Compartmental models that dynamically divide the host population in categories such as susceptible, infected and immune constitute the mainstream of epidemiological modelling. Effectively such models treat infection and immunity as binary variables. We constructed an individual based stochastic model that considers immunity as a continuous variable and incorporates factors that bring about small changes in immunity. The small immunity effects (SIE) comprise cross immunity by other infections, small increments in immunity by sub clinical exposures and slow decay in the absence of repeated exposure. The model makes qualitatively different epidemiological predictions including repeated waves without the need for new variants, dwarf peaks (peak and decline of a wave much before reaching herd immunity threshold), symmetry in the upward and downward slopes of a wave, endemic state, new surges after variable and unpredictable gaps, new surge after vaccinating majority of population. In effect the SIE model raises alternative possible causes of the universally observed dwarf and symmetric peaks and repeated surges, observed particularly well during the Covid-19 pandemic. We also suggest testable predictions to differentiate between the alternative causes for repeated waves. The model further shows complex interactions of different interventions that can be synergistic as well as antagonistic. The model suggests that interventions that are beneficial in the short run can also be hazardous in the long run.

Herd immunity happens when a relatively large proportion of a population becomes infected by an agent, subsequently recovers, and attains immunity against the same agent. That proportion thus indirectly protects the naïve population by preventing the spread of the infection. Herd immunity has been suggested to interrupt and control the COVID-19 pandemic. However, relying on establishing herd immunity can be catastrophic considering the virulence and lethality of SARS-CoV-2. Meanwhile our understanding of the pathogenesis, case-fatality rate, transmission routes, and antiviral therapy for COVID-19 remains limited now. Interrupting or slowing the COVID-19 transmission seems more opportune than vaccination, antiviral therapy, or herd immunity, all of which will take some time to yield. Thus, social distancing, face-masking, and hygiene are the most appropriate immediate countermeasures. Because the social fabrics, economic implications, and local demands of various nations are unique, early relaxation of restrictions may seem hasty particularly when fatality rates are high, or when the healthcare systems could be inadequate or become inundated. Conclusively, avoiding any overwhelmingly risky approach in fighting the pandemic is prudent.

The pandemic associated with SARS-CoV-2 is a worldwide public health challenge. The WHO has proposed to achieve 70% COVID-19 vaccination coverage in all countries by mid-2022. Nevertheless, the prevention strategy based on COVID-19 vaccination and other applied prevention measures have not been sufficient to prevent SARS-CoV-2 epidemic waves. The study assessed the vaccination coverage that would be required to establish herd immunity against SARS-CoV-2 by taking into account virus transmissibility (Ro values from 1.1 to 10) and Covid-19 vaccination effectiveness. The study found that Covid-19 vaccination programs could establish herd immunity against SARS-CoV-2 with Ro < 3 with levels of Covid-19 vaccination effectiveness of 10−100% and against viruses with Ro values ranging from 3 to 10 with levels of Covid-19 vaccination effectiveness of 70−100%. Factors reducing Covid-19 vaccination effectiveness (emergent variants, reinfections, high risk individuals) and factors increasing SARS-CoV-2 transmissibility (close settings) increased percentages of vaccination coverage that would be required to establish herd immunity. The vaccination coverage objective of 70% could be adequate against SARS-CoV-2 with Ro values of 1.1−2.5, while percentages of vaccination coverage of 80% and 90% could be more adequate against viruses with Ro values of 2.5−3.5 and >3.5, respectively. On February 2022, the vaccination coverage for complete vaccination was lower than 70% in 73.2% of the countries of the world. Percentages of Covid-19 vaccination coverage must be increased in most countries of the world to increase individual and herd immunity levels in the population.

