Abstract
The initial transient phase of an emerging epidemic is of critical importance for datadriven model building, modelbased prediction of the epidemic trend, and articulation of control/prevention strategies. Quantitative models for realworld epidemics need to be memorydependent or nonMarkovian, but this presents difficulties for data collection, parameter estimation, computation, and analyses. In contrast, such difficulties do not arise in the traditional Markovian models. To uncover the conditions under which Markovian and nonMarkovian models are equivalent, we develop a comprehensive computational and analytic framework. We show that the transientstate equivalence holds when the average generation time matches the average removal time, resulting in minimal Markovian estimation errors in the basic reproduction number, epidemic forecasting, and evaluation of control strategy. The errors depend primarily on the generationtoremoval time ratio, while rarely on the specific values and distributions of these times. Overall, our study provides a general criterion for modeling memorydependent processes using Markovian frameworks.
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Introduction
When an epidemic emerges, the initial transient phase of the disease spreading dynamics before a steady state is reached is of paramount importance, for two reasons^{1,2,3,4,5,6,7,8,9,10,11}. First, estimating key indicators or parameters, e.g., the generation time, serial intervals, and basic reproduction number, is crucial for predicting and formulating control strategies when the underlying dynamical process has not reached a steady state. Second, it is during the transient phase that control and mitigation strategies can be effectively applied to prevent a largescale outbreak. Prediction and control depend, of course, on a quantitative model of the epidemic process, which can be constructed based on the key parameters estimated from data collected during the transient phase. In principle, since the dynamical processes underlying realworld epidemics are generally memorydependent in the sense that the state evolution depends on the history, a rigorous modeling framework needs to be of the nonMarkovian type, but this presents great challenges in terms of data collection, parameter estimation, computation and analyses^{12,13,14}. The difficulties can be alleviated by adopting the traditional simplified memoryless Markovian framework, which, however, may potentially result in deviations^{15,16,17,18,19,20,21,22,23,24,25,26}. An outstanding question is, are there specific conditions under which a Markovian epidemic outbreak could be equivalent to the nonMarkovian counterpart during the transient phase, enabling an accurate description of memorydependent transmission within the Markovian framework? Additionally, another important issue remains, how are the errors of Markovian estimation determined? The purpose of this paper is to provide a comprehensive answer to these questions.
The COVID19 pandemic has highlighted the need and importance of understanding disease spreading and transmission to accurately predict, control, and manage future outbreaks through nonpharmacological interventions and vaccine allocation strategies^{1,2,3,4,5,6,7,8,9,10}. To accomplish these goals, accurate mathematical modeling of the diseasespreading dynamics is key. In a general population, epidemic transmission occurs via some kind of point process, where individuals become infected at different points in time. It has been known that point processes in the real world are typically nonMarkovian with a memory effect in which the distribution of the interevent times is not exponential^{2,23,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41}. For example, the interevent time distribution arising from the virus transmission with COVID19 is not of the memoryless exponential type but typically exhibits memorydependent features characterized by the Weibull distribution^{2}. Strictly speaking, from a modeling perspective, disease spreading should be described by a nonMarkovian process. A nonMarkovian approach takes into account historical memory of disease progression, mathematically resulting in a complex set of integrodifferential equations in the form of convolution. (Note that besides the transmission process, where the infection capacity is highly dependent on the elapsed time from the infection, there are other types of memorydependent factors. For instance, in certain metapopulation network models, the infection capacity of a node is influenced by the transmission path leading to that node, rather than solely relying on the time of infection^{42}. Thus, it is important to mention that we only take the elapsed time into consideration in this work.)
