Transmission Dynamics and Control of Infectious Diseases


We invited Mr. Keita Wagatsuma, a data science expert, who has been engaged in the COVID-19 response of Niigata Prefecture, Japan to discuss transmission dynamics. While a number of articles have explained the usefulness of reproduction note and the effective reproduction number using relatively layman’s terms, his article also gives us the extent of the concept with very basic mathematical equations to help understand the concept. Of note, this website recently included the effective reproduction number in Japan. We hope that this article will be of value to understand to interpret this numbers.

What is the basic reproduction number, R0 (pronounced as R nought)?

The basic reproductive number R0, pronounced “R nought“, is the most basic indicator of transmissibility in infectious disease epidemiology and is the term used to describe the ability of an infectious disease to be transmitted.

Theoretically, R0 is defined as the average number of secondary cases generated by a single primary case during its entire period of infectiousness in a fully susceptible population. In other words, it is the average number of people who will contract an infectious disease from one infected person. This is particularly true to population of previously uninfected and unvaccinated people.

For instance, consider a case where the basic reproductive number, R0, is 5. A specific infectious disease with R0 of about 5 (e.g., smallpox). An R0 of 5 for a disease means that a person with that disease will infect an average of 5 other people. An important point of view in the interpretation of this indicator is that the replication will continue, as long as the concerning population is not immune to the disease.

What does the basic reproduction number, R0 mean?

Three possibilities exist for the transmission or reduction of infectious diseases, depending on their R0 value. For simplicity, it is easier to look at them in the following situations.

  • If R0 < 1, each existing infection causes less than one new infection. The disease declines and the epidemic eventually comes to an end.
  • If R0 = 1, each existing infection causes one new infection. The disease survives and stabilizes, showing a linear increasing trend, but no epidemic occurs.
  • If R0 > 1, each existing infection causes one or more new infections. The disease is transmitted among people and an epidemic may occur.

It is important to note that the R0 value for a disease only applies if all at risk are completely vulnerable to the disease. In other words, it represents a very limited situation: no one has been vaccinated, no one has ever had the disease, and there is no way to control the spread of the disease. Also, the R0 value is related to the population density, social structure, and the mode of contact between individuals, so the estimates may vary according to the local conditions in which the analysis is carried out.

Transmissibility of infectious diseases

The R0 can be used to measure any contagious disease that can be spread in susceptible populations and has been estimated for a variety of viral infections. Highly contagious diseases include measles, whereas the Middle East respiratory syndrome-related coronavirus (MERS) and seasonal influenza do not spread very easily among people. Indeed, it is estimated that the R0 of COVID-19 is approximately 2.2 (uncertainty range, 2–3) (Figure 1).

Basic mathematical models of infectious diseases

There are various methods for estimating R0 under infectious disease epidemics, but the simplest method is to use a basic mathematical model. In particular, the stochastic SIR model (i.e., Kermack-Mckendrick type model), which captures the epidemic dynamics of infectious diseases that are transmitted directly from person to person, is often used in infectious disease epidemiology. It was proposed to explain the rapid rise and fall in the number of infected patients observed in epidemics such as the plague (London 1665-1666, Bombay 1906) and cholera (London 1865). It assumes that the population size is fixed (i.e., no births, deaths due to disease, or deaths by natural causes), the incubation period of the infectious agent is instantaneous, and the duration of infectivity is the same as the length of the disease. It also assumes a completely homogeneous population with no age, spatial, or social structure. The SIR model divides a population into three compartments according to the stage of infection: susceptible, infected, and recovered/removed, and models changes in infectious status over time in a bottom-up fashion (Figure 2). Generally, the SIR model is represented by the following system of ordinary differential equations:  

where S(t), I(t),and R(t) are the proportions of susceptible, infected, and recovered/removed persons in the population at time t. β is the coefficient of the infection rate per unit time, while βI(t) gives the force of infection at time t. γ is the rate of elimination by recovery/removal per unit of time, and its inverse γ−1 gives the average value if the duration of infection between infection and recovery/removal is assumed to follow an exponential distribution. Transforming the second equation, we can obtain the follows:

If the number of newly infected people is increasing, βS(t)−γ >0, which is a condition for an infectious disease epidemic (since in the early stages of an infestation a small number of infected individuals follow an exponential increase (i.e., Malthusian law) with I(t) ≈ I(0) exp{(βγ)t}). Furthermore, this equation can be transformed to βS(t)γ−1 >1. Here, we can assume that S(0) = 1 (∵ N = S(t)+I(t)+R(t)), since at time 0 all members of the population are susceptible and that βγ−1 is the threshold for infectious disease prevalence. Therefore, this value can be rephrased as the basic reproduction number, R0, in the SIR model as follows:

In practice, we will assess the R0 value, the number of new infections, and public policy interventions by simulating this simultaneous differential equation. Figure 3 shows a typical simulation result, where the epidemic size and its speed are characterized by β and γ. For instance, in the simulation of a pilot influenza epidemic, we can read for example (Figure 3).

  • The peak number of infected people is about 1/5 of the total population
  • More than 85% of the population will eventually be infected (Suspected: 25% of the uninfected population after 40 days).
  • It takes about 40 days to end the disease.

Because the model considers short-term trends, it assumes that background demographics such as births and deaths can be largely ignored. Essentially, the SIR model is often used as a basis for more realistic mathematical modeling (e.g., SEIR (Suspected–Exposed–Infected–Recovered) model), simulation and forecasting.

How well are we responding to the pandemic and what next?

Next, let us consider the relationship between infectious disease control and the number of reproductions. While the basic reproduction number R0 can be formulated by the SIR model as described above, R0 can be decomposed as shown on the right-hand side of the following equation:

where c is the average number of contacts per unit of time that a person in the population makes that result in infection, known as effective contacts; b is the probability of contact from an infected person to a susceptible person if one effective contact is made, and D is the average infectivity period. Generally, infectiousness can be reduced by reducing c by contact reduction measures, b by wearing masks and hand washing, or D by early detection and treatment. Additionally, vaccination may be used to reduce the number of susceptible people in the population. The effective reproductive number, Rt, is a measure of the effectiveness of interventions, and keeping it below 1 is a guideline for infectious disease control. Rt is sometimes written Re or Reff, but they are essentially the same and either expression is acceptable. Indeed, R0 represents the infectiousness of a population in which everyone is susceptible, but if we assume that some are immune in the population, the effective reproductive number can be formulated by the following equation, where x is the percentage of immune carries:

If x is described in the setting so that Rt is 1, it is called the critical immunization rate, and this is the target value of the vaccination rate to be achieved. In particular, since no one was immune to the coronavirus disease 2019 (COVID-19) at first, the way to reduce Rt is to take measures against infectious diseases, such as physical distancing, staying at home, wearing masks, and washing hands (i.e. nonpharmaceutical interventions). When the reduction of infectivity by these interventions is defined as p, Rt can be formulated as follows:

The interpretation of Rt is the same as that of R0.

If the R0 of COVID-19 is 2.2, Rt is effectively less than 1 if the reduction in infectivity of these measures is 60%. Specifically, we solve for Rt = (1−p)R0 < 1 in order to compute p which can achieve Rt less than 1; thus the condition is that p > 1−(1/R0) = 1−(1/2.2) > 0.6. Therefore, we can conclude that the spread of infection can be diminished if we reduce our contact with people, through physical distancing, by 60%. In the current COVID-19 measures, considering the heterogeneity of transmission (e.g., behavior of infectious diseases based on intrinsic and extrinsic assumptions such as contact patterns, disease onset, and population movements), it was taken into account to shorten the state of emergency by keeping the Rt under epidemic control as low as possible below 1, and a reduction in the number of contacts by 80%, which is about 20% more than 60%, was sought through a simulation that added various factors to the SIR model. On the other hand, the development of vaccines, immune responses, and mutant strains in 2021 is also possible influences, factors that should be considered in future mathematical model projections. Several fine review articles concerning this topics have been published in major journals for further reading (Homadhl et al., Vespignani et al, and Borchering et al.)  

Practically, when we observe the estimation methods of reproduction numbers in various countries, we find that models based on the SIR model described in this article with extended componentry (e.g., models modified by adding terms for asymptomatic and exposed persons) are often used in the United Kingdom, the United States, and China. Additionally, statistical models that take into account the time delay between a positive test and is reported as an infected person –usually approximated by the serial interval, that is, the time interval between the onset time of the primary infected person and the onset time of the secondarily infected person– are often constructed. On the contrary, in the initial stage of outbreak, it is possible not only to estimate R0 basing such a model but also to formulate it using epidemic growth rates and renewal equations.

