Preface to the special issue "Complex dynamics of COVID-19: modeling, prediction and control (part II)".

The world has been severely upended by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) since the end of 2019 when the first case was diagnosed and its potential harm was announced in Wuhan, China. As of now, several mutations have been identified in the SARS-CoV-2 genome with different degrees of pathogenicity, infectivity, transmissibility, thus putting a continuous pressure on diagnostics, therapeutics, vaccines, and policy decision making processes. As a result, developed countries prefer possibly achieving herd immunity while a few countries such as China have maintained a dynamic disease control policy, according to which all citizens are required to wear masks in public places and practice rapid safety isolation in mobile hospitals or at their homes. Random traveling and cargo transportation can induce unpredictable spreading in different regions. All residents and citizens have been required to be vaccinated more than twice to prevent harmful effects from COVID-19 disease. In some metropolitan areas severely hit by COVID-19 and the Omicron variants, due to the strict home isolation policy and traffic control, more and more companies have had to announce layoffs due to the impact on their businesses. Therefore, people have increasingly lost their jobs and cannot ensure their livelihoods. To prevent the spread of the epidemic, it has become difficult for migrant workers in cities in China hit by the epidemic to return to their hometowns. They can only leave the cities where the epidemic has broken out after obtaining a safety permit. It has become the dream of many migrant workers to successfully buy air tickets or high-speed rail tickets, or drive back to their hometowns.

People are questioning why the travel of citizens covered by three vaccination doses should be restricted and wonder when the epidemic will eventually be overcome. At the same time, more people hope that effective drugs can be found to cure or prevent the disease and help the anxiety ease off.

Scientific research is using exponentially growing data to analyze and predict the end of the epidemic; in fact, as of now, more than 35,737 papers about COVID-19 are indexed in the Web of Science. That is, the dynamics of COVID-19 has received increasing attention and a formidable stream of works has confirmed the complexity in the dynamics of viral infection and recurrence. NONLINEAR DYNAMICS has published several interesting papers sharing different aspects and views. This Special Issue features fifteen papers about Covid-19. We believe that these scientific contributions can be helpful to shape prevention strategies especially if they underpin government policies. A brief summary is provided next.

Badfar et al. [1] propose a model to estimate the behavior of coronavirus in the Iranian and Russian societies by using a set of ordinary differential equations (ODEs), in which the control input signals are vaccination, social distance and facial masks, and medical treatment. A sliding mode is also proposed to control the dynamics of Covid-19. Cooper et al. [2] propose a SIR model to estimate the spread of COVID-19 in Germany, Japan, India and highly impacted states in India, and a complete dynamic analysis is carried out. Debbouche et al. [3] explore the nonlinear dynamic behavior of a novel COVID-19 pandemic SIR model defined by commensurate and incommensurate fractional-order derivatives, which seems to be more accurate in predicting daily new cases than applying integer-order models. Ghosh et al. [4] investigate the characteristics of a multi-wave SIR model, in which the origin of the multi-wave pattern in the solution of this model is explained, and the features of these pandemic waves in India are explained successfully. Hametner et al. [5] propose a methodology to predict the COVID-19 case numbers, case-specific hospitalization and ICU admission rates as well as hospital and ICU occupancies, and differential flatness is used to provide estimates of the states of an epidemiological compartmental model and estimates of the unknown exogenous inputs driving its nonlinear dynamics. Hoque et al. [6] introduce a SEIATR compartmental model to analyze and predict the COVID-19 outbreak in seriously affected countries in the world including USA, India, Brazil, France, and Russia. Wei et al. [7] present a stochastic reaction–diffusion epidemic model and a control scheme is suggested to control the infectious disease by exploring the stationary distribution and Turing instability, which provides sufficient criteria for the persistence and extinction of the disease. Yu et al. [8] claim that the multi-wave peaks for the spread of COVID-19 offer new insights to understand a nonlocal SIHRDP epidemic model with long memory in dynamics. Majumder et al. [9] introduce an HIV-TB co-infection model by considering the treatment provision limitation induced by the COVID-19 pandemic that greatly impacts this dual epidemic. Mondal et al. [10] establish a compartmental model to predict the course of the pandemic and come up with a strategy to control it effectively; it is confirmed that the disease transmission rate has an impact in mitigating the spread of diseases. Rabiu et al. [11] develop an endemic model of COVID-19 to assess the impact of vaccination and immunity waning on the dynamics of the disease, which exhibits distinct backward bifurcation and bi-stability. Saadatmand et al. [12] study the reliability of non-pharmaceutical interventions in managing the current Coronavirus pandemic and predict the next wave of infection in Iran. Temerev et al. [13] improve a COVID-19 model and stochastic numerical simulation is used to predict the transmission of similar infectious diseases that accounts for the geographic distribution of population density, detailed down to the level of location of individuals, and age-structured contact rates. Based on the architecture of Graph Convolutional Neural Networks, Tomy et al. [14] analyze the capability of deep neural networks and algorithmic solutions are used to infer the state of the whole population from a limited number of measures. Yang et al. [15] propose a mathematical spreading model based on strict epidemic prevention measures and the known spreading characteristics of Covid-19.

Reliable theoretical models are critical for predicting and proposing feasible schemes to prevent further diffusion of COVID-19 and similar infectious diseases. Besides these ODE models, perhaps diffusive networks can be more suitable for uncovering complete spatio-temporal dynamics of the COVID-19 by exploring pattern diffusion and pattern stability. The relevant papers in this SI can be helpful for a broad readership to develop new models or refinements of the existing models, and control techniques.

Last but not least, this Special Issue was launched with the enthusiastic support of Prof. Tenreiro Machado who passed away unexpectedly on October 6, 2021 (https://link.springer.com/article/10.1007/s11071-021-07162-z). We are very much indebted to Tenreiro for accepting the challenge of launching in March 2020 the first Special Issue on “Nonlinear dynamics of COVID-19 pandemic: modeling, control, and future perspectives” (https://link.springer.com/journal/11071/volumes-and-issues/101-3) which was published in August 2020, followed by a second Special Issue titled “Complex dynamics of COVID-19: modeling, prediction and control” published as Part 1 in October 2021 (https://link.springer.com/journal/11071/volumes-and-issues/106-2) and in the present issue as Part 2. Several papers in this SI were editorially handled by Tenreiro. He will be truly missed by the community for his dedication and drive.

Guest Editors of the Special Issue.

W. Lacarbonara, Sapienza University of Rome, Italy.

Jun Ma, Lanzhou University of Technology, China.

C. Nataraj, Villanova University, USA.