There are difficulties with nonMarkovian modeling of memorydependent disease spreading. The foremost is data availability. In particular, while standard epidemic spreading models are available, the model parameters need to be estimated through data. A nonMarkovian model often requires detailed and granular data that can be difficult to get, especially during the early stage of the epidemic where accurate modeling is most needed^{12}. From a theoretical point of view, it is desired to obtain certain closedform solutions for key quantities such as the onset and size of the epidemic outbreak, but this is generally impossible for nonMarkovian models^{15,43}. Computationally, accommodating memory effects in principle makes the underlying dynamical system infinitely dimensional, practically requiring solving an unusually large number of dynamical variables through a large number of complex integrodifferential equations^{13,14}. In contrast, in an idealized Markovian point process, events occur at a fixed rate, leading to an exponential distribution for the interevent time intervals and consequently a memoryless process. If the spreading dynamics were of the Markovian type, the aforementioned difficulties associated with nonMarkovian dynamics no longer exist. In particular, a Markovian spreading process can be described by a small number of ordinary differential equations with a few parameters that can be estimated even from sparse data, and the numerical simulations can be carried out in a computationally extremely efficient manner^{15,16,17,18,19,20,21,22,23,24,25,26}. For these reasons, many recent studies of the COVID19 pandemic assumed Markovian behaviors to avoid the difficulties associated with nonMarkovian modeling^{3,4,5,6,7,8,9,10}. The issue is whether such a simplified approach can be justified. Addressing this issue requires a comprehensive understanding of the extent to which the Markovian approach represents a good approximation to model nonMarkovian type of memorydependent spreading dynamics, and specifically of the conditions under which the Markovian theory can produce accurate results that match those from the nonMarkovian model. This highlights the importance of studying the equivalence between nonMarkovian and Markovian dynamics.
There were previous studies of the socalled steadystate equivalence between Markovian and nonMarkovian modeling for epidemic spreading. In particular, when the system has reached a steady state, such an equivalence can be established through a modified definition of the effective infection rate^{18,21,22}. From a realistic point of view, the equivalence limited only to steady states may not be critical as the transient phase of the spreading process before any steady state is reached is more relevant and important. For example, when an epidemic occurs, it is of fundamental interest to estimate the key indicators such as the generation time (the time interval between the infections of the infector and infectee in a transmission chain), the serial interval (the time from illness onset in the primary case to illness onset in the secondary case), and the basic reproduction number (the average number of secondary transmissions from one infected person), but they are often needed to be estimated when the dynamics have not reached a steady state^{2,3,4,5,6,7,8,9,10,11}. It is the equivalence in the transient dynamics rather than the steady state that determines whether the transmission features in the early stages of a memorydependent disease outbreak can be properly measured through Markovian modeling. Moreover, it is only during the transient phase that control and mitigation strategies can be effective in preventing a largescale outbreak. Discovering when and how a nonMarkovian process can be approximated by a Markovian process during the transient state is thus of paramount importance. To our knowledge, such a “transientstate equivalence”, where the Markovian and nonMarkovian transmission models produce similar behaviors over the entire transient transmission period, has not been established. In fact, the conditions under which the transient equivalence may hold are completely unknown at present.
In this paper, we present results from a comprehensive study of how memory effects impact the Markovian estimations in terms of the errors that arise from the Markovian hypothesis. We consider both the steadystate and transientstate equivalences between nonMarkovian and Markovian models. We first rigorously show that, in the steady state, a memorydependent nonMarkovian spreading process is always equivalent to certain Markovian (memoryless) ones. We then turn to the transient states and find that an approximate equivalence can still be achieved but only if the average generation time matches the average removal time in the memorydependent nonMarkovian spreading dynamics. Qualitatively, the equality of the two times gives rise to a memoryless correlation between the infection and removal processes, thereby minimizing the impact of any memory effects. We establish that equality gives the condition under which Markovian theory accurately describes memorydependent transmission.
One fundamental quantity underlying an epidemic process is the basic reproduction number R_{0}. Our theoretical analysis indicates that, when the average generation and removal times are equal, the transientstate equivalence between memorydependent and memoryless transmissions will minimize the error of the Markovian approach in estimating R_{0} and lead to its accurate epidemic forecasting and prevention evaluation. Another finding is that the generationtoremoval time ratio plays a decisive role in the accuracy of the Markovian approximation. Specifically, if the average generation time is smaller (greater) than the average removal time, the Markovian approximate will lead to an overestimation (underestimation) of R_{0} and epidemic forecasting as well as the errors of the prevention evaluation, which can also be verified based on readily accessible clinical data of four types of real diseases.
