Computational analysis on Covid-19 transmission escalates into new imuuno-epidemiology considerations.

Background For Severe Acute Respiratory Syndrome Coronavirus-2, the investigation of the heterogeneity of individual infectiousness is important due to the recorded widespread cross reactive immunity of general population that can alter transmission dynamics. We therefore aimed to understand how SARS-COV-2 transmits in the general population in relation to age. Design Using a sample of infected population with SARS-COV-2 in close geographical proximity to the initial Severe Advanced Respiratory Syndrome-1 (SARS-1) outbreak, we explored the association between infector’s age and dispersion (or heterogeneity) of individual infectiousness (k) in order to investigate the relatedness with the age of an individual’s capability to disperse SARS-COV-2. Results We have found a negative association between k and increase of infector’s age. Signicantly this becomes more evident for the age group of 20-60 years comparing with the infectors of younger age.


Introduction
The resistance of the "host-donor" de nes the e cient transmission of the infectious agents [1]. This resistance can be described as the immunity of individuals raised to disable transmission [1][2]. The previous Severe Acute Respiratory Syndrome-1 (SARS-1) patients, even after 17 years post infection poses a sustained-well-developed, speci c T cell memory response against SARS Coronavirus-1 (SARS COV-1). This speci c immunity is intensi ed according to the increased severity of previous SARS-1 clinical condition [2]. These SARS-1 patients whilst all poses a long term cross reactive immunity against SARS-1-COV, also all (to a tested sample of 23), poses reacting T cells to the N peptides of SARS-COV-2 [3]. Further, the unexposed individuals with no previous history of SARS-1 infection, and negative for Nucleocaspid protein (N) antibodies and neutralizing antibodies of SARS-COV-2, to a 51.35 % (19/37), have also a reactive T cell immunity against SARS-COV-2 N, and non structural proteins (nsp) proteins of SARS-COV-2 (50 % and 50 % respectively) [3]. Also a widely distributed T cell cross reactivity (35 % -60 %) amongst unexposed individuals extends against to the SARS-COV-2 spike (S) protein [4][5].
The speci c memory immunity of unexposed individuals is attributed either to immune cross reactions with common u coronavirus infections [4][5] or to the involvement of animal species cross transmitting coronaviruses to human [3]. Since there is a 17 year interval from SARS-1 epidemic and the developmental stages of immunity can be severely in uenced [6][7] by the cross barrier transmission of coronaviruses (including common cold coronaviruses) between animal species and human individuals (as shown in gure 1), we have selected speci cally to investigate the variation of dispersion parameter k with age increase, bearing in mind the concept of higher k à less heterogeneity of age group à more di cult to control the epidemic [8]. We considered that investigation on this route could identify distinct variation in individual infectiveness with the new virus and thus identify variations on the e ciency of individuals to spread SARS-COV-2 between speci c age groups. Figure. 1. Cross species barrier transmission between human and animals by coronaviruses. The cross species barrier infection is achieved by coronaviruses using suitable receptors that enable this transmission. Molecules like the carcinoembryonic antigen-related adhesion molecules (CEACAM) comprise a family of antigens which are highly preserved between animal and human species and their participation can lead to a wide immune cellular dispersal of coronaviruses throughout the human organism [9].
Patients And Methods
The dataset contains 1407 transmission pairs that are identi ed and reconstructed according to the previous studies [8,21], governmental new release, and o cial situation reports.
We identi ed 807 infectors, and who act as source of infection to transmit to the infectee. We extract the information, including age and gender, of each infector as well as the number of offspring infectees generated by each infector. After excluding the infector with missing information on age, we collected 777 infectors for further analysis.

Heterogeneity of individual infectiousness: a statistical modeling perspective
We consider the variation in the individual-level infectiousness as a quanti able scale that affects the distribution of offspring infectee generated by an infector. Following the previous study [21] , we introduce the number of offspring infectee generated by an infector, denoted by r, as a random variable from a Gamma distribution, denoted by h(), with mean R (> 0) and dispersion parameter k (> 0). Thus, we have r ~ h(R, k). Here, R is the reproduction number that is de ned as the expected (or average) number of secondary cases caused by one typical infected individual.
The dispersion parameter k governs the dispersiveness of the Gamma distribution. As demonstrated theoretically in the previous study [21], with R xed, a larger k results in a lower effectiveness of non-pharmaceutical interventions in controlling the epidemics, which is also discussed in another study [8].
Poisson process with rate r, denoted by f(r), is adopted to address the stochastic effects in transmission, and to govern the number of infectee caused by each infector, denoted by Z (≥ 0). [22]. Thus, we have Z ~ f(r) = f(R, k).
Straightforwardly, f(R, k) is an Negative Binomial (NB) distribution with mean R and variance R•(1 + R/k). By the de nition of NB distribution, the probability that one infector generates (j−1) offspring infectees, i.e., cluster size of j (≥ 1), which is denoted by Pr(Z = j−1) = L j , is given in Eqn (1).
Here, Γ(•) denotes the Gamma function. Specially, the NB distribution f(•) reduces to a Geometric distribution when k = 1, and it reduces to a Poisson distribution when k approaches in nity. Importantly, a smaller value of k indicates larger heterogeneity in individual infectiousness.
By tting distribution f(•) to the real-world observations, we may estimate value of dispersion parameter k, and explore the determinants of k.