Background. The course of the COVID-19 epidemic process depends on population immunity which prevents pathogen spread among the population. Aim: to study the evolution of SARS-CoV-2 humoral immunity in the Belarusian population relative to COVID-19 pandemic dynamics. Materials and methods. The work was carried out according to a methodology for assessing population immunity developed by Rospotrebnadzor (Russia) and the Belarusian Ministry of Health with the participation of the St. Petersburg Pasteur Institute (SPPI), taking into account WHO recommendations. The study was approved by the Bioethics Committee of Belarus and the SPPI Bioethics Committee. Participant selection was carried out by questionnaire using a cloud (internet server) service. To monitor population immunity, a cohort of 4,661 people (participating in all stages of seromonitoring) was formed from the overall volunteer group. Volunteers were randomized by age group (1-17, 18-29, 30-39, 40-49, 50-59, 60-69, 70+ years), region, and professional group. For the detection of antibodies (Abs) to nucleocapsid (Nc) and S glycoprotein receptor-binding domain (RBD), corresponding assay systems were used following manufacturer instructions. The study was conducted in 4 stages according to a single scheme. Results. In the 1st stage (pandemic month 15), collective immunity was due mainly to Nc+RBD+ Ab status alone. By the 2nd stage (carried out after 4 months), their share decreased 1.2-fold, while the share of volunteers who had only RBD Abs increased 1.7-fold. In the 3rd and 4th stages (carried out after 9 and 19 months), the share of persons with RBD+Nc‒ compared to the 2nd stage decreased by 3.5%; the proportion of persons with Nc+RBD‒ Abs increased by 1.5-fold. The most important factor in population immunity was vaccination of the population, the coverage of which reached 70% by the 4th stage. Among vaccines, the Sputnik V and Sputnik Light vector designs were used most often. The whole-virion, inactivated BIBP-CorV vaccine was used less often. Conclusion. The evolution of collective SARS-CoV-2 humoral immunity included a set of changes in circulating Ab levels (Nc, RBD). The hybrid immunity formed helped to reduce the incidence to nearly zero.

In spite of numerous predictions, the natural herd immunity for covid-19 visru had not been approahed anywhere in the world. Thus, the traditional mathematical models of disease spread demonstrated their inability to describe adequately the covid-19 pandemic. In author's works, the novel model of the disease spread was developed. This model reflects the basic features of the covid-19 pandemic: a) the social clustering character of virus spread, b) . Social clustering is mathematically modelled with ultrametric spaces having the treelike geometry encoding hierarchy of the regulation constraints. The virus spread is described by ultrametric diffusion or random walk on the hierarchic energy landscape. In contrast to the standard models which are characterized by the exponential decrease of the probability to become infected - at the stage of approaching of the herd immunity, the ultrametric model is characterized by the power law. Moreover, the model gives the possibility to quantify the influence of restriction measures up to the lockdown. Our main result is that the play with restrictions, including lockdowns, is counterproductive and leads to the essential slowdown of approaching the herd immunity or even makes this impossible.

We present a model of infection dynamics that might explain slower approaching the herd immunity during the covid-19 epidemy in Sweden than it was predicted by a variety of other models; see graphs Fig. \ref{GROWTH2}. The new model takes into account the hierarchic structure of social clusters in the human society. We apply the well developed theory of random walk on the energy landscapes represented mathematically with ultrametric spaces. This theory was created for applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, the virus should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider linearly increasing (with respect to hierarchy's levels) barriers. This structure of barriers matches with a rather soft regulations imposed in Sweden in March 2020. In this model, the infection spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This model's feature matches with the real situation during the covid-19 epidemy, with its cluster spreading structure. Clusters need not be determined solely geographically, they are based on a number of hierarchically ordered social coordinates. The model differs crucially from the standard models of spread of disease, such as the SIR-model. Our model describes such a specialty of spread of covid-19 virus as the presence of ``super-infectors'' who by performing a kind of random walk on a hierarchic landscape of social clusters spreads infection. In future, this model will be completed by adding the SIR-type counterpart. But, the latter is not a specialty of covid-19 spreading.