The estimation accuracy is largely determined by the time ratio but rarely depends on the particular forms of time distributions or the specific values of the average generation and removal times. This property is of great practical significance because it is in general challenging to obtain the detailed distributions of the generation and removal times in the early stages of the epidemic^{44}, but their average values can be reliably estimated even during the transient phase^{45,46,47,48,49}. Moreover, based on this property, we have developed a semiempirical mathematical relationship that connects the errors in estimating R_{0} with the generationtoremoval time ratio. This relationship holds practical value as it can be utilized to rectify errors in realworld scenarios. The rectification of R_{0} and epidemic forecasting can be accomplished through our webbased application^{50}.
Overall, our study establishes a general criterion for modeling memorydependent processes within the context of Markovian frameworks. Once the condition for the existence of a transientstate equivalence between Markovian and nonMarkovian dynamics is fulfilled, epidemic forecasting and prevention evaluation can be carried out using the Markovian model, again based solely on the data collected from the transient phase.
Results
The overall structure of this work is depicted in Fig. 1. The section titled “Model” presents the Model building of the agestratified SusceptibleInfectedRemoved (SIR) spreading (Fig. 1a), highlighting the difference between the Markovian (memoryless) and nonMarkovian (memorydependent) dynamics (Fig. 1b). In the “Dynamical equivalence” section, we demonstrate the equivalence between Markovian (memoryless) and nonMarkovian (memorydependent) dynamics for steady state and transient dynamics, which will further lead to the accurate description of memorydependent dynamics by the Markovian theory (Fig. 1c). The section “Markovian approximation of memorydependent spreading dynamics” analyzes the errors of the Markovian approach in estimating R_{0}, epidemic forecasting, and prevention evaluation (Fig. 1d). (Note that this study involves numerous parameters and variables, all of which have been comprehensively detailed and listed for reference in Supplementary Table S1 of Supplementary Note 1.)
Model
We articulate an agestratified SIR spreading dynamics model, in which the entire population is partitioned into various age groups with intricate agespecific contact rates among them. The distribution of population across different age groups is represented by an age distribution vector (p), with the agedependent contact matrix (A) quantifying the transmission rates between different age groups. Both p and A can be constructed from empirical data^{51,52}. For convenience, to distinguish between the actual dynamical process and its theoretical treatment, throughout this paper we use the terms “memorydependent” and “memoryless” to describe actual spreading processes and Monte Carlo simulations, while in various theoretical analyses, the corresponding terms are “nonMarkovian” and “Markovian.”
The mechanism of disease transmission across different age groups and the recovery or death of infected individuals can be described by the SIR compartmental model, as illustrated in Fig. 1a, where the individuals possess three types of states: susceptible (S), infected (I), and removed (R). Susceptible individuals (S) have not contracted the disease and are at risk of being infected. Infected individuals (I) have contracted the disease and can infect others. Removed individuals (R) who have recovered or died from the disease. There are two dynamical processes: (1) infection during which susceptible individuals become infected by others and transition to the I state so as to become capable of infecting others, as shown in Fig. 1a(i–iii), and (2) removal during which infected individuals recover or die from the disease transmission and transition to the R state, as shown in Fig. 1a(iv). The ability to infect others of an infected individual can be characterized by the infection time distribution, \({\psi }_{\inf }(\tau )\) where τ denotes the time elapsed between the time the individual is infected and the current time, and the probability of the infection process occurring during the time interval [τ,τ + dτ) is given by \({\psi }_{\inf }(\tau )d\tau\), as shown in Fig. 1b(i). Likewise, the removal process is described by the removal time distribution, ψ_{rem}(τ), where the probability of a removal occurring within the time interval [τ,τ + dτ) is given by ψ_{rem}(τ)dτ, as shown in Fig. 1b(ii). The time distributions of the infection and removal processes with memory effects are general, with the exponential distributions associated with the memoryless process being a special case of the memorydependent process.