Likelihood inference framework and subgrouping by infector's age
We considered observed samples of number of offsprings from N infectors. We denoted the number of infectors who have j infectees associated by n j (≥ 1). Note that all j > 1 in our dataset though j may be 1 theoretically or observed in other datasets, and thus we adjusted for this truncation in our likelihood framework.
Straightforwardly, we have . Then, following the previous studies [23,24], we constructed the likelihood function, denoted by L, as in Eqn (2).
We estimated the dispersion parameter k using the maximal likelihood estimation approach. To explore the association between the infector's age and k, we repeated the above tting and estimation procedure after subsetting the dataset into subgroups by the infector's age. We considered 76 age bins, and they include 5-25, 6-26, …, 79-99, and 80+ years. We estimate the value of k for each age bin to examine the association between the infector's age and k.

Results
We estimated the k ranges from 0.4 to 1.5 for different age bins, which is line with previous studies [8,23,25]. We observe an evident downward trend in k as the infector's age increases, with p-value < 0.001 for linear trends testing using Student's t test. We detect a structural break [26] at age bin 20-40 years, in which k drops 47% with pvalue < 0.001 comparing with the infector with younger age, presented in gure 2.

Discussion
We have employed the metric named dispersion of individual infectiousness, denoted by k, which was rst proposed for SARS-1 [21] because it was used to quantify the role of heterogeneity of individuals in transmitting Covid-19 as well as the di culty in controlling the epidemics with Non Pharmaceutical Interventions (NPIs) at population scale [23,25]. We performed a statistical calculation of k, and explored to de ne possible relatedness of infector's age as a determinant of k. We estimated the k ranges from 0.4 to 1.5 for different age bins in line with the previous studies [8,23,25]. We observe an evident downward trend in k as the infector's age increases, with pvalue < 0.001 for linear trends testing using Student's t test. Moreover, we detect a structural break at age bin 20-40 years, in which k drops 47% with p-value < 0.001 comparing with the infector with younger age (see gure 2).
The dataset [see [20] for full list of transmission pairs] we used to investigate variation in heterogeity of individual infectiveness with age initially contained 1407 transmission pairs [25], out of which we have identi ed 807 infectors. Out of these infectors, we selected 777 pairs with adequate with information about age.
In the investigation of individual infectiveness we kept the reproduction number R constant in order to measure the variation of individual dispersiveness, k, with age. The k value decreases with age increase, with the important difference being between the two age groups 0-20, and 20-60 years of age (see gure 2). This is important, as their difference in k value lies in the heterogeneity of each sub-population group [8,21]. For youth age as k is larger, the heterogeneity of subgroup population is smaller and for the older age group as k, decreases the heterogeity between individual increases [21,25,27]. As presented in gure 3, when hypothesizing that R equals to 2, which is a realistic scenario for SARS-COV-2 [28], for four seed cases of youth and old age respectively, the offspring cases will be eight for each age group. This means that almost all youth age infectors will produce two offspring cases, whereas for old age infectors, almost half will not be able to produce any offspring case, one will be able to produce six offspring cases and one two offspring cases.
Re ecting the change of individual infectiveness with age and thus their heterogeneity, NPIs, are more applicable to the sub-population older than 20 years of age [21], whereas NPIs are not expected to provide an adequate solution [21,29], for disease spread containment from younger age infectors. Given the situation that a high proportion of youth age remains asymptomatic but highly infectious [6,30], this makes the contact tracing effort in this group even more di cult but urgent. Speci c screening strategy for the youth population to identify as more possible positivity will make restriction contact measures more e cient as almost all young infected individuals are likely to transmit the disease. The notable differences in the heterogeity of individuals across the age groups of 0-20, 20-60 and slightly over 60-90 years old may re ect de-similarities between developments of immune surveillance mechanisms due to environmental cross transmission reactions [10][11][12][13][14][15][16][17][18][19]. These are encountered in previous SARS-1 patients and unexposed individuals [3][4][5]. Due to Covid-19 overspread and consistence, the targeted pharmaceutical interventions may be appropriate to lower seed cases in population groups with small heterogeneity. As our results show, by focusing on youth age population to prevent from spreading SARS-COV-2 to rest of general population, this will help to contain pandemic in a similar way to SARS-1.

Conclusions
NPI's are expected to work e ciently for population between ages of 20-60 years old as with SARS-1. However, this seems not to be the case for younger population infecting SARS-COV-2. For this population other measures are needed to contain infection to general population. For older than 60 years of age sub-population the heterogeneity of infectiousness also decreases but not in the same way as for youngsters. The differences of dispersion ability between age groups can re ect differences in memory immunity acquired post the SARS-COV-1 outbreak as time interval matches. Special cross reactive immunity studies to detect memory immunity in the young population group are needed. Similar testing of dispersion parameter (k) in European and USA are needed to discover trends of hererogeneity across age groups.