COVID-19 SIR model: Bifurcation analysis and optimal control

This paper proposes a SIR epidemic model with vital dynamics to control or eliminate the spread of the COVID-19 epidemic considering the constant population, saturated treatment, and direct-indirect transmission rate of the model. We demonstrate positivity, boundness and calculate the disease-free equilibrium point and basic reproduction number from the model. We use the Jacobian matrix and the Lyapunov function to analyze the local and global stability, respectively. It is observed that indirect infection increases the basic reproduction number and gives rise to multiple endemic diseases. We perform transcritical, forward, backward, and Hopf bifurcation analyses. We propose two control parameters (Use of face mask, hand sanitizer, social distancing, and vaccination) to minimize the spread of the coronavirus. We use Pontryagin's maximum principle to solve the optimal control problem and demonstrate the results numerically.

Efficient Direct and Limited Environmental Transmission of SARS-CoV-2 Lineage B.1.22 in Domestic Cats.

The susceptibility of domestic cats to infection with SARS-CoV-2 has been demonstrated by several experimental studies and field observations. We performed an extensive study to further characterize the transmission of SARS-CoV-2 between cats, through both direct and indirect contact. To that end, we estimated the transmission rate parameter and the decay parameter for infectivity in the environment. Using four groups of pair-transmission experiment, all donor (inoculated) cats became infected, shed virus, and seroconverted, while three out of four direct contact cats got infected, shed virus, and two of those seroconverted. One out of eight cats exposed to a SARS-CoV-2-contaminated environment became infected but did not seroconvert. Statistical analysis of the transmission data gives a reproduction number R0 of 2.18 (95% CI = 0.92 to 4.08), a transmission rate parameter ß of 0.23 day-1 (95% CI = 0.06 to 0.54), and a virus decay rate parameter µ of 2.73 day-1 (95% CI = 0.77 to 15.82). These data indicate that transmission between cats is efficient and can be sustained (R0 > 1), however, the infectiousness of a contaminated environment decays rapidly (mean duration of infectiousness 1/2.73 days). Despite this, infections of cats via exposure to a SARS-CoV-2-contaminated environment cannot be discounted if cats are exposed shortly after contamination. IMPORTANCE This article provides additional insight into the risk of infection that could arise from cats infected with SARS-CoV-2 by using epidemiological models to determine transmission parameters. Considering that transmission parameters are not always provided in the literature describing transmission experiments in animals, we demonstrate that mathematical analysis of experimental data is crucial to estimate the likelihood of transmission. This article is also relevant to animal health professionals and authorities involved in risk assessments for zoonotic spill-overs of SARS-CoV-2. Last but not least, the mathematical models to calculate transmission parameters are applicable to analyze the experimental transmission of other pathogens between animals.

Understanding the Role of Environmental Transmission on COVID-19 Herd Immunity and Invasion Potential.

COVID-19 is caused by the SARS-CoV-2 virus, which is mainly transmitted directly between humans. However, it is observed that this disease can also be transmitted through an indirect route via environmental fomites. The development of appropriate and effective vaccines has allowed us to target and anticipate herd immunity. Understanding of the transmission dynamics and the persistence of the virus on environmental fomites and their resistive role on indirect transmission of the virus is an important scientific and public health challenge because it is essential to consider all possible transmission routes and route specific transmission strength to accurately quantify the herd immunity threshold. In this paper, we present a mathematical model that considers both direct and indirect transmission modes. Our analysis focuses on establishing the disease invasion threshold, investigating its sensitivity to both transmission routes and isolate route-specific transmission rate. Using the tau-leap algorithm, we perform a stochastic model simulation to address the invasion potential of both transmission routes. Our analysis shows that direct transmission has a higher invasion potential than that of the indirect transmission. As a proof of this concept, we fitted our model with early epidemic data from several countries to uniquely estimate the reproduction numbers associated with direct and indirect transmission upon confirming the identifiability of the parameters. As the indirect transmission possess lower invasion potential than direct transmission, proper estimation and necessary steps toward mitigating it would help reduce vaccination requirement.

Effect of Human Mobility on the Spatial Spread of Airborne Diseases: An Epidemic Model with Indirect Transmission.

We extended a class of coupled PDE-ODE models for studying the spatial spread of airborne diseases by incorporating human mobility. Human populations are modeled with patches, and a Lagrangian perspective is used to keep track of individuals' places of residence. The movement of pathogens in the air is modeled with linear diffusion and coupled to the SIR dynamics of each human population through an integral of the density of pathogens around the population patches. In the limit of fast diffusion pathogens, the method of matched asymptotic analysis is used to reduce the coupled PDE-ODE model to a nonlinear system of ODEs for the average density of pathogens in the air. The reduced system of ODEs is used to derive the basic reproduction number and the final size relation for the model. Numerical simulations of the full PDE-ODE model and the reduced system of ODEs are used to assess the impact of human mobility, together with the diffusion of pathogens on the dynamics of the disease. Results from the two models are consistent and show that human mobility significantly affects disease dynamics. In addition, we show that an increase in the diffusion rate of pathogen leads to a lower epidemic.

Impact of environmental transmission and contact rates on Covid-19 dynamics: A simulation study.

The emergence of the COVID-19 pandemic has been a major social and economic challenge globally. Infections from infected surfaces have been identified as drivers of Covid-19 transmission, but many epidemiological models do not include an environmental component to account for indirect transmission. We formulate a deterministic Covid-19 model with both direct and indirect transmissions. The computed basic reproduction number R 0 represents the average number of secondary direct human-to-human infections, and the average number of secondary indirect infections from the environment. Using Partial Rank Correlation Coefficient, we compute sensitivity indices of the basic reproductive number R 0 . As expected, the most significant parameter to reduce initial disease transmission is the natural death rate of pathogens in the environment. Variation of the basic reproduction number for different values of direct and indirect transmissions are numerically investigated. Decreasing the effective direct human-to-human contact rate and indirect transmission from human-to-environment will decrease the spread of the disease as R 0 decreases and vice versa. Since the effective contact rate often accounted for as a factor of the force of infection and other interventions measures such as treatment rate are prominent features of infectious diseases, we consider several functional forms of the incidence function, and numerically investigate their potential impact on the long-term dynamics of the disease. Simulations results revealed some differences for the time and infection to reach its peak. Thus, the choice of the functional form of the force of infection should mainly be influenced by the specifics of the prevention measures being implemented.

Using the SEIR model to constrain the role of contaminated fomites in spreading an epidemic: An application to COVID-19 in the UK.

The use of the SEIR model of compartmentalized population dynamics with an added fomite term is analysed as a means of statistically quantifying the contribution of contaminated fomites to the spread of a viral epidemic. It is shown that for normally expected lifetimes of a virus on fomites, the dynamics of the populations are nearly indistinguishable from the case without fomites. With additional information, such as the change in social contacts following a lockdown, however, it is shown that, under the assumption that the reproduction number for direct infection is proportional to the number of social contacts, the population dynamics may be used to place meaningful statistical constraints on the role of fomites that are not affected by the lockdown. The case of the Spring 2020 UK lockdown in response to COVID-19 is presented as an illustration. An upper limit is found on the transmission rate by contaminated fomites of fewer than 1 in 30 per day per infectious person (95% CL) when social contact information is taken into account. Applied to postal deliveries and food packaging, the upper limit on the contaminated fomite transmission rate corresponds to a probability below 1 in 70 (95% CL) that a contaminated fomite transmits the infection. The method presented here may be helpful for guiding health policy over the contribution of some fomites to the spread of infection in other epidemics until more complete risk assessments based on mechanistic modelling or epidemiological investigations may be completed.

Porous surfaces: stability and recovery of coronaviruses.

The role of indirect contact in the transmission of SARS-CoV-2 is not clear. SARS-CoV-2 persists on dry surfaces for hours to days; published studies have largely focused on hard surfaces with less research being conducted on different porous surfaces, such as textiles. Understanding the potential risks of indirect transmission of COVID-19 is useful for settings where there is close contact with textiles, including healthcare, manufacturing and retail environments. This article aims to review current research on porous surfaces in relation to their potential as fomites of coronaviruses compared to non-porous surfaces. Current methodologies for assessing the stability and recovery of coronaviruses from surfaces are also explored. Coronaviruses are often less stable on porous surfaces than non-porous surfaces, for example, SARS-CoV-2 persists for 0.5 h-5 days on paper and 3-21 days on plastic; however, stability is dependent on the type of surface. In particular, the surface properties of textiles differ widely depending on their construction, leading to variation in the stability of coronaviruses, with longer persistence on more hydrophobic materials such as polyester (1-3 days) compared to highly absorbent cotton (2 h-4 days). These findings should be considered where there is close contact with potentially contaminated textiles.

Assessing the impact of adherence to Non-pharmaceutical interventions and indirect transmission on the dynamics of COVID-19: a mathematical modelling study.

Adherence to public health policies such as the non-pharmaceutical interventions implemented against COVID-19 plays a major role in reducing infections and controlling the spread of the diseases. In addition, understanding the transmission dynamics of the disease is also important in order to make and implement efficient public health policies. In this paper, we developed an SEIR-type compartmental model to assess the impact of adherence to COVID-19 non-pharmaceutical interventions and indirect transmission on the dynamics of the disease. Our model considers both direct and indirect transmission routes and stratifies the population into two groups: those that adhere to COVID-19 non-pharmaceutical interventions (NPIs) and those that do not adhere to the NPIs. We compute the control reproduction number and the final epidemic size relation for our model and study the effect of different parameters of the model on these quantities. Our results show that there is a significant benefit in adhering to the COVID-19 NPIs.

Community Transmission via Indirect Media-To-Person Route: A Missing Link in the Rapid Spread of COVID-19.

To prevent the spread of coronavirus disease 2019 (COVID-19), stringent quarantine measures have been implemented so that healthy people and virus carriers have isolated themselves in the same community owing to the limit capacity of healthcare facilities. With the exponential growth of the infected population, the residential environment is contaminated by fomites from the infected residents and consequently threating the health of susceptible residents. Till now, little has been acknowledged on this indirect transmission route and its role on community transmission. Here we address the impact of self-isolated virus carriers on the residential environment and elucidate the potential transmission pathways via contaminated environment in communities. We urge further investigation on the superspreading cases in communities and hope to arouse the attention to evaluate the potential risk of indirect transmission route as well as the corresponding control measures.

A dynamic model for the COVID-19 with direct and indirect transmission pathways.

Two common transmission pathways for the spread of COVID-19 virus are direct and indirect. The direct pathway refers to the person-to-person transmission between susceptibles and infectious individuals. Infected individuals shed virus on the objects, and new infections arise through touching a contaminated surface; this refers to the indirect transmission pathway. We model the direct and indirect transmission pathways with a S A D O I R ode model. Our proposal explicitly includes compartments for the contaminated objects, susceptible individuals, asymptomatic infectious, detected infectious, and recovered individuals. We compute the basic reproduction number and epidemic growth rate of the model and determine how these fundamental quantities relate to the transmission rate of the pathways. We further study the relationship between the rate of loss of immunity and the occurrence of backward bifurcation. An efficient statistical framework is introduced to estimate the parameters of the model. We show the performance of the model in the simulation scenarios and the real data from the COVID-19 daily cases in South Korea.

Mathematical analysis for COVID-19 resurgence in the contaminated environment.

A mathematical model is proposed that incorporates the key routes of COVID-19 resurgence: human-to-human transmission and indirect transmission by inhaling infectious aerosols or contacting public facilities with the virus. The threshold condition for the disease invasion is established, and the relationships among the basic reproduction number, peak value and final size are formulated. The model is validated by matching the model with the data on cases of COVID-19 resurgence in April of 2020 from Heilongjiang province in China, which indicates that the predictive values from the mathematical model fit the real data very well. Based upon the computations from the model and analytical formulae, we reveal how the indirect transmission from environmental pathogens contribute to the disease outbreak and how the input of asymptomatic individuals affect the disease spread. These findings highlight the importance of mass detection and environmental disinfection in the control of COVID resurgence.

The evidence of indirect transmission of SARS-CoV-2 reported in Guangzhou, China.

BACKGROUND: More than 2 months have passed since the novel coronavirus disease 2019 (COVID-19) first emerged in Wuhan, China. With the migration of people, the epidemic has rapidly spread within China and throughout the world. Due to the severity of the epidemic, undiscovered transmission of COVID-19 deserves further investigation. The aim of our study hypothesized possible modes of SARS-CoV-2 transmission and how the virus may have spread between two family clusters within a residential building in Guangzhou, China. METHODS: In a cross-sectional study, we monitored and traced confirmed patients and their close contacts from January 11 to February 5, 2020 in Guangzhou, China, including 2 family cluster cases and 61 residents within one residential building. The environmental samples of the building and the throat swabs from the patients and from their related individuals were collected for SARS-CoV-2 and tested with real-time reverse transcriptase polymerase chain reaction (RT-PCR). The relevant information was collected and reported using big data tools. RESULTS: There were two notable family cluster cases in Guangzhou, which included 3 confirmed patients (family No.1: patient A, B, C) and 2 confirmed patients (family No.2: patient D, E), respectively. None of patients had contact with other confirmed patients before the onset of symptoms, and only patient A and patient B made a short stop in Wuhan by train. Home environment inspection results showed that the door handle of family No.1 was positive of SARS-CoV-2. The close contacts of the 5 patients all tested negative of SARS-CoV-2 and in good health, and therefore were released after the official medical observation period of 14-days. Finally, according to the traceability investigation through applying big data analysis, we found an epidemiological association between family No.1 and family No.2, in which patient D (family No.2) was infected through touching an elevator button contaminated by snot with virus from patient A (family No.1) on the same day. CONCLUSIONS: Contaminants with virus from confirmed patients can pollute the environment of public places, and the virus can survive on the surface of objects for a short period of time. Therefore, in addition to the conventional droplet transmission, there is also indirect contact transmission such as snot-oral transmission that plays a crucial role in community spread of the virus.