The generic memorydependent SIR spreading dynamics can be described by a set of deterministic integrodifferential equations:
where s_{l}(t), i_{l}(t), and r_{l}(t), respectively, denote the fractions of the susceptible, infected, and removed individuals in age group l. ŕ_{l} = r_{l}(0) represents the initial fraction of the removed individuals in age group l. The term c_{l}(t) = 1 − s_{l}(t) = i_{l}(t) + r_{l}(t) represents the fraction of cumulative infections (including both infections and removals; note that the physical meaning of dc_{l}(t) is the new infected fraction in age group l at time t, which can further spread the infection to others, so the value of dc_{l}(0) is considered as the fraction of infection seeds, where the initial removed fraction is not taken into account) in age group l, while p_{m} denotes the mth element of the vector p, A_{lm} denotes the lmth element of matrix A, k is a parameter to adjust the overall contacts and n is the total number of age groups. The quantity \({\omega }_{\inf }(\tau )\) represents the hazard function of \({\psi }_{\inf }(\tau )\), meaning the rate at which infection happens at τ, given that the infection has not occurred before τ. Ψ_{rem}(τ) is the survival function of ψ_{rem}(τ), meaning the probability that the removal has not occurred by τ (see Method for detailed calculations for hazard and survival functions). When the infection and removal time distributions are known, Eqs. ((1)–(3)) provide an accurate description of generic (including memorydependent and memoryless) SIR spreading Monte Carlo simulations in an agestratified populationbased system. As shown in Fig. 2a–c, the theory is validated by the agreement between the numerical solutions of Eqs. ((1)–(3)) and the results from direct Monte Carlo simulations (see Method for a detailed description of the Monte Carlo simulation procedure).
Equations (1–3) provide a general framework encompassing both nonMarkovian and Markovian descriptions. If the infection and removal time distributions are exponential: \({\psi }_{\inf }(\tau )=\gamma {e}^{\gamma \tau }\) and ψ_{rem}(τ) = μe^{−μτ}, Eqs. (1–3) can be reformulated into a Markovian theory and further simplified into a set of ordinary differential equations with the constant infection and removal rates γ and μ (a detailed derivation of these equations is presented in Supplementary Note 2) :
While the nonMarkovian and Markovian theories [Eqs. (1–3) and Eqs. (4–6), respectively] accurately describe the memorydependent and memoryless Monte Carlo simulations, we focus on whether the Markovian theory can accurately capture memorydependent dynamics and how memory effects influence its accuracy. For this purpose, we seek to establish the equivalence between Markovian and nonMarkovian approaches for describing spreading dynamics.
Dynamical equivalence
Equations (1–6) provide a base to study the steadystate equivalence and transientstate equivalence between nonMarkovian and Markovian theories, where a steady state characterizes the longterm dynamics of disease spreading and a transient state is referred to as the shortterm behavior prior to system’s having reached the steady state. As illustrated in Fig. 1c, note that steadystate equivalence means that the two types of spreading dynamics attain identical steady states^{18,21,22}, whereas transientstate equivalence implies that two types of dynamics are consistent throughout the entire transmission period. Transientstate equivalence thus implies steadystate equivalence, but not vice versa. Since Eqs. (1–6) also provides a numerical framework for Monte Carlo simulations, the terms “steadystate equivalence” and “transientstate equivalence” not only describe the connection between nonMarkovian and Markovian theories, but also illustrate the relationship between memorydependent and memoryless processes. Therefore, the equivalence between the two theories implies the equivalence between the two corresponding processes, and vice versa.
Steadystate equivalence
Equations (1)–(3) give the following transcendental equation for determining the steady state (see Supplementary Note 3 for detailed derivation):
where \({\tilde{s}}_{l}=\mathop{\lim }\nolimits_{t\to +\infty }{s}_{l}(t)\) and \({\tilde{r}}_{l}=\mathop{\lim }\nolimits_{t\to +\infty }{r}_{l}(t)\) denote the fractions of the susceptible and removed individuals in age group l at the steady state (note that \({\tilde{s}}_{l}=1{\tilde{r}}_{l}\), because at steady state, no infection exists), while ś_{l} = s_{l}(0) and ŕ_{l} = r_{l}(0) represent the initial fractions of the susceptible and removed individuals in this age group. For nonMarkovian dynamics, basic reproduction number R_{0} can be determined by:
For Markovian dynamics, R_{0} is given by:
where \({{{\Lambda }}}_{\max }\) is the maximum eigenvalue of the matrix kA ∘ p, and ∘ denotes a rowwise Hadamard product between a matrix and a vector (see Supplementary Note 3 for a detailed description). Since Eq. (7) applies to both nonMarkovian and Markovian dynamics, an identical R_{0} value in the two cases will result in equivalent steady states from the same initial conditions. Consequently, for a given nonMarkovian spreading process, there exists an infinite number of Markovian models with the same steady state, as the R_{0} value is only determined by the ratio of γ to μ, but not by either value.
