1. Introduction
Coronaviruses (CoV) are a broad family of viruses, with symptoms that vary from those of the common cold to those of more serious diseases, e.g., Middle East Respiratory Syndrome (MERS-CoV) and Severe Acute Respiratory Syndrome (SARS-CoV). Several coronaviruses can spread between animals and humans, which means they are zoonotic viruses. According to Haider et al. [
1], COVID-19 should be classified as an “emerging infectious disease (EID) of probable animal origin”. Based on detailed investigations, SARS-CoV was transmitted from civet cats to humans, whereas MERS-CoV was spread from camels to humans [
2,
3,
4]. Furthermore, several coronaviruses that have not yet infected humans have been identified in animals.
Coronavirus disease 2019 (COVID-19) is a contagious disease caused by the SARS-CoV-2 virus [
5]. The disease spreads mainly through human-to-human transmission; however, there have been several reports of disease spread between humans and some animals as well. SARS-CoV-2 ribonucleic acid (RNA) has been identified in animals that have had contact with infected humans, such as owners, caregivers, or anyone who came into close contact with the animals. Animals infected with the virus have been documented all over the world, including minks on mink farms such as American mink (
Neogale vison), dogs, domestic cats, hyenas, snow leopards, lions, tigers, a binturong, raccoon dogs, non-human primates, otters, a fishing cat, hippopotamuses, a coatimundi, manatees, a giant anteater, white-tailed and mule deer, a black-tailed marmoset, and wild mink near mink farms [
6,
7,
8,
9,
10,
11,
12,
13,
14]. However, thus far, animal-to-human transmission has been observed in the cases of farmed mink in Europe and the US, pet hamsters in Hong Kong, white-tailed deer in Canada, and a cat in Thailand [
6,
12,
15,
16,
17]. To the best of our knowledge, the World Organisation for Animal Health (WOAH) has received reports of SARS-CoV-2 in farmed mink from the following countries: Denmark, the Netherlands, France, Latvia, Lithuania, Poland, Greece, Italy, Spain, Sweden, Canada, and the USA [
12,
18,
19,
20,
21]. The possibility of human-to-mink and mink-to-human transmission has also been established [
12,
15,
16,
17]. The modes of SARS-CoV-2 virus transmission between minks and humans are presented in
Figure 1.
Denmark is the world’s largest mink producer, and the country has 1500 farms that produce mink skins valued at EUR
billion [
22]. In the summer of 2020, a farm in the Danish region of North Jutland announced the first case of COVID-19 infection in farmed mink in Denmark [
23]. Despite the fact that the animals were symptom-free, three of the first farms to be identified were culled [
22,
24]. Since the decision to cull all minks in Denmark to prevent infection to humans was announced on 4 November 2020, over 17 million minks have been culled [
25]. About
of farms were infected with COVID-19 during the first cull period [
15]. Denmark’s authorities commanded a provisional ban on mink farming in December 2020 (initially, until the end of 2021, and then later extended until the end of 2022) [
13]. Given the way mink farming was discontinued, the unclear situation surrounding the pandemic’s course, and the efforts of animal rights activists, it seems doubtful that mink farming in Denmark will be able to return to its full potential once the COVID-19 epidemic is under control [
22].