Deciding when to replace dairy bulls presents a complex challenge for artificial insemination (AI) companies. These decisions encompass multiple factors, including a bull's age, predicted semen production, and estimated genetic merit. This study's purpose was to provide a practical, objective tool to assist in these decisions. We utilized a Markov Chain model to calculate the economic valuation of dairy bulls, incorporating key factors such as housing costs, collection and marketing expenses, and the bull's probable tenure in the herd. Data from a leading AI company were used to establish baseline values. The model further compared a bull's net present value to that of a potential young replacement, establishing a relative valuation (BullVal$). The range of BullVal$ observed spanned from -$316,748 to $497,710. Interestingly, the model recommended culling for 49% of the bulls based on negative BullVal$. It was found that a bull's net present value was primarily influenced by market allocation and pricing, coupled with the interaction of semen production and genetic merit. This study offers a robust, data-driven model to guide bull replacement decisions in AI companies. Key determinants of a bull's valuation included market dynamics, semen production rates, and genetic merit.

Nietzsche is almost always regarded as one of the thinkers who advocate extreme individualism, totally indifferent to or exclusively polemical towards the public human dimension. While this is very difficult to contradict, if we read his texts carefully we can see how his constant celebration of the individual runs parallel to an acute awareness of living in a new era, which he defined as ‘the century of the multitude and the masses’. The herd, conformism, mediocrity, public opinion: a civilisation in which community attempts suffocate all individual inspiration, and which therefore seems to row in the opposite direction. Although Nietzsche often uses collective life merely as a negative pole for more effectively emphasising the individual, his provocative words—pushed to the limits of the inexorable victory of the herd and of the paradoxical impossibility of all that is ‘public’—offer us a direct testimony of the tragic way of life of the man of his time. This provides us with an extremely clear and interesting phenomenological cross-section of the social sphere, as well as a very finely tuned and valuable seismograph for the continual monitoring of our everyday coexistence with and perception of the constantly incumbent dangers of its degeneration.

In this article we have presented a new perception of herd immunity threshold (HIT) which considers that only a “band of population” are susceptible to any pathogenic infection. This is termed as the “effective herd immunity threshold” (EHIT) and the progression of the disease (caused by this pathogenic infection) is mainly determined by this EHIT value. We have argued here that this EHIT value (considering the immunity band picture in the population) will be substantially lower than the estimated canonical HIT values obtained from various existing models. We propose that the actual prediction of the disease progression should now be calculated using the EHIT values.

Herdsmen use different techniques, as per varying geographies and cultures, to keep the cohesion within herds and avoid animals getting lost or predated. However, there is no study on the social behaviour of yaks and on herdsmen management practices. Therefore, this ethology study was initiated by ethnographic inquiries. In Manang, the success of the shepherd is dictated by his personal attribute of “Khula man” or open-heartedness. This attribute refers to good intentions and emotions such as empathy that allows the shepherd to focus more on others than on ownself. This cultural method of assessing the skills required to become a successful and knowledgable shepherd guided us to study the effect of cultural values on the herd’s social behavior. We collected data from two herds living at the same settlement (Yak kharka, 4,100 m altitude, Nepal) by equipping them with loggers. One of the herdsman used the tether rope while other did not. Moreover, the Thaku herd had a more proactive shepherd than the Phurba one. In each herd, 17 animals were equipped with one actigraph wgt3x-BT to measure activity using accelerometer and spatial associations using proximity recorder. One of the herds was equipped with GPS (N=11) as well. Using GPS locations and activity, we showed that the two herds were cohesive and synchronised their activities but the herd with the tether rope was more cohesive. The shepherds also have personal knowledge of the social relationships of their herds and use these relationships to keep the group cohesive and to well manage cattle.

To date, viruses closely related to SARS-CoV-2 have been reported in four bat species: Rhinolophus acuminatus, Rhinolophus affinis, Rhinolophus malayanus, and Rhinolophus shameli. Here, we analysed 343 sequences of the mitochondrial cytochrome c oxidase subunit 1 gene (CO1) from georeferenced bats of the four Rhinolophus species identified as reservoirs of SARS-CoV-2-like viruses. Haplotype networks were constructed in order to investigate patterns of genetic diversity among bat populations of Southeast Asia. No strong geographic structure was found for the four Rhinolophus species, suggesting high dispersal capacity. The ecological niche of SARS-CoV-2 like viruses was predicted using the four localities of bat SARS-CoV-2-like viruses and the localities where bats showed identical or very similar CO1 haplotypes than virus-positive bats. The ecological niche of SARS-CoV-like viruses was deduced from the localities where bat SARS-CoV-like viruses were previously detected. The results show that the ecological niche of SARS-CoV2-like viruses includes several regions of mainland Southeast Asia whereas that of SARS-CoV-like viruses is mainly restricted to China. In agreement with these results, human populations in Laos, Vietnam, Cambodia, and Thailand appear to be much less affected by the Covid-19 pandemic than other countries of Southeast Asia. In the climatic transitional zone between the two ecological niches (southern Yunnan, northern Laos, northern Vietnam, and possibly Hainan and Taiwan), genomic recombination between highly divergent viruses is more likely to occur. Since recombinant viruses can threaten the benefit of vaccination campaigns, these regions should be under surveillance.