As shown in Fig. 2d, memorydependent and memoryless spreading dynamics that reach the same steady state with the identical R_{0} value confirm the steadystate equivalence. Figure 2e demonstrates that, even for R_{0} ranging from 0.023 to 4.63, the equivalent memorydependent and memoryless spreading dynamics still produce highly consistent steady states that can be calculated from Eq. (7), which share the same critical point of phase transition at R_{0} = 1.
Transientstate equivalence
From the preceding section, we used the basic reproduction number R_{0}, a fundamental metric quantifying the number of secondary infections generated by a single individual, to characterize the steadystate equivalence. Here, we propose to quantify the transientstate equivalence through the average generation time T_{gen} that measures the “velocity” at which secondary infections occur. This time can be calculated as^{2,53}:
where
is the generation time distribution. Effectively, T_{gen} measures the average duration of disease transmission from an infected individual to the next generation of individuals. Likewise, the average infection time \({T}_{\inf }\) and the average removal time T_{rem} are defined as the mean values of the infection and removal time distributions:
In calculating T_{gen}, the individual’s removal is taken into account, while \({T}_{\inf }\) measures the average time of the first disease transmission of an infectious individual without factoring in removal. In the classical memoryless transmission with exponential distributions \({\psi }_{\inf }(\tau )\) and ψ_{rem}(τ), the equality T_{gen} = T_{rem} holds. However, for memorydependent spreading, the possible scenarios are: T_{gen} = T_{rem}, T_{gen} < T_{rem}, or T_{gen} > T_{rem}. Specifically, because T_{gen}, \({T}_{\inf }\) and T_{rem} all represent the mean values of distributions, it is possible for T_{rem} to be shorter than T_{gen} or \({T}_{\inf }\) in some situations. And our webbased application demonstrates the impact of parameters on the time distributions (infection, removal, and generation) as well as their average times^{50}.
For a nonMarkovian spreading process, if the equality T_{gen} = T_{rem} holds, it can be approximately equivalent to a Markovian one in the transient state, because it satisfies the following equation:
which exhibits a memoryless transmission pattern similar to Markovian dynamics (see Supplementary Note 4 for a detailed analysis). Intuitively, the equality T_{gen} = T_{rem} signifies that the infection and removal processes occur concurrently, which in turn leads to a memoryless relationship between the two processes, thereby minimizing the memory effects. Furthermore, to determine the corresponding Markovian parameters γ and μ of the Markovian transmission which is equivalent to the nonMarkovian dynamics in the transient state, we need to utilize the EulerLotka equation^{45,53,54}:
where g denotes the growth rate of the nonMarkovian dynamics and is another measure of how quickly the epidemic is spreading within a population. Therefore, we can calculate the values of the basic reproduction number, R_{0}, and growth rate, g, in the nonMarkovian dynamic by using Eqs. (8), (10), and (12). Additionally, the Markovian form of ψ_{gen}(τ) according to Eq. (10) is μe^{−μτ}, and the equivalent Markovian and nonMarkovian dynamics in the transient state have the same values of R_{0} and the equal values of g. By substituting ψ_{gen}(τ) = μe^{−μτ} and the calculated R_{0} and g into Eq. (12), we can determine the value of μ. Furthermore, using Eq. (9), we can find the value of γ based on μ. Hence, the Markovian parameters γ and μ are determined as follows:
And we provide visualizations that illustrate how the values of γ and μ are influenced by the distribution parameters in our webbased application^{50}.