A considerable number of scientific works have appeared that assess the epidemiological characteristics of COVID-19 in order to reduce its burden on public health (e.g., [
26,
27,
28,
29,
30]). Rasmussen et al. [
31] developed an SEIRS model that included deaths outside of hospitals, as well as independent assessments of cases with and without symptoms, with varied immune memories. The model was adjusted to account for the progression of the epidemic observed in Denmark. According to the findings, COVID-19 has a low mortality rate since most of the infected individuals are either symptom-free or have mild symptoms. As a result, only a small number of affected people require hospitalization. Valentin et al. [
32] used an SEIR-type model to identify the basic reproduction number of the epidemic in Denmark prior to and following the implementation of lockdowns, revealing a considerable drop from 3.32 to 0.92. The lockdown, which began on 18 March 2020, had an effect after a few days. Gumel [
33] established a deterministic two-strain model for the dynamics of the transmission of bird flu between birds and humans. The model included the spread of an avian strain and its mutant (which can be transmitted among humans), as well as the isolation of those with symptoms from either strain. The reproduction number determines the system’s global dynamics. Numerical simulations suggested that the disease burden increases as the avian strain’s mutation rate increases. Agusto [
34] improved the model of Gumel [
33] by adding control over the isolation rate of humans infected with avian and mutant strains. Rashkov and Kooi [
35] developed a host-vector model for dengue fever, considering two strains of the virus, allowing temporary cross-immunity for the hosts, and the possibility of secondary infections. Royce and Fu [
36] presented a model for transmission among three species that accounts for a zoonotic disease, which mutates in an intermediate host. They found that with realistic parameters of interspecies transmission, a zoonosis with the ability to mutate in an intermediate host species can establish itself in humans, even if the basic reproduction number in humans is lower than 1. Sardar et al. [
37] developed three different two-strain MERS-CoV models that take into account human-to-human transmission in the community and hospitals, as well as passive zoonotic transmission, to predict past outbreaks from 2012 to 2016 and obtain key epidemiological information for the following Saudi cities: Mecca, Medina, and Riyadh. They examined infection variability using disease incidence functions of three different forms, capturing social behavior triggered by an epidemic. In their recent research de León et al. [
38], a new mathematical model was proposed to account for two virus strains, along with a vaccination program. By applying this model to the pandemic in the United States, the authors accurately forecasted the rise of the alpha variant and highlighted the possible impact of the delta variant in the year 2021. Additionally, they determined the lowest percentage of the fully vaccinated population required, along with other intervention strategies, to effectively reduce the spread of the variants and mitigate the multi-strain pandemic. Tchoumi et al. [
39] developed a mathematical model to analyze the transmission dynamics of COVID-19, considering different strains and vaccination effects. The model demonstrated stability and identified the conditions for strain persistence and dominance. Strains would persist if their reproductive numbers exceeded 1. Strain 2 could become dominant if its reproductive number surpassed strain 1’s or if strain 1’s reproductive number was below 1. However, strain 2 would not establish itself if strain 1’s vaccination generated herd immunity and the transmission threshold for strain 2 remained low. Several studies have investigated the COVID-19 pandemic using two-strain models [
40,
41,
42,
43].
In this work, we establish two mathematical models to study the COVID-19 outbreak in Denmark, taking into consideration human-to-human, human-to-mink, and mink-to-human transmission. The human population is partitioned into two groups based on the individuals’ contact with minks: humans in direct contact with minks and humans in indirect contact with minks. We construct a two-strain compartmental model by considering the virus mutation in the mink population, as well as the spread of the new SARS-CoV-2 mink variant to humans. Also, we consider a single-strain model by neglecting the virus mutation in minks and, in view of the ongoing development of vaccines for animals, we include a mink vaccination compartment. The purpose of this research is to assess the possibility of the human population being infected by a new mutant virus originating from minks and study the effect of different control measures, such as mink culling or vaccination, on disease transmission between humans and minks. Using numerical simulations, we estimate the parameters of both models using data on the daily number of COVID-19-infected cases among humans in Denmark in order to obtain the best investigation strategies and sensitivity analysis.
4. Discussion and Conclusions
COVID-19 is mostly transmitted from person to person, although it has also been known to be transmitted from humans to minks. Mink-to-human transmission, on the other hand, has been documented in the cases of farmed mink in Europe and the United States. In this work, we developed two compartmental models to investigate SARS-CoV-2 virus transmission between humans and minks in Denmark, taking into consideration human-to-human, human-to-mink, mink-to-human, and mink-to-mink transmission of SARS-CoV-2. In the presented new models, we split the human population into two categories based on their level of contact with minks. In the mink population, new SARS-CoV-2 virus strains have been discovered. These variants have been observed to be able to be transmitted back to humans through close contact with infected minks. Therefore, we established a novel two-strain compartmental model, taking into account the possibility of the virus mutation in minks and the spread of the newly mutated virus among humans.
To the best of our knowledge, the models presented in this work are the first compartmental models for SARS-CoV-2 transmission that, in addition to the original virus transmission, take into account the mink mutant strain transmission in both human and mink populations. Our results indicate that if the disease contact rates between humans and minks are high and the disease incubation time is short, this will significantly increase the number of infected minks and, as a result, the overall number of infected humans. We also investigated the possibility that the mutated virus in minks may be transmitted to humans. Moreover, the findings indicate that the mutant strain can invade the original strain under two scenarios: either when the transmission rate is high and the infection period is long, or when the transmission rate and the virus mutation rate are both high. However, whereas mutations in minks increase transmission rates in minks, this does not always translate to increased transmission rates from minks to humans or from humans to humans. Two simulation scenarios are presented to investigate the impact of mink culling on SARS-CoV-2 transmission in mink and humans. The findings support Denmark’s decision to cull approximately 17 million minks on 4 November 2020.
To demonstrate the impact of mink vaccination on SARS-CoV-2 disease transmission between minks and humans, we developed a novel compartmental single-strain model with a mink-vaccinated class. In the absence of an animal vaccine, the findings suggest that the mink vaccination strategy would be effective in suppressing the pandemic, i.e., in decreasing the number of infections in humans. As a consequence, mink vaccination may be another solution instead of killing minks. A sensitivity analysis was carried out to compare the effects of the single-strain model parameters on the number of human-infected cases. We found that the transmission rates from human-to-human and mink-to-human are the most important factors regarding disease transmission. Also, both killing minks and establishing a vaccination plan can considerably reduce the number of infected cases.
Obviously, there are limitations to our models. Concerns that the SARS-CoV-2 mutation might create a risk to human health led to the shutdown of approximately 1500 mink farms in Denmark. Since there are no precise data, or at least no data from all countries that experienced a COVID-19 pandemic in mink farms, on the number of infected minks or mink farms infected with SARS-CoV-2, there are also no precise data on the number of humans who have been affected as a result of the COVID-19 mink mutation. We selected Denmark as an example and utilized our models to study SARS-CoV-2 transmission between minks and humans; however, we found some data in the literature, such as data on the number of mink farm workers or caregivers and the total mink population in Denmark. Unfortunately, there were insufficient data in the literature on SARS-CoV-2 infections in Denmark’s mink farms; thus, we used our models to study SARS-CoV-2 infection in the country’s mink farms with the total mink population, instead of focusing on a specific farm. Due to a lack of data and sources, we found the mink population to be complicated, and it was hard to estimate the appropriate values for different parameters based on the existing literature. Furthermore, we estimated a large number of parameters, which naturally adds complexity because different parameter values may produce equivalent results. Even in cases where multiple parameter sets offered equally good fits, there was a very small difference that enabled us to identify the most optimal one among those that were closely comparable. We conducted extensive research in the literature to acquire the majority of the parameters, and when specific values were unavailable, we determined realistic ranges for these parameters. These ranges formed the foundation for fitting the daily number of COVID-19-infected cases among humans with the most appropriate values. As a result, while there may exist different feasible values within the ranges that yield similar fits, the variations between them should not be significantly distant from one another. Therefore, we determined the best-fit parameter values for COVID-19-infected humans from Denmark, and the best-fit solution is considered as the baseline.
Another limitation of our model was that we were unsure which mink culling strategy was used in Denmark, so we applied mink culling to the whole mink population at the same time, either by using mink culling ratios or increasing the mink death rate. However, Denmark decided to kill all minks by 4 November 2020, and the implementation took some time, which is consistent with our findings. Due to the economic impact on the mink industry, culling minks may not be the best solution. We searched for other possible control measures, such as decreasing mink-to-human transmission by using a single strain model that included a vaccination compartment. Given the lack of an animal vaccine and the uncertainty as to when vaccination should be administered in order to maximize its chances of success, we decided to vaccinate a fraction of the mink population (this is another limitation of our model).
The overall findings of this study suggest that to control the disease and spread of the COVID-19 mutant strains among human and mink populations, we must minimize the transmission and contact rates between mink farmers and other humans through quarantining such individuals. In addition, culling or vaccination strategies for infected mink farms must be implemented in order to reduce the virus mutation rate in minks.