We analyze the Covid-19 mortality scenario in India and compare it with those in other large-population regions such as Asia-excluding-China, Africa, European Union, South America, and USA. We compare existing fatality data and offer an interpretation for low fatality based on immunity due to endemic malaria and TB. We identify the hot climate in the past summer as a possible cause for low death count in southern-hemisphere countries without endemic malaria and TB. We also make India-specific observations for easing the lockdown and estimations for the time required to attain herd immunity. Whatever optimism we present should be viewed as a guarded optimism. There should not be room for complacency.

Since December 2019, a rapid increase in the number of SARS-CoV-2 (COVID-19) cases was reported worldwide, despite strict infection control and lock down measures. Current paper investigated the actual facts behind this rapid increase in the number of cases. Study of genomic sequence reveals that domestic and wild animals were likely ancestors and zoonotic source for SARS-CoVs, MERS-CoVs, and SARS-CoV-2. Strong evidence suggest that these viruses already existed and replicated in animals and humans during past several decades, exhibiting diverse mutations, evolutions and self-limiting diseases, except during outbreaks. Serious zoonotic reservoir investigations are required to investigate animal transmission of SARS-CoVs and SARS-CoV-2 to limit current pandemic. This might be the reason of increasing number of cases via animals. SARS-CoV-2 has been retrospectively isolated in different studies in August 2019, several months before Wuhan announced. Hence, there is a possibility that viruses existed, went undetected, infecting subclinically, in past several years, and SARS-CoV-2 antigens and neutralizing antibodies may have been present in humans since long time. This might be another reason of increasing number of cases by screening as mass screening and antigen or antibody testing was not carried out in the past years. Randomized controlled trials are required to investigate human to human transmission by touch, as the current evidence is limited with conflicting results. As all SARS-CoVs are basically respiratory viruses, droplet precautions and infection control measures are essential, especially for hospital staff. Increased number of SARS-CoV-2 asymptomatic, or subclinical cases are detected worldwide. This silent phase of transmission can be beneficial for humans. Lack of symptoms eventually lessen virus transmission and reduce the pathogen's long-term survival and provide humoral herd immunity up to several years. Hence, seropositivity with diverse antibodies develops against mutating SARS-CoVs which will confer strong immunity during epidemics. Strategies such as identification, contact tracing and quarantine are costly and practically difficult. Hence, asymptomatic persons can continue their work with droplet precautions and standard infection control procedures, while symptomatic or sick persons can isolate themselves in their homes without the need for strict quarantine until clinical recovery, with reduced hospital visits and minimizing chances of hospital acquired infections. RT-PCR has low sensitivity and specificity, carries a high risk of handling live virus antigens, and requires difficult protocols. As viral load also sharply declines after few days of onset of infection, this technique might overlook infection. Furthermore, SARS-CoV-2 infection may be present in blood when oropharyngeal swabs are negative by RT-PCR. Additionally, RT-PCR usually gives false negative and false positive results and must be interpreted cautiously. This might be again a reason of increasing number of cases by false positive RT-PCR reporting. Moreover, antibodies against SARS-CoVs develop robustly in serum even by reduced amount of antigens. In contrast to RT-PCR, ELISA for diagnosing antibodies against SARS-CoV-2 demonstrates 100% specificity and 100% sensitivity, even in clinically asymptomatic individuals. These antibodies can be used for serologic surveys, monitoring and screening. However, screening tests for SARS-COV-2 should be avoided in unhygienic public places by nasopharyngeal swabs, which carry a high risk of further transmission, co-infection or superinfection. Such highly infectious virus must be isolated and tested in highly sterilized laboratory. Further strict international laws and policies are required to stop the possible spread of experimental viruses, biological warfare and bioterrorism.

We have analysed the death and recovery rate of Covid-19 disease progression. From the analysis, we have argued that the pandemic is over in certain countries (labelled as group-A) and for other countries (labelled as group-B) the disease appears to remain as endemic. Taking into account the serological survey (sero-survey) test results obtained by certain groups and comparing it with herd immunity threshold value one can infer that the low number of infection for group-B is either due to acquired immunity by some previous infection by other coronavirus or due to innate immunity towards this infection. This effect is stronger for group-B to slow the progress of the disease to such an extent resulting in flattening of the disease progression curve compared to group-A.

As an on-going pandemic caused by the out-break of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) or simply COVID-19 sweeps through the globe at an unprecedented rate leaving behind trails of high infection and mortality, it is crucial to understand the propagation dynamics of the virus in a host population in order to take urgent and effective remedial and mitigating steps to save life. It is already observed in many countries and communities that accurate and timely testing, tracing, and tracking of the infection lead to better containment and slowing down of the spread. In this exploratory research, the early growth dynamics of infection within a population is pursued based on real data. The study posits that the early growth in a homogenous population follows an exponential pattern motivating further rigorous quantitative treatment based on a number of analytical models such as logistic model, Richard’s model, and Gompertz model– the acceleration pattern of the outbreak is ascertained from the daily inflection data, and regression analysis against population models yields dynamic growth indices which allow very accurate prediction of the successive outbreak size when calibrated continually with updated data. The performance of the various models is evaluated with the real dataset. More, the basic reproduction number of the COVID-19 virus propagation in the community is estimated based on the on-set phase dataset using multi-compartmental epidemiological model. Also, the maximum infection size, infection doubling time and the scope of the herd immunity are also inferred for COVID-19 in a population.

We present a general approach to define reproduction ratios or numbers to monitor the outbreak of epidemics that are modeled by mathematical evolution equations. This provides a solution to an important topic that has not been completely settled in the literature, especially in the case of complex epidemiological models. We illustrate our procedure with a full implementation of a standard deterministic SEIR model that is applied to examine the Covid-19 outbreaks and the effects of intervention measures in several countries in America (Argentina, Brazil, Mexico, USA) and Europe (France, Italy, Spain and UK) in 2020. Our code is also used to investigate herd immunity levels for Covid-19, indicating values between 85% and 90%.

We present a new mathematical model of disease spread reflecting specialties of covid-19 epidemic by elevating the role social clustering of population. The model can be used to explain slower approaching herd immunity in Sweden, than it was predicted by a variety of other mathematical models; see graphs Fig. \ref{GROWTH2}. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider homogeneous trees with $p$-branches leaving each vertex. Such trees are endowed with algebraic structure, the $p$-adic number fields. We apply theory of the $p$-adic diffusion equation to describe coronavirus' spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling {\it dynamics on energy landscapes.} To move from one social cluster (valley) to another, the virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider {\it linearly increasing barriers.} This structure matches with mild regulations in Sweden. The virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the covid-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. We present socio-medical specialties of the covid-19 epidemic supporting our purely diffusional model.

Severe acute respiratory syndrome coronavirus 2 (SARS-Cov-2/2019-nCoV) infection is an emerging pandemic. The virus binds to angiotensin converting enzyme 2 (ACE2) and TMPRSS2 which are abundantly expressed on various human cells including lung epithelial cells and intestinal cells and the virus can infect these cells. Currently no specific treatments or vaccines are available for this disease. A per oral live attenuated vaccine can be beneficial in SARS-Cov-2 infection because the attenuated virus initially infects the gut, stimulates the mucosa associated immune system sparing the respiratory system during the initial immune response. The live virus can also spread in the community boosting herd immunity.