As illustrated in Fig. 2f–g, when the equality T_{gen} = T_{rem} holds for the nonMarkovian dynamics, Eq. (11) holds, which can be seen by comparing the susceptible curve calculated from Eq. (1) to that inferred from Eq. (11), as shown in Fig. 2f. In this case, the Markovian spreading curves deduced from Eqs. ((13) and (14)) closely align with the nonMarkovian transient curves, as shown in Fig. 2g. However, as shown in Fig. 2h–i, if the equality does not hold, the equivalence in transient states breaks down. It is important to note that the EulerLotka equation assumes an exponential growth of a disease outbreak and is only reasonable at the initial stage. Consequently, as the cumulative infections increase (Fig. 2g), the Markovian curves will exhibit slight deviations from the nonMarkovian counterparts. Meanwhile, because the equivalent dynamics share the same R_{0}, they will ultimately reach the same steady state, ensuring that deviations will diminish while they approach the steady state.
To evaluate, under different values of the generationtoremoval time ratio η ≡ T_{gen}/T_{rem} for nonMarkovian dynamics we introduce a metric, ε, to quantify the difference from the corresponding Markovian results calculated from Eqs. ((13) and (14)) (see “Method” for detailed definition of ε). Figure 2j shows, for nonMarkovian numerical calculations, five scenarios under various forms of time distributions \({\psi }_{\inf }(\tau )\) and ψ_{rem}(τ) constrained by certain average infection and removal times: Weibull, \({T}_{\inf }=5\), T_{rem} = 7; Weibull, \({T}_{\inf }=7\), T_{rem} = 7; Weibull, \({T}_{\inf }=5\), T_{rem} = 5; lognormal, \({T}_{\inf }=5\), T_{rem} = 7; and gamma, \({T}_{\inf }=5\), T_{rem} = 7 (see “Method” for detailed definitions of Weibull, lognormal, and gamma distributions). The T_{gen} value is adjusted to obtain different values of \(\ln \eta\). For \(\ln \eta =0\), i.e., T_{gen} = T_{rem}, the “distance” ε between the transient states of nonMarkovian and Markovian dynamics with parameters determined from Eqs. ((13)–(14)) is minimal. Otherwise, ε increases as \(\ln \eta\) deviates from zero. Meanwhile, Fig. 2j shows that ε depends primarily on the ratio of T_{gen} to T_{rem}, but rarely on the values of T_{gen}, \({T}_{\inf }\), or T_{rem}. Meanwhile, the specific form of the time distributions also has limited influence on it.
Furthermore, it is important to note that the condition where T_{gen} = T_{rem} in a nonMarkovian dynamic ensures transientstate equivalence between this nonMarkovian transmission and a Markovian one, but according to Eqs. (13) and (14), it does not imply that the average generation and removal times of the nonMarkovian dynamic must be equal to those of the equivalent Markovian one. For instance, if a nonMarkovian dynamic satisfies the condition of transientstate equivalence and we keep its average generation and removal times fixed, altering the shape of the corresponding time distributions will change the transmission speed^{53}. This change, in turn, affects the infection and removal rates of the equivalent Markovian dynamic, leading to different average generation and removal times for the Markovian equivalent dynamic (see Supplementary Fig. S1 and Supplementary Notes 5 and 6 for a detailed analysis).
Markovian approximation of memorydependent spreading dynamics
As illustrated in Fig. 1d, testing the applicability of Markovian theory for memorydependent spreading dynamics requires three steps. The first step is fitting, where the memorydependent Monte Carlo simulation data are divided into two parts: (a) a short early stage used as the training data for fitting the Markovian parameters in Eqs. ((4)–(6)), and (b) the remaining testing data for evaluating the performance of the Markovian model (see “Method” for details of the fitting procedure). The second step is to employ the Markovian model, equipped with fitted Markovian parameters, to accomplish various tasks, such as estimating R_{0}, predicting outbreaks and assessing the prevention effects of different vaccination strategies. The third step is testing, i.e., evaluating the accuracy of the Markovian model, e.g., by comparing the estimated and actual R_{0} values, disease outbreaks and prevention effects. As realworld disease spreading is subject to environmental, social and political disturbing factors, for the fitting and testing steps, we conduct Monte Carlo simulