Epidemiological dynamics with fine temporal resolution
EEpidemiological dynamics with fine temporal resolution
Yaroslav Ispolatov ∗ Departamento de Fisica, Universidad de Santiago de Chile,Casilla 302, Correo 2, Santiago, Chile
Abstract
To better predict the dynamics of spread of COVID-19 epidemics, it is important not only to investigatethe network of local and long-range contagious contacts, but also to understand the temporal dynamicsof infectiousness and detectable symptoms. Here we present a model of infection spread in a well-mixedgroup of individuals, which usually corresponds to a node in large-scale epidemiological networks. Themodel uses delay equations that take into account the duration of infection and is based on experimentally-derived time courses of viral load, virus shedding, severity and detectability of symptoms. We show thatbecause of an early onset of infectiousness, which is reported to be synchronous or even precede the onset ofdetectable symptoms, the tracing and immediate testing of everyone who came in contact with the detectedinfected individual reduces the spread of epidemics, hospital load, and fatality rate. We hope that this moreprecise node dynamics could be incorporated into complex large-scale epidemiological models to improvethe accuracy and credibility of predictions. ∗ Electronic address: [email protected] a r X i v : . [ q - b i o . P E ] M a y . INTRODUCTION In less than 3 months the COVID-19 pandemics has brought the entire world to a halt. To anextend, such a rapid spread of infection is caused by its peculiar dynamics. According to the gen-eral paradigm of epidemiology, a severe or deadly disease usually does not spread too far as theinfected individuals quickly become incapacitated or die, thus limiting the secondary infections.Apparently, such was the case of recent epidemics of SARS, MERS, and Ebola. Conversely, thediseases with less severe symptoms, such as common colds, spread wider as the symptoms do notlimit the mobility and social contacts of the bearers. From what has already become known aboutthe COVID-19, the infection exhibits both those attributes: At the early stages of infection, an al-ready highly contagious bearer may still remains asymptomatic [1–4]. In addition, an infected andinfectious individual may stay completely asymptomatic through the course of the disease. Thecomplications that severely limit mobility and sometimes lead to death usually arrive in the secondweek after the onset of symptoms or even later ([5]). This fairly unusual disease dynamics necessi-tates more complex mitigation strategies and, thus, epidemiological models. For example, variouslockdown criteria and schedules for imposing and exiting quarantine, such as cyclic lockdown-exitscenario proposed in [6] , require accurate predictors of disease dynamics and contagiousness inboth the symptomatic and asymptomatic parts of population, especially given the current lack ofpopulation-wide testing for active viruses and antibodies.Here we propose an extension of the traditional SEIR epidemiological node dynamics thattakes into account the actual time courses of viral load, detectable symptoms, and virus shedding.It serves to better quantify the processes of infection, possible quarantine, and recovery or death.To do so, we replace the continuous or discrete transition from Exposed to Infected, traditionallyused in the epidemiological models, by keeping track of the progress of individual disease and ex-pressing the probability of infection or detection as functions of the disease duration. This resultsin replacing the traditional SEIR ordinary differential equations by the delay differential equationssimilar to [7, 8]. The temporal changes in the infectiousness and severity of symptoms determinethe number of secondary infection and serve as an indicator detectability of symptoms and sub-sequent restrictions in contacts. The time courses of infectiousness and severity of symptoms aretaken directly from clinical data or approximated by functional fits to that data. The model con-firms that a rapid tracing and testing the contacts, rather than reacting to the detectable symptoms,noticeably restricts the spread of the infection and makes quantitative predictions about level of2erd immunity and fatality rate.
II. THE MODEL
We consider a single “node” of epidemiological models, which represents a well-mixed partof the global population and consists of Susceptible (S), Infected (I), Quarantined (Q), Recovered(R), and deceased (D) individuals. The susceptible individuals, whose concentration at time t is denoted by S ( t ) , may pick up a disease from an infected individuals, whose infection could beeither undetected or detected. The individuals with detected symptoms are considered quarantined,so the infectiousness of detected individuals is less than that of the undetected ones by factor χ < . The concentrations of undetected and quarantined infected individuals are denoted as I ( τ, t ) and Q ( τ, t ) , where the variable τ , ≤ τ ≤ t , is the time when the individual was infected.Quantitatively, the infection of susceptibles occurs at the rate dS ( t ) dt = − S ( t ) C I (cid:90) t Γ I ( t − τ ) [ I ( τ, t ) + χQ ( τ, t )] dτ. (1)The empirically derived function Γ I ( t − τ ) , which depends on the duration of a particular infection t − τ , is one of the key features of our model. It shows how contagious an individual is at aparticular stage of infection. This information is based on clinical data (viral load measured viaswabs and blood tests) and can be entered the form of a table or histogram (normally with dailybins [2, 5]) or a functional fit to the data, e.g. Gamma distributions used in [1, 2].The gain in the number of undetected infected individuals, described by the first term in (2),comes from the new infections described above. The delta-function δ ( t − τ ) indicates that thebeginning of a new infection occurs at the current time t . The loss term describes the detection ofinfection and thus the conversion of detected individuals into the quarantined ones. It is assumedto be happening with the rate dependent on the severity of symptoms given by the function Γ S ( t − τ ) . This function is another key element of our model and is based on the empirical data of thetemporal evolution of severity of symptoms, and most importantly, on the time of the onset ofdetectable symptoms. The epidemiological dynamics as well as the clinical data [1, 2, 9] indicatethat the transmissivity of the infection may start before the symptoms onset, which means that theinfection function Γ I ( t − τ ) may start to increase and even reach its maximum earlier than thesymptom function Γ S ( t − τ ) does. Similarly to Γ I ( t − τ ) , the symptom function Γ S ( t − τ ) canbe tabulated based on clinical data or approximated by a fit to that data. The last term in Eq. (2)3escribes resolution of a disease after a typical duration of τ ∗ ∼ days, resulting in recovery ordeath of a patient. ∂I ( τ, t ) ∂t = δ ( t − τ ) C I S (cid:90) t Γ I ( t − τ (cid:48) ) × [ I ( τ (cid:48) , t ) + χQ ( τ (cid:48) , t )] dτ (cid:48) − (2) − I ( τ, t ) [ C S Γ S ( t − τ ) + δ ( t − τ − τ ∗ )] The concentration of quarantined individuals increases via the detection of so far undetectedinfected individuals and decreases on the τ ∗ th day of the disease as individuals recover or die (3), ∂Q ( τ, t ) c∂t = I ( τ, t ) C S Γ S ( t − τ ) − Q ( τ, t ) δ ( t − τ − τ ∗ ) . (3)The approximation of a “sudden recovery” or a possible “sudden death” on the τ ∗ th day of thedisease does not affect the dynamics of infection as it has been observed that individuals destinedeither to recover or die loose infectiousness in about a week after the onset of symptoms, which iswell in advance of τ ∗ [1–3]. The approximation does slightly affect the death and recovery runningstatistics but not the final number of recovered and dead by the end of epidemics. We assume thatthe quarantined individuals receive better care and have lower death rate than the undetected ones.This is reflected by the coefficient ν < . dD ( t ) dt = C D δ ( t − τ − τ ∗ ) [ I ( τ, t ) + νQ ( τ, t )] . (4)The individuals who do not die, recover (5), dR ( t ) dt = δ ( t − τ − τ ∗ ) { I ( τ, t ) [1 − C D ] + Q ( τ, t ) [1 − C D ν ] } (5)The constants C I in (1,2), C S in (2,3), and C D in (4,5) are fitted to reproduce the empiricalinfection, detection of infection (quarantining), and death rates. As in [1, 3], we use Gammadistributions to fit the temporal evolution of infectiousness Γ I ( t − τ ) and severity of symptoms Γ S ( t − τ ) , Γ( y ) = x k − ( k − θ k exp( − y/θ ) , (cid:104) y (cid:105) = kθ. (6)We choose to set k = 3 to mimic that both the infectiousness and symptoms grow slower thanlinearly immediately after the infection and assign the values of scale parameters θ I and θ S in therange derived from epidemiological and clinical data in [1, 9]. Specifically, the mean serial interval (cid:104) T I (cid:105) , defined as the duration between symptom onsets of successive cases in a transmission chain,4nd the incubation period (cid:104) T S (cid:105) , defined as the time between infection and onset of symptoms,were reported in [1] as (cid:104) T I (cid:105) = 5 . and (cid:104) T S (cid:105) = 5 . , and in [9] as (cid:104) T I (cid:105) = 7 . and (cid:104) T S (cid:105) = 4 . . In theexample below we use the average times from [1]. From the definition, the mean serial intervalcorresponds to the mean infectiousness time, so we find it natural to assume that θ I = (cid:104) T I (cid:105) / ≈ . . The factor 3 appears because the mean of the 3-rd order ( k = 3 in (6)) Gamma-distributionis triple the scaling parameter. The evaluation of θ S , the scale parameter in the distribution ofseverity of the symptoms, requires a different argument. We assume that the onset of symptoms isregistered when their severity reaches its maximum, which for the 3-rd order Gamma distributionoccurs at y = 2 θ . Hence, we assume that θ S = (cid:104) T S (cid:105) / ≈ . . III. RESULTS
In Fig. 1 we show an example of temporal dynamics of the model. The values of constants C I , C S , C R were chosen to reproduce the reported doubling period of infection at its early stages ≈ days and mortality ≈ . Now we show how the timing of detection and quarantining affectsthe spread of infection and the eventual outcome of the epidemics. Keeping the rest of parametersthe same, we vary the parameter θ S , which controls the time when the detectability rate is at itsmaximum. In Fig. 2 we show the fractions of dead, recovered, and remaining susceptible afterthe infection is over, that is, when the sufficient herd immunity is reached. These fractions in Fig.2 correspond to the respective fractions at the final time ( t = 200 days) in Fig. 1. Also similarlyto Fig. 1, we show what fraction of infected individuals were detected as infected through thecourse of their disease, and what fraction of infections was never detected. It follows from Fig.2 that the earlier detection reduces the death rate severalfold and almost doubles the number ofnever infected susceptible.A quick detection of new infection is normally done through contact tracing and testing. Weaccount for these measures by simply shifting the maximum of detectability of symptoms to earliertime, leaving the detection efficiency at the postulated sub-ideal level. Nevertheless, the beneficialepidemiological effect of even such sub-optimal early detection is clearly pronounced. Conversely,even a short delay in contact tracing or processing of the test results, corresponding to an increasein θ S , brings in more infections and higher fatality rate.5IG. 1: Temporal dynamics of fractions of dead (black), recovered (red), undetected infected(green), detected infected (blue), and susceptible (orange) individuals for the followingparameters: The infection rate coefficient C I = 2 , the coefficient in the rate of detection ofinfected C S = 0 . , the death rate coefficient C R = 0 . , the attenuation of death rate andinfectiousness after the detection of the disease ν = 0 . and χ = 0 . . The characteristic times ofdistribution of infectiousness θ I = 1 . and symptom severity θ S = 2 . were derived from thedata presented in [1]. IV. CONCLUSION
In this note we suggest to improve the current epidemiological models by incorporating thehistory of infections into a single node dynamics. The resulting delay equations are only slightlymore complex than the traditional SLIR ordinary differential equation models, yet they allowone to make precise and credible predictions of the effects of early detection and isolation of in-fected individuals, including pre-symptomatic ones, on the spread of infection. This is especiallyrelevant to the current COVID-19 epidemics, where the infectiousness was shown to increase syn-6IG. 2: The fraction of susceptible (black), recovered (red), never detected infected (green),detected infected (blue) and never infected susceptible (orange) individuals as the functions of thetime of maximal detectability of infection (maximum of symptoms) (cid:104) T S (cid:105) = 3 θ S . The remainingparameters are the same as in Fig. 1.chronously or even ahead of the visible onset of symptoms. We believe that this simple approachcould fitted to the clinical data and the data from detailed epidemiological case studies, and thenincorporated as the modified single node dynamics into large-scale network-based epidemiologicalmodels. 7 . APPENDIX: SIMILARITY OF OF SAIR MODEL AND DELAY EQUATION SOLUTIONS. The standard Susceptible ( S ( t ), Latent ( L ( t ) ), Infected ( I ( t ) ), and Removed ( R ( t ) ) or “SLIR”model is usually formulated in terms of ordinary differential equations, dSdt = − αSL (7) dLdt = αSL − βLdIdt = βL − γIdRdt = γI (8)(9)At the early stages of infection the majority of population is susceptible S ( t ) ≈ S = 1 . Here S, L, I, and R denote fractions of populations, so S + L + I + R = 1 . The system 9 becomesa linear system with the solution presented by a sum of increasing (first term) and decreasing(second term) exponential functions, L ( t ) I ( t ) = I α √ ( β − γ ) +4 βα √ ( β − γ ) +4 βα +( β − γ )2 √ ( β − γ ) +4 βα exp (cid:32) (cid:112) ( β − γ ) + 4 βα − ( β + γ )2 t (cid:33) + (10) + I − α √ ( β − γ ) +4 βα √ ( β − γ ) +4 βα − ( β − γ )2 √ ( β − γ ) +4 βα exp (cid:32) − (cid:112) ( β − γ ) + 4 βα − ( β + γ )2 t (cid:33) . Here I is the initial number of infected individuals, and L ( t = 0) = 0 . By adjusting constants α, β, γ the model can be fitted to the observed exponential growth of infection at its early stage.The delay equation takes into account that an infected individual becomes infectious only aftertime τ I and recovers or dies around time τ R > τ I , dIdt = S ( t ) βI ( t − τ I ) − γI ( t − τ R ) (11)Assuming again that we analyze the early stages of infection, S = S = 1 , we seek the solutionin the exponential form, I ( t ) = I exp( δt ) , where I is the initial number of infected individuals.Taking into account that β > γ for the infection growth, the resulting transcendental equation for δ always has a solution, δ = β exp( − δτ I ) − γ exp( δτ R ) ≡ φ ( δ ) . (12)8his can be concluded by observing that the function φ ( δ ) , defined by the right hand side of(12) changes from β − γ > for δ = 0 to for δ = + ∞ either monotonously decreasing when βτ I > γτ R , or going through a maximum in the opposite case. It is possible to find the approximatesolution of the Eq. (12). For example, at early stages of infection when recovery and death are notyet happening (i.e. for t < τ R ) or for a rapidly developing infection, the recovery term vanishes orbecomes insignificant. In the limits of short ( βτ I (cid:28) ) and long ( βτ I (cid:29) ) infectiousness delay, δ can be approximated as δ ≈ β βτ I , when βτ I (cid:28) , (13)or δ ≈ ln( βτ I ) − ln(ln( βτ I )) τ I when βτ I (cid:29) . Since both the ODE and delay equations can exhibit any desirable exponential growth for theearly stages of infection, the former is used much more often as it is simpler mathematicallyand can be implemented utilizing standard ODE integrators. However, as shown in our text, thedelay equations provide the additional information about the age of infection. Such information isessential for making correct predictions of the effects and timing of detection and quarantining onthe spread of infection [7, 8].
VI. ACKNOWLEDGMENTS
This work was supported by FONDECYT (Chile) grant 1200708. [1] X. He, E. H. Lau, P. Wu, X. Deng, J. Wang, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, et al.,medRxiv (2020).[2] L. Zou, F. Ruan, M. Huang, L. Liang, H. Huang, Z. Hong, J. Yu, M. Kang, Y. Song, J. Xia, et al., NewEngland Journal of Medicine , 1177 (2020).[3] K. K.-W. To, O. T.-Y. Tsang, W.-S. Leung, A. R. Tam, T.-C. Wu, D. C. Lung, C. C.-Y. Yip, J.-P. Cai,J. M.-C. Chan, T. S.-H. Chik, et al., The Lancet Infectious Diseases (2020).[4] A. Ling and Y. Leo, Emerging Infectious Diseases , 1052 (2020).
5] F.-X. Lescure, L. Bouadma, D. Nguyen, M. Parisey, P.-H. Wicky, S. Behillil, A. Gaymard,M. Bouscambert-Duchamp, F. Donati, Q. Le Hingrat, et al., The Lancet Infectious Diseases (2020).[6] O. Karin, Y. M. Bar-On, T. Milo, I. Katzir, A. Mayo, Y. Korem, B. Dudovich, A. J. Zehavi, N. Davi-dovich, R. Milo, et al., medRxiv (2020).[7] S. Ruschel, T. Pereira, S. Yanchuk, and L.-S. Young, Journal of mathematical biology , 249 (2019).[8] L.-S. Young, S. Ruschel, S. Yanchuk, and T. Pereira, Scientific reports , 1 (2019).[9] H.-Y. Cheng, S.-W. Jian, D.-P. Liu, T.-C. Ng, W.-T. Huang, H.-H. Lin, et al., medRxiv (2020)., 1 (2019).[9] H.-Y. Cheng, S.-W. Jian, D.-P. Liu, T.-C. Ng, W.-T. Huang, H.-H. Lin, et al., medRxiv (2020).