Transmission Dynamics and Control of COVID-19 Pandemic: A Mathematical Modelling Study

Background: The coronavirus disease 2019 (COVID-19) pandemic remains a major global public health concern. It is assumed that the COVID-19 outbreak originated in Wuhan, China and spread worldwide as a result of international travel. Here, I used data from China to assess the impact of contact rate for COVID-19 transmission and decay of transmission rate due to implemented interventions for disease control. Method: I developed a generalized susceptible-exposed-infectious-recovered (SEIR) compartmental model based on the disease clinical progression and interventions to explore the transmission dynamics of the global COVID-19 pandemic. I computed the basic reproduction number from the model to assess the epidemiology of the disease. Results: I found that the number of asymptomatic was higher than symptomatic cases, and that COVID19 transmission is directly linked with contact rate between infectious and susceptible individuals. It shows that at high contact rate, disease peaked in a short period of time, and doubling time decrease with increasing contact rate. The curve of COVID-19 transmission was found to flatten with increasing decay of transmission rate due to interventions that have been implemented. The basic reproduction number computed in this study without intervention at the contact rate of 1.137 day-1 was found to be 4.9976, suggesting that the number of secondary cases resulted from introduction of a single infectious individual into naive susceptible population was increasing. At the decay transmission rate of 0.02 day-1, the basic reproduction number was 0.4534 and 0.04113 at the decay transmission rate of 0.04 day-1, implying that disease can be controlled with implementation of interventions. Conclusion: The study shows that COVID-19 transmission was amplified by high contact rates between infectious and susceptible individuals, and that asymptomatic individuals might be super spreaders of the disease. It shows that the number of cases decreases with increasing decay of transmission rate due to earlier implementation of interventions.


Introduction
Coronavirus disease 2019 (COVID-19) is a major public health concern worldwide [1,2]. It is not well understood how the COVID-19 outbreak started and became the global pandemic as declared by the World Health Organization (WHO). However, it is assumed that COVID19 originated from seafood market in Wuhan, Hubei province in China where the outbreak began [3]. At the beginning of the outbreak, it was assumed that COVID-19 was only transmitted from animal-to-human, and that there was no proof of huma-to-human transmission. Apart from animal-to-human transmission as suggested previously, epidemiological data show that COVID-19 is transmitted by aerosols from human-to-human through breathing, talking, coughing and sneezing. The first case of COVID-19 was confirmed in December 2019 in the city of Wuhan

Model design and development
To explore the transmission dynamics of the global COVID-19 pandemic, I developed a generalized susceptible-exposedinfectious-recovered (SEIR) compartmental model based on the disease clinical progression and interventions, such as home quarantine and hospital isolation ( Figure 1). In the model design in terms of interventions, I made real-world assumptions that proportions of exposed-susceptible, exposed (latent) and asymptomatic individuals are quarantined at home, and that severe symptomatic, critical symptomatic and quarantined individuals who develop severe or critical symptoms become hospitalized. I parameterized the model using data from published studies that used data of confirmed cases of COVID-19 [14]. I computed the basic reproduction number using a next generation matrix approach to assess disease transmission. In the design of the model, I stratified the populations as susceptible individuals (S) who are at a high risk but not yet infected, exposed individuals (E) who are latently infected but not infectious, home quarantined individuals, including exposed-susceptible, exposed (latent) and asymptomatic or with mild infection but not yet severe or critical symptomatic (Q). Furthermore, I included asymptomatic individuals (I a ) who are infectious but with no symptoms, severe symptomatic individuals (I s ) who are infectious with symptoms and may require oxygen, critical symptomatic individuals (I c ) who are infectious with symptoms and may require ventilation, hospitalized individuals (H) who are either critically or severely ill and recovered individuals (R) (Figure 1).
Based on the design of the model (Figure 1), recruitment due to either migration or birth move to a susceptible state at a rate Λ.
Susceptible individuals are infected at a rate λ, which is the force of infection. A proportion of susceptible individuals, b become latently infected and move to an exposed state, and another proportion, 1-b is quarantined at home as a result of contact tracing. Exposed or latently infected individuals develop disease at a rate θ, of which 80% become asymptomatic, 15% severely symptomatic and 5% critically symptomatic infected individuals as suggested by the World Health Organization (WHO)0020 [1,15]. Additionally, based on contact tracing, exposed individuals become home quarantined at a rate q. Asymptomatic or individuals with mild symptoms become quarantined at rate r or recover at a rate f. Home quarantined individuals move to hospital isolation at a rate x if they become either severely or critically infected. Severely infected individuals become either hospital isolated at a rate z or recover at a rate y. Critically infected individuals move to either a hospitalized state at a rate ω or a recovered state at a rate k. Hospitalized individuals with either severe or critical infection recover and move to recovered state at a rate δ. I assume that natural death occurs in all compartments at a rate µ, and death rate due to disease (d) occurs in asymptomatic, severe, critical and hospitalized states.
Since the waning period of acquired immunity for COVID-19 is not well known, I assumed that recovered individuals become immune.
From the model in Figure 1, I developed the mathematical model as follows:

Basic reproduction number
The basic reproduction number (R 0 ) is the most important quantity in the predition of infectious disease epidemiology. If R 0 >1 it means disease transmission is increasing, and if R 0 <1 disease transmission is decreasing and it can be controlled [3,5].
In    implying that disease transmission can be controlled (Table 1).

Results
Contact rate defined as the number of contacts per unit time   The number of exposed individuals was observed to be the highest when transition time (incubation period) from infection to disease is estimated to be between 2 and 14 days. In this study, incubation period in this group (exposed) was estimated to be 5.1 days before transition to either asymptomatic or symptomatic disease (Table 1). Since incubation period seems to be long time, this group might be spreading disease before showing symptoms.
The number of asymptomatic cases was observed to be higher than severe and critical symptomatic cases, suggesting that the vast majority of COVID-19 might be asymptomatic. Since they do not show any symptoms and interact with susceptible individuals on a daily basis, asymptomatic cases might be super-spreaders of the disease in the community. Many countries have implemented contact tracing and mass testing, but asymptomatic individuals may have been missed and the spread of the disease continued increasing. The number of severe and critical symptomatic individuals were observed to be lower than home quarantined and hospitalized individuals probably because the majority of severely and critically ill patients might be hospitalized as they may require oxygen or ventilation.  [2,17] p a Proportion of exposed becoming asymptomatic 0.8 [1,15] p s Proportion of exposed becoming severe symptomatic 0.15 [1,15] p c Proportion of exposed becoming critical symptomatic 0.05 [1,15] θ Incubation/progression rate from exposed to disease 0.196 day -1 [16,14] µ Natural death rate 0.002 [1] d Disease related death rate 0.043 day -1 [1] z Hospital isolation rate for severe symptomatic 0.095 day -1 [7] ω Hospital isolation rate for critical symptomatic 0.143 day -1 [7] x Hospital isolation rate for home quarantined individuals 0.111day -1 [7] r Home quarantined rate for mild or asymptomatic individuals The basic reproduction number computed in this study without intervention at the contact rate of 1.137 day -1 was found to be 4.9976, suggesting that the number of secondary cases resulted from a single infectious individual introduced into susceptible population was increasing. Additionally, when intervention was implemented, I found the basic reproduction number of 0.4534 at the decay transmission rate of 0.02day -1 and 0.04113 at the decay transmission rate of 0.04 day -1 implying that disease transmission can be decreased.
Here, I have shown that implementation of effective interventions has high impact for disease control by amplifying the decay of transmission rate (Figure 3). This implies that implementation of more than one intervention, such as combination of mitigation may flatten the curve faster than a single intervention