of the individual; it may also relate to his/hers biological properties (e.g., his/herbody reproduces a virus faster and therefore he/she is more infectious).We assume that the likelihood parameters of each individual consist each of the sum of twoparameters. (1) S p ( a ) and I p ( a ) which are the personal-trait susceptibility and infectiousnessparameters of a , respectively. Those values reflect personal traits and are drawn once (pan-demic beginning) and remain constant throughout, exactly as in [8]. (2) S e and I e which arethe event-based (cultural) susceptibility and infectiousness parameters. Those values reflectoccasional event-based spreading which is probabilistically redrawn for each individual at everystep of the pandemic. We assume that the likelihood of a to be infected is: S ( a ) = p · S p ( a ) + (1 − p ) · S e and the likelihood of a to infect others is I ( a ) = p · I p ( a ) + (1 − p ) · I e , where p determines the weight of each infection type. The symmetry between S(a) and I(a) issimilar to that of [8] and stems from the assumption that infectiousness level and susceptibilitylevel are proportional to the level of interaction the individual has with others or to its biologicalproperties.We use this model to analyze the progression of an infectious disease and the value of R ( n ),the effective reproduction number, as a function of the fraction of population infected. Weshow that the contribution of the personal-trait spreading drops sharply at early stages of thedisease, as the super-spreaders are likely to contract the disease and develop immune at earlystages of the process. On the other hand, the contribution of the event-based spreading dropsmuch more slowly, and is affected very little at early stages, as its reduction is proportionalto the decrease of the susceptible population size. In other words – when super-spreaders are3ut (event-based) super-spreading is in. Hence, R ( n ) may remain at high values even after thesuper-spreaders population is fully immune, as opposed to [8]. This results in a slower decayof R ( n ) and leads to a higher value of the Herd Immunity Threshold.We show that for COVID-19 the Herd Immunity Threshold depends on the mix ( p ) betweenthe weights of the personal-trait and event-based spreading. In particular, we prove that the HIT ≈
5% estimate of [8] holds when assuming only personal-trait spreading, and that thetraditional prediction of
HIT ≈
67% holds when assuming only event-based spreading.Having established a formula expressing R ( n ), we address operational aspects and analyzethe effects of lockdowns on the Herd Immunity Threshold. Lockdowns, of a variety of variants,have been enforced worldwide in order to fight the COVID-19 pandemic. While lockdownsmight have immediate impact such as collapse of the effective reproduction and suppressionof infections and mortality, they have long-term impact as well. We discuss two differentlockdown policies: (1) An Event-based spreading targeted lockdown (e.g shut down of culturalevents). (2) A Personal-trait spreading targeted lockdown (e.g. restricting daily/professionalactivities). We analyze the effect of these policies on the composition of the infected population,on the effective reproduction number, and on the Herd Immunity Threshold (HIT). We showthat while a lockdown (or a sequence of lockdowns) which is targeted at event-based spreadingreduces the disease spread by decreasing the HIT, a lockdown (or a sequence of lockdowns)which is targeted at personal-trait spreading will act adversely and will increase the diseasespread by increasing the HIT.The rest of the paper in organized as follows: In Section 2 we formally describe our model,and present the effective reproduction number which plays a major role in the analysis. InSection 3 we develop an expression for R ( n ), and calculate the Herd Immunity Threshold fora general-case disease. We examine the result in a numerical discussion based on COVID-19spreading distributions. Then, in Section 4, we study various lockdown policies, and analyzetheir effect on the Herd Immunity Threshold. Finally, concluding remarks are given in Section5. In this section, we present our model for the spreading of infectious disease accounting forheterogeneity of infectiousness/susceptibility. We extend the model of [8] and propose thatthere exists an event-based infection factor in addition to the factors described in their model.Our analysis begins with a certain number of infected individuals. We measure the spread ofthe disease as a function of the number of individuals who got infected. Specifically, we indexthe individuals by the order they are infected and have R ( n ) denote the effective reproduction4umber associated with the n th infected individual. Namely, the event whereby the n thindividual gets infected is the n th event (or step n ). We use n also to denote the step of thedisease .Measuring the spread of the disease as a function of the infected population size will beuseful in deriving the Herd Immunity Threshold (HIT) of the disease, namely the fraction ofthe population that gets infected prior to reaching R ( n ) ≤ We follow the model of [8] and assign