Important factors for the prevention of infectious diseases

In this article, I have introduced the basic concept of reproductive numbers (R0 and Rt) of infectious diseases. In particular, the reproductive number introduced in this article is one of the most effective calculations for predicting and controlling the transmission of infectious diseases. Generally, the terminology of infectious disease epidemiology is difficult to understand for people outside the field (in my eyes, it may be a niche compared to other fields of epidemiology) despite its sudden popularity in this pandemic. This is why it is particularly important to describe the meaning of such terms from the point of view of risk communication.

Globally, the unexpected emergence and outbreak of COVID-19 have resulted in a significant tremendous health disease burden and impact on the world economy, whereas it is also true that recent advances in medicine have led to the development of new treatments and vaccines for a wide range of diseases at a much faster rate than expected. However, it is still important that each of us does what we can and consciously takes the infection control measures including avoiding 3Cs situation.

References and further reading

  1. Heymann DL, Shindo N; WHO Scientific and Technical Advisory Group for Infectious Hazards. COVID-19: what is next for public health?. Lancet. 2020;395(10224):542-545. doi:10.1016/S0140-6736(20)30374-3
  2. Legido-Quigley H, Asgari N, Teo YY, et al. Are high-performing health systems resilient against the COVID-19 epidemic?. Lancet. 2020;395(10227):848-850. doi:10.1016/S0140-6736(20)30551-1
  3. Kucharski AJ, Russell TW, Diamond C, et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling. Lancet Infect Dis. 2020;20(5):553-558. doi:10.1016/S1473-3099(20)30144-4
  4. Liu Y, Gayle AA, Wilder-Smith A, Rocklöv J. The reproductive number of COVID-19 is higher compared to SARS coronavirus. J Travel Med. 2020;27(2):taaa021. doi:10.1093/jtm/taaa021
  5. Delamater PL, Street EJ, Leslie TF, Yang YT, Jacobsen KH. Complexity of the Basic Reproduction Number (R0). Emerg Infect Dis. 2019;25(1):1-4. doi:10.3201/eid2501.171901
  6. Adam D. A guide to R – the pandemic’s misunderstood metric. Nature. 2020;583(7816):346-348. doi:10.1038/d41586-020-02009-w
  7. Anderson RM, May RM. Population biology of infectious diseases: Part I. Nature. 1979;280(5721):361-367. doi:10.1038/280361a0
  8. May RM, Anderson RM. Population biology of infectious diseases: Part II. Nature. 1979;280(5722):455-461. doi:10.1038/280455a0
  9. Mahase E. Covid-19: What is the R number?. BMJ. 2020;369:m1891. doi:10.1136/bmj.m1891
  10. Brett TS, Rohani P. Transmission dynamics reveal the impracticality of COVID-19 herd immunity strategies. Proc Natl Acad Sci U S A. 2020;117(41):25897-25903. doi:10.1073/pnas.2008087117
  11. Inaba H, Nishiura H. The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model. Math Biosci. 2008;216(1):77-89. doi:10.1016/j.mbs.2008.08.005
  12. Holmadahl I, Buckee C. Wrong but useful-what COVID19 epidemiologic models can and cannot tell us. N Engl J Med 2020;383:303-305
  13. Brett T, Rohani P. Transmission dynamics reveal the impracticality of COVID-19 herd immunity strategies. PNAS 117(41)25897-25903
  14. Borchering RK, Viboud C, Howerton E, et al. Modeling of Future COVID-19 Cases, Hospitalizations, and Deaths, by Vaccination Rates and Nonpharmaceutical interventions Scenarios-United States, April-September 2021. MMWR Morb Mortal Wkly Rep 2021;70:719-724
  15. Nishiura H. Correcting the Actual Reproduction Number: A Simple Method to Estimate R0 from Early Epidemic Growth Data. Int. J. Environ. Res. Public Health 2010, 7(1), 291-302

About the author

Mr. Keita Wagatsuma is a Ph.D. candidate and research assistant of Division of International Health (Public Health) and Infectious Diseases Research Center (IDRC), Graduate School of Medical and Dental Sciences, Niigata University, Niigata, Japan. His research interests include infectious disease epidemiology, biostatistics, travel medicine, and data science.

Mr. Keita Wagatsuma