to each individual a a personal-trait-susceptibility pa-rameter S p ( a ) and a personal-trait-infectiousness parameter I p ( a ) drawn from some probabilitydistributions. Those values quantify how likely a is to be infected and infect others, respec-tively, according to its personal traits. The values of S p ( a ) and I p ( a ) accompany a throughoutthe entire progress of the disease, and remain at the same values. We follow [8] and define theaverage conditional infectiousness ϕ ( s ). In our case, it is logical to parametrize ϕ ( s ) only bythe personal-trait susceptibility and infectiousness: ϕ ( s ) := E S p ( a )= s [ I p ( a )] . (1)As was discussed, the heterogeneity of the spreading values of the population will play amajor role in our analysis. Hence, we will measure: ρ ( s, n ) := Pr h S p ( a ) = s (cid:12)(cid:12)(cid:12) a ∈ H n i (2)where H n is the healthy population at step n .In addition, and beyond the model of [8], we assign an event-based infectiousness param-eter and event-based susceptibility parameter to each individual. Those values are subject tochange through the progress of the disease. At step i we assign to a S ie ( a ), the event-based-susceptibility parameter and I ie ( a ), the event-based-infectiousness parameter, both are randomvariables whose realizations hold only for iteration i . Since the values of S e , I e measure theevent-based-spreading of the society (assigned to its individuals at a given time), they aredrawn from probability distributions that are common for the entire population, denote themby Λ S and Λ I .The susceptibility of a at step i , which is the likelihood of a to be infected, is: S i ( a ) = p · S p ( a ) + q · S ie ( a ) , (3)and the infectiousness of a , which is the likelihood of a to infect others is I i ( a ) = p · I p ( a ) + q · I ie ( a ) (4) For continuous s , Eq. (2) should be considered as a density function. < p < q = 1 − p . The value of p determines the mix between the spreading types.We call p (and respectively, q ) the weights of the personal-trait (and respectively, event-based)spreading of the disease. As will be seen later, the value of p will have a drastic effect on theHerd Immunity Threshold. Note that the special case where p = 1 gives exactly the model of[8]. Under this model, the probability that a will be infected at step n , assuming that a washealthy at step n − h a is the n th infected (cid:12)(cid:12)(cid:12) a is health in step n − i = S n − ( a ) P b ∈ H n − S n − ( b ) (5) The basic reproduction number , R , is a measure of how transferable a disease is. It is defined asthe expected number of secondary cases produced by a single (typical) infection in a completelysusceptible population (whose size is N ).In reality, varying proportions of the population are immune to any given disease at anygiven time. Hence, as in [8], we will measure the effective reproduction number , R ( n ), which isdefined as the expected number of infections directly generated by the n th infected individual. R ( n ) = E [ of individuals that will be infected by the n th infected] (6)where the expectation is taken over the n th individual to be infected. As in [8], Eq. (6) equalsto R ( n ) = E [ I n ( a ) · X b = a S n ( b )] == X a ∈ H n − S n − ( a ) P b ∈ H n − S n − ( b ) · I n ( a ) · X b = a ∈ H n − S n ( b ) == X a ∈ H n − P b = a ∈ H n − S n ( b ) P b ∈ H n − S n − ( b ) · S n − ( a ) · I n ( a )this can be approximated by: R ( n ) ≈ X a ∈ H n − S n − ( a ) · I n ( a ) . (7)Using Eq. (3), (4) and (7) we have: R ( n ) ∼ N ( n ) · Z ρ ( σ, n ) · ( p · σ + q · λ S ) · ( p · ϕ ( σ ) + q · λ I ) dσ (8)where N ( n ) is the size of the healthy population at step n and λ I , λ S are the means of Λ I , Λ S ,respectively. The expectation is taken over all possible scenarios of infection. Reaching Herd Immunity
In this section, we analyze the changes in the composition of the population through the spreadof the disease, and the decrease of R ( n ) as the fraction of the population that contracted withthe disease increases. We prove the following theorem: Theorem 3.1 (General Case Herd Immunity Threshold) . For any δ when − Z ρ ( σ ) · exp ( − δ · ( p · σ + q · λ S )) dσ (9) fraction of the population is infected, the effective reproduction number, R () , will be reduced bya factor of R ρ ( σ, · exp ( − δ · ( p · σ + q · λ S )) · ( p · σ + q · λ S ) · ( p · ϕ ( σ ) + q · λ I ) dσ R ρ ( σ, · ( p · σ + q · λ S ) · ( p · ϕ ( σ ) + q · λ I ) dσ (10) relatively to the basic reproduction number, R . The threshold for herd immunity is when thevalue of the effective reproduction number is . Having this expression for the change in the effective reproduction number for a generaldistribution, we will later use the special case of the Gamma distribution with estimatedparameters for COVID-19 [3, 5, 6] and inspect the HIT values for different p values. In order to prove Theorem 3.1, we establish the following claim:
Claim 3.2 (The likelihood of an individual to be infected) . For a person a , Pr[ a is healthy at round n ] ≈ exp ( − β ( n ) · ( p · S p ( a ) + q · λ S )) (11) where β ( n ) = n − X i =0 N ( i ) · E b ∼ H i [ S i ( b )] . (12) Proof of Claim 3.2.
The proof follows the proof of Claim I provided in [8] with modificationsrequired for our extended model. The proof is based on using Eq. (5) and obtaining:Pr[ a is healthy at step n ] = − S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] ! · Pr[ a is healthy at step n − . (13)The rest of the proof consists of algebraic manipulations of (13) and the full details are givenin Appendix A. (cid:3)
7e next establish two supporting lemmas, corresponding to equations (3.6) and (3.7) in[8], and conclude with the proof of Theorem 3.1.
Lemma 3.3 (Heterogeneity of the population during the process) . For any s ∈ Supp( S p ) , ρ ( s, n ) ≈ ρ ( s, · exp ( − β ( n ) · p · s ) R ρ ( σ, · exp ( − β ( n ) · p · σ ) dσ Proof of Lemma 3.3.
By definition ρ ( s, n ) = ρ ( s, · Pr[ a is healthy at round n | S ( a ) = s ] R ρ ( σ, · Pr[ a is healthy at round n | S ( a ) = σ ] dσ . (14)Using Eq. (11), ρ ( s, n ) ≈ ρ ( s, · exp ( − β ( n ) · ( p · s + q · λ S )) R ρ ( σ, · exp ( − β ( n ) · ( p · σ + q · λ S )) dσ = ρ ( s, · exp ( − β ( n ) · p · s ) R ρ ( σ, · exp ( − β ( n ) · p · σ ) dσ . (cid:3) Lemma 3.4 (The size of the susceptible population) . For any n ∈ [ N ] , N ( n ) ≈ N · Z ρ ( σ, · exp ( − β ( n ) · ( p · σ + q · λ S )) dσ. (15) where N ( n ) is the size of the susceptible population at step n , and N is the total size of thepopulation.Proof of Lemma 3.4. By definition, N ( n ) ≈ N · Z ρ ( σ, · Pr[ a is healthy at round n | S p ( a ) = σ ] dσ. (16)Using Eq. 11, we have Eq. (15). (cid:3) Proof of Theorem 3.1.
Denote r ( σ ) := ( p · σ + q · λ S ) · ( p · ϕ ( σ ) + q · λ I ) . Using Eq. (8) we know that: R = N · Z ρ ( σ, · r ( σ ) dσ. We develop the ratio: R ( n ) R = N ( n ) · R ρ ( σ, n ) · r ( σ ) dσN · R ρ ( σ, · r ( σ ) dσ . ρ ( s, n ) and the ration N ( n ) /N and have: R ( n ) R = R ρ ( σ, · exp ( − β ( n ) ( p · σ + q · λ S )) dσ · R ρ ( σ, · exp( − β ( n ) · p · σ ) R ρ ( σ , · exp( − β ( n ) · p · σ ) dσ · r ( σ ) dσ R ρ ( σ, · r ( σ ) dσ == R ρ ( σ, · exp ( − β ( n ) · p · σ + q · λ S ) · r ( σ ) dσ R ρ ( σ, · r ( σ ) dσ . Since the value of expression (9) is given by:1 − N ( n ) N we have that Eq. (10) holds. (cid:3) We move to demonstrate the results of Theorem 3.1 on a Gamma distribution with shape andscale parameters k and θ , respectively. The Gamma distribution was previously attributedto the infectiousness of COVID-2 [5]. We substitute the estimates for COVID-19: R ≈ k ≈ .
1. [3, 6, 8]. We assume that the personal-trait-susceptibility and personal-trait-infectiousness of the population are highly correlated. I.e., we set ϕ ( s ) = s . This stemsfrom assuming that both personal-trait infectiousness and susceptibility levels are correlatedto social interaction levels of the individual or to its biological properties.In Figure 1 we demonstrate the decay of the effective reproduction number, R ( n ), andits contributing factors, classified by their spreading types. We plot the following values as afunction of the fraction of the infected population. R ( n ) = N ( n ) · Z ρ ( σ, n ) · ( p · σ + q · λ S ) · ( p · ϕ ( σ ) + q · λ I ) dσ (17) R p ( n ) = N ( n ) · Z ρ ( σ, n ) · p · σ · ϕ ( σ ) dσ (18) R e ( n ) = N ( n ) · Z ρ ( σ, n ) · q · λ S · λ I dσ (19) R mix ( n ) = R ( n ) − R p ( n ) − R e ( n ) (20)9igure 1: The over-time reduction in the effective reproduction number, R ( n ), and its con-tributing factors as a function of n , assuming p = 0 . k = 0 . R = 3. Note that n (horizontal-axis) is normalized to percentage. In red - R ( n ) (Eq. (17)); In blue - R p ( n ) (Eq.(18)); In Green - R e ( n ) (Eq. (19)); In yellow - R mix ( n ) (Eq. (20));The blue curve in Figure 1 depicts the contribution of the personal-trait spreading to R ( n ), while the green curve depicts the contribution of the event-based spreading. As canbe seen, the contribution of the personal-trait spreading drops at early stages of the disease.This results from the assumption that infectiousness is positively correlated with susceptibility,since if more infectious people are also more susceptible, then they have higher probability tobe infected and develop natural immunity much sooner (than the less infectious individuals).On the other hand, the contribution of the event-based spreading is affected very little at earlystages. Its reduction is proportional to the decrease of the susceptible population, which islinear in n . The weight of each spreading type, which is determined by p , determines thecombined behaviour of R ( n ) .In Figure 2 we plot the Herd Immunity Threshold as a function of p . This demonstratesthe effect of the weight of the personal-trait spreading on the decay of R ( n ).10igure 2: The Herd-Immunity Threshold (HIT) as a function of p assuming Gamma distribu-tion with shape parameter k = 0 .
1. In green - R = 3. In red - R = 9.When assuming only event-based-spreading (i.e., p = 0) the HIT is approximately (cid:18) − R (cid:19) , (21)i.e., 67% when R ≈ R ≈
9. This follows the classical models [1, 2, 4, 10]. Onthe other hand, assuming only personal-trait-spreading (i.e., p = 1) the HIT is approximately5% when R ≈ R ≈
9. The 5% matches the expected threshold given by [8].In reality, the two types of spreading contribute to infections and hence 0 < p <
1; Inorder to predict the Herd Immunity Threshold, one has to estimate the value of p , given apopulation. In Figure 3 we plot the effective reproduction number, R ( n ), throughout thespread of the disease for a number of p values.11igure 3: The effective reproduction number R ( n ) as a function of n throughout the diseasespread for different p values where R = 3 and k = 0 .
1. Note that n (horizontal-axis) isnormalized to percentage.Figure 3 and the shapes of the various curves can be used to assist in predicting the value of p at early stages of a disease. As can be seen, even when a small percentage of the populationis infected ( < R ( n ) values for different p ’s.Such prediction can assist in predicting the Herd Immunity Threshold.The current data available to us for the current-spreading COVID-19 is biased due tolockdowns of many types, which makes it challenging to estimate p and q . This estimationproblem is left open for forthcoming research.The effect of lockdown policies on the value of R ( n ) and the Herd Immunity Threshold isthe subject of our next section. Lockdown strategies, of a variety of variants, have been used worldwide as a major meansto fight an epidemy, specifically COVID-19. The question addressed in this section is howsuch strategies affect the scope of the disease, namely how they affect the Herd Immunity12hreshold. We show that, depending on its type, a lockdown may either increase or decreasethe size of the population infected prior to reaching herd immunity. In particular, we show thata personal-trait spreading targeted lockdown acts adversely on the efforts to reduce the spreadof a disease since it increases the threshold. In contrast, an event-based spreading targetedlockdown affect positively the disease blocking and reduces the Herd Immunity Threshold.
Large scale physical distancing measures and moving-around restrictions, often referred to as lockdowns , can slow disease transmission by limiting contacts between people. These days,many restrictions are used worldwide in order to slow down the spread of the COVID-19pandemic. We classify the restrictions into two inherently different types: (1) a Personal-trait spreading targeted lockdown, for example by closing or restricting workplaces. (2) anEvent-based spreading targeted lockdown, for example by prohibition on cultural events.Formally, these lockdowns will be defined as follows:
Definition 4.1 (Personal-Trait Spreading Targeted Lockdown) . During a personal-traitspreading targeted lockdown, the infectiousness and susceptibility of an individual a is: I i ( a ) = p L · I p ( a ) + q · I ie ( a ) , S i ( a ) = p L · S p ( a ) + q · S ie ( a ) , where q remains the same and p L = 0 . Definition 4.2 (Event-Based Spreading Targeted Lockdown) . During an event-based spread-ing targeted lockdown, the infectiousness and susceptibility of an individual a is: I i ( a ) = p · I p ( a ) + q L · I ie ( a ) , S i ( a ) = p · S p ( a ) + q L · S ie ( a ) , where p remains the same and q L = 0 . We will use the following notation. Assume that a lockdown starts at the n b th step ofthe disease and ends at step n e . For any n ∈ [ n b , n e ], let R L ( n ) be the expected value of theeffective reproduction number during the lockdown; For any n > n e let R L ( n ) be its expectedvalue after the lockdown is released and p (or q ) returns to its original value. Similarly, let HIT be the Herd Immunity Threshold assuming a ”natural” spread of the disease (i.e., nolockdown), and
HIT L be the threshold assuming a lockdown was performed. Note that thecomparison between the natural evolution and the lockdown evolution is based on the numberof individuals that contract the disease during those evolutions. I.e., the coupling between theevolutions is done user by user by the order of infection rather than by the time period inwhich the policies are compared. This coupling method is useful in deriving HIT and
HIT L .13t n b we set p L = 0 or q L = 0. Hence, the value of the effective reproduction number islikely to drop during the lockdown. The question we answer in this section is if – and what –will be the long-term impact of the lockdown, after it is released (at n > n e ). Since at n e thevalue of p (or q ) will increase, R L ( n ) is most likely to increase as well. Yet, will it pass R ( n )?Will it stay below it? Consequently, what will be the impact on the Herd Immunity Threshed?The next two theorems establish that any personal-trait spreading targeted lockdown willincrease the HIT while any event-based spreading targeted lockdown will decrease the HIT. Theorem 4.3 (HIT of Personal-Trait Spreading Targeted Lockdown) . Assume that a personal-trait spreading targeted lockdown was performed for n ∈ [ n b , n e ] . Then for any n > n e , R ( n ) < R L ( n ) . (22) Consequently,
HIT < HIT L . Theorem 4.4 (HIT of Event-Based Spreading Targeted Lockdown) . Assume that an event-based spreading targeted lockdown was performed for n ∈ [ n b , n e ] . Then for any n > n e , R L ( n ) < R ( n ) . (23) Consequently,
HIT > HIT L . Due to the generality of Theorems 4.3 and 4.4, we state the following corollary:
Corollary 4.5.
Any sequence of personal-trait spreading targeted lockdowns will result with
HIT < HIT L . Similarly, any sequence of event-Based spreading targeted lockdowns will resultwith HIT > HIT L . The following claim will be useful in proving Theorems 4.3 and 4.4.
Claim 4.6.
Let S , S be continuous random variables. Let ρ () , ρ () be their pdfs, and P () , P () their CDFs, respectively. Let r () be an injective monotone function. If for any s > s : ρ ( s ) ρ ( s ) > ρ ( s ) ρ ( s ) or ρ ( s ) ρ ( s ) < ρ ( s ) ρ ( s ) then E [ r ( S )] ≤ E [ r ( S )] . Proof of Claim 4.6.
The proof is given in Appendix A. (cid:3) .2 Personal-Trait Lockdown: Proving Theorem 4.3 Recall that ρ ( s, n ) is the normalized susceptibility distribution (density) at step n assuming a”natural” spread of the disease (i.e., no lockdown). Let ρ L ( s, n ) be the normalized susceptibilitydistribution (density) at step n assuming that a lockdown was performed. Proof of Theorem 4.3.
Resulting from Lemma 3.3 and by the assumption that p L = 0 for n b ≤ n ≤ n e , at the end of the (personal-trait) lockdown, i.e. at n e , it holds that ρ L ( s, n e ) = ρ L ( s, n b ) = ρ ( s, n b ) . (24)However, ρ ( s, n e ) ≈ ρ ( s, n b ) · exp ( − β ( n b , n e ) · ( p · s )) R ρ ( σ, n b ) · exp ( − β ( n b , n e ) · ( p · σ )) dσ . (25)Therefore, ρ L ( s, n e ) ρ ( s, n e ) ≈ exp ( β ( n b , n e ) · ( p · s )) · c (26)where c = R ρ ( σ, n b ) · exp ( − β ( n b , n e ) · ( p · σ )) dσ . Therefore, for any s > s : ρ L ( s , n e ) ρ ( s , n e ) > ρ L ( s , n e ) ρ ( s , n e ) . (27)Note that: r ( s ) = ( p · s + q · λ S ) is an injective monotone function. Hence, by the definitionof R ( n ) (Eq. (8)) and from Claim 4.6: R ( n e ) < R L ( n e ) . (28)This is demonstrated in Figure 4.More specifically, Since ρ L ( s, n e ) = ρ ( s, n b ) (Eq. (24)), and by Eq. (8), it holds that R L ( n e ) = N − n e N − n b R ( n b ) . (29)We move to measure the value of the effective reproduction number after the lockdownends. Let n > n e , and denote x = n − n e N − n e . Let n = n b + x · ( N − n b )(note that n = n e + x · ( N − n e )). n and n are depicted in Figure 5.At the end of the lockdown p returns to its original value and the disease spreads naturally,with its original parameters. At that point ( n e ), according to Eq. (24), the distribution ofthe population is the same as it was before the lockdown began, where only the size of the15igure 4: The expected value of R L ( n ) under a personal-trait spreading targeted lockdown(vs. R ( n ) in a natural evolution) assuming p = 0 . R = 3 and k = 0 .
1. The lockdown beginsat n b = 5% and ends at n e = 30%.susceptible population was changed. In other words, R L (red) behave to the right of n e exactlyas R (blue) behaves to the right of n b (see Figure 5). Therefore, R L ( n ) R L ( n e ) = R ( n ) R ( n b ) . (30)This is demonstrated in Figure 5. Combining Eq. (30) with Eq. (29), R L ( n ) R ( n ) = N − n e N − n b . (31)Let us look at ˜ r := N ( n ) · Z ρ ( σ, n ) · ( p · σ + q · λ S ) · ( p · ϕ ( σ ) + q · λ I ) dσ. (32)Hence, ˜ rR ( n ) = N − nN − n . Note that: N − nN − n = N − ( n e + x · ( N − n e )) N − ( n b + x · ( N − n b )) =16igure 5: The expected value of R L ( n ) under a personal-trait spreading targeted lockdown(vs. R ( n ) in a natural evolution) assuming p = 0 . R = 3 and k = 0 .
1. The lockdown beginsat n b = 5% and ends at n e = 30%. We use x = 10%. The dashed purple line demonstrates alinear reduction in the value of R ( n ). As proven, the purple and red curves intersect at n .= N − n e − x · N + x · n e N − n b − x · N + x · n b = (1 − x ) · ( N − n e )(1 − x ) · ( N − n b ) = N − n e N − n b . Using Eq. (31) ˜ r = R L ( n ) . In Eq. (32) ˜ r was calculated using ρ ( σ, n ). Hence, as in Eq. (27) and (28), R ( n ) < ˜ r, and we have that R ( n ) < R L ( n ). This holds for any n > n e , and we conclude that HIT Under an event-base lockdown, for any n ≥ n e and for any s > s , ρ ( s , n ) ρ L ( s , n ) > ρ ( s , n ) ρ L ( s , n ) . (33) Proof of Claim 4.7. When the lockdown begins, we have: ρ ( s, n b ) = ρ L ( s, n b ) . (34)For any n b ≤ n ≤ n e , ρ ( s, n ) ≈ ρ ( s, n b ) · exp ( − β ( n b , n ) · ( p · s )) R ρ ( σ, n b ) · exp ( − β ( n b , n ) · ( p · σ )) dσ (35)and ρ L ( s, n ) ≈ ρ L ( s, n b ) · exp ( − β L ( n b , n ) · ( p · s )) R ρ L ( σ, n b ) · exp ( − β L ( n b , n ) · ( p · σ )) dσ . (36)The ratio between the density functions is: ρ ( s, n ) ρ L ( s, n ) = ρ ( s, n b ) ρ L ( s, n b ) · exp ( − β ( n b , n ) · ( p · s ))exp ( − β L ( n b , n ) · ( p · s )) · R ρ L ( σ, n ) · exp ( − β L ( n b , n ) · ( p · σ )) dσ R ρ ( σ, n ) · exp ( − β ( n b , n ) · ( p · σ )) dσ . Hence: ρ ( s, n ) ρ L ( s, n ) = exp( p · s · ( β L ( n b , n ) − β ( n b , n )) · c (37)where c = R ρ L ( σ,n e ) · exp( − β L ( n b ,n e ) · ( p · σ )) dσ R ρ ( σ,n e ) · exp( − β ( n b ,n e ) · ( p · σ )) dσ . We establish the following claim. Claim 4.8. For any n b < n ≤ n e , β L ( n b , n ) > β ( n b , n ) . Proof of Claim 4.8. We will prove the claim using induction on n .18 (Base case). Since q L = 0 ( < q ) we have that: S n b L ( a ) < S n b ( a ) for any a . Hence, β L ( n b , n b + 1) > β ( n b , n b + 1).• (Inductive step). Let n ∈ ( n b , n e ). Assume that β L ( n b , n ) > β ( n b , n ) . By Eq. (37), for any s > s , ρ ( s , n ) ρ L ( s , n ) > ρ ( s , n ) ρ L ( s , n ) , and using Claim 4.6, E [ S nL ( a )] < E [ S n ( a )]Using the definition of β and by our assumption, β L ( n b , n + 1) > β ( n b , n + 1) . we conclude that for any n b < n ≤ n e , β L ( n b , n ) < β ( n b , n ). (cid:3) Next, we apply Claim 4.8 for n e and have that β L ( n b , n e ) − β ( n b , n e ) > 0. Therefore, byEq. (37) for any s > s ρ ( s , n e ) ρ L ( s , n e ) > ρ ( s , n e ) ρ L ( s , n e ) , (38)and according to Claim 4.6, R L ( n e ) < R ( n e ), where R and R L are calculated using the originalvalues of p and q . This is demonstrated in Figure 6.In a similar develop as in Eq. (37), we have that for any n > n e , ρ ( s, n ) ρ L ( s, n ) = exp( p · s · (( β L ( n b , n e ) + β L ( n e , n )) − ( β ( n b , n e ) + β ( n e , n ))) · c · c . (39)where c = R ρ L ( σ,n ) · exp( − β L ( n e ,n ) · ( p · σ )) dσ R ρ ( σ,n ) · exp( − β ( n e ,n ) · ( p · σ )) dσ . We establish the following claim, similar to Claim4.8: Claim 4.9. For any n e < n , β L ( n e , n ) > β ( n e , n ) . Proof of Claim 4.9. The proof follows the same idea of the proof of Claim 4.8.The full proofis given in Appendix A. (cid:3) We conclude the proof of the claim using Claim 4.9 and Eq. (39). It holds that ∀ s > s , ∀ n ≥ n e , ρ ( s , n ) ρ L ( s , n ) > ρ ( s , n ) ρ L ( s , n ) . (cid:3) R L ( n ) under an event-based spreading targeted lockdown (vs. R ( n ) ina natural evolution) assuming p = 0 . R = 3 and k = 0 . 1. The lockdown begins at n b = 5%and ends at n e = 30%. Proof of Theorem 4.4. Combining Claim 4.7 with Claim 4.6 we have that for any n ≥ n e : R L ( n ) < R ( n ) . (40)Consequently, HIT L < HIT , and Theorem 4.4 follows. (cid:3) .Note that the proof of Theorem 4.4 was not based on the assumption that q L = 0, but onthe fact that q L < q (see the proof of Claim 4.8). Hence, Theorem 4.4 can be generalized tohold for any 0 ≤ q L < q , namely for Partial Lockdown . Corollary 4.10 (Generalization of Theorem 4.4) . Any event-based spreading targeted partiallockdown, namely where p remains at the same level and q L < q , results in HIT L < HIT . Remark 4.11 (Generalization of Theorem 4.3) . We conjecture that the generalization of The-orem 4.3 for personal-trait spreading targeted partial lockdown holds as well. Conclusions and Discussion In this work we studied the effects of infectiousness heterogeneity (overdispersion) and lock-downs on herd immunity. Recent literature suggests that COVID-19 is characterized by suchheterogeneity which affects dramatically herd immunity. We proposed that infectiousness (andsusceptibility) should be classified into two inherently different types which we called personal-trait and event-based spreading.We followed up [8] and proposed a model that accounts for both of these types. Under thisnew model we showed that herd immunity and the HIT strongly depend on the mix between thetwo types of spreading. We analyzed the decay of the effective reproduction number, R (), andshowed that the contribution of the personal-trait spreading drops sharply at early stages of thedisease while the contribution of the event-based spreading drops much more slowly. That is -the super-spreaders ”leave the game” at early stages, while the super-spreading events remainactive.We demonstrated the results on a (Gamma) distribution which previously attributed tothe infectiousness of COVID-2 [5] and COVID-19 [3, 8]. We showed that in order to predictthe mix between the spreading types, only a small fraction of the population is needed to getinfected. Such prediction can help in calculating the HIT in an early stage of the disease. Thisestimation problem for COVID-19 is left open for forthcoming research.We addressed operational aspects of disease blocking and analyzed the effect of lockdownson the HIT. We showed that different lockdown strategies, targeting different spreading types,result in opposite effect on the HIT. In particular, a lock-down which targets personal-traitspreading would act adversely and reduces herd immunity. This seems to fit a lockdown thatfocuses on daily/professional activities. In contrast, a lock-down which targets event-basedspreading will increase herd immunity. This may fit a lockdown that focuses on sports/socialevents.Of course, a lockdown may have other objectives such as achieving temporary slow down ofthe disease spreading (to allow handling the patients masses) which may justify the lockdownstrategy. Yet – the effect on herd immunity requires consideration, especially due to theopposite effects of various lockdowns. References [1] The second (1926) milroy lecture on experimental epidemiology. The Lancet , 207(5350):531 – 537, 1926. ISSN 0140-6736. doi: https://doi.org/10.1016/S0140-6736(00)92941-6.Originally published as Volume 1, Issue 5350.212] R. M. Anderson and R. M. May. Infectious diseases of humans: dynamics and control .Oxford university press, 1992.[3] A. Endo, S. Abbott, A. J. Kucharski, S. Funk, et al. Estimating the overdispersion incovid-19 transmission using outbreak sizes outside china. Wellcome Open Research , 5(67):67, 2020.[4] P. Fine, K. Eames, and D. L. Heymann. “herd immunity”: a rough guide. 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Reviews of modern physics , 87(3):925, 2015.[10] H. E. Randolph and L. B. Barreiro. Herd immunity: Understanding covid-19. Immunity ,52(5):737–741, 2020.[11] K. Rock, S. Brand, J. Moir, and M. J. Keeling. Dynamics of infectious diseases. Reportson Progress in Physics , 77(2):026602, 2014.[12] Rt.live. Current r value in the us, 2020. URL https://rt.live .[13] A. V. Tkachenko, S. Maslov, A. Elbanna, G. N. Wong, Z. J. Weiner, and N. Goldenfeld.Persistent heterogeneity not short-term overdispersion determines herd immunity to covid-19. arXiv preprint arXiv:2008.08142 , 2020.[14] W. Topley and G. Wilson. The spread of bacterial infection. the problem of herd-immunity. Epidemiology & Infection , 21(3):243–249, 1923.2215] Worldometers.info. Covid-19 coronavirus pandemic, 2020. URL . A Proofs Proof of Claim 3.2 (continued). The proof follows the proof of Claim I provided in [8]. Takingnatural log of Eq. (13),log(Pr[ a is healthy at round n ]) − log(Pr[ a is healthy at round n − − S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] ! . (41)It holds that: log − S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] ! == − S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] − O S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] ! . We attempt to find the herd-immunity threshold, and hence can bound number of steps, n , by(1 − /R ) · N . Hence, log − S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] ! == − S n − ( a ) N ( n − · E b ∼ H n − [ S n − ( b )] − O max b S n − ( b ) N · E b [ S n − ( b )] ! . Counting over the steps 1 , . . . , n we have:log(Pr[ a is healthy at round n ]) = − n − X i =1 S i ( a ) N ( i ) · E b ∼ H i [ S n ( b )] − O (cid:18) max b S n ( b ) E b [ S n ( b )] (cid:19) · nN ! . And as long as max b S n ( b ) E b [ S n ( b )] (cid:28) √ N we have that:log(Pr[ a is healthy at round n ]) ≈ − n − X i =1 S i ( a ) N ( i ) · E b ∼ H i [ S n ( b )] . (42)Note that for any i , S i ( a ) = p · S p ( a ) + q · S ie ( a ) . a is healthy at round n ]) ≈ − β ( n ) · ( p · S p ( a ) + q · λ S ) (43)Hence, Pr[ a is healthy at round n ] ≈ exp ( − β ( n ) · ( p · S p ( a ) + q · λ S )) (44)and the proof is complete. (cid:3) Proof of Claim 4.6. Let s > 0. By our assumption, for any s < s : ρ ( s ) ρ ( s ) > ρ ( s ) ρ ( s ) . Hence, integrating s over (0 , s ) we have: Z s ρ ( s ) ρ ( s ) − ρ ( s ) ρ ( s ) ds > . Therefore: ρ ( s )P ( s ) = Z s ρ ( s ) ρ ( s ) ds > Z s ρ ( s ) ρ ( s ) ds = P ( s ) ρ ( s ) . (45)By the same ideas we have that:(1 − P ( s )) ρ ( s ) = Z ∞ s ρ ( s ) ρ ( s ) ds > Z ∞ s ρ ( s ) ρ ( s ) ds = ρ ( s )(1 − P ( s )) . (46)Using Eq. (45) and Eq. (46) we have:1 − P ( s )1 − P ( s ) > ρ ( s ) ρ ( s ) > P ( s )P ( s )and hence: ∀ s, P ( s ) < P ( s ) . (47)Since r is an injective monotone function, it holds that:Pr[ S > s ] = Pr[ r ( S ) > r ( s )] . (48)In the same way: Pr[ S > s ] = Pr[ r ( S ) > r ( s )] . (49)Using Eq. (47) combined with Eq. (48) and 49):Pr[ r ( S ) > r ( s )] < [ r ( S ) > r ( s )] . (50)24or any non-negative random variable X it holds that: E [ X ] = Z ∞ − F X ( s ) ds. (51)Therefore, by Eq. (50) combined with 51 we know that: E [ r ( S )] ≤ E [ r ( S )] . And the proof is complete. (cid:3) Proof of Claim 4.9. We will prove the claim using induction on n .• (Base case). Using Eq. (38), we have that: E [ S n e L ( a )] < E [ S n e ( a )] . Hence, β L ( n e , n e + 1) > β L ( n e , n e + 1) . • (Inductive step). Let n e < n . Assume that β L ( n e , n ) > β ( n e , n ) . By Claim 4.8 and Eq. (39), for any s > s , ρ ( s , n ) ρ L ( s , n ) > ρ ( s , n ) ρ L ( s , n ) , and using Claim 4.6, E [ S nL ( a )] < E [ S n ( a )]Using the definition of β and by our assumption, β L ( n e , n + 1) > β ( n e , n + 1) . we conclude that for any n e < n , β L ( n e , n ) < β ( n e , n ). (cid:3)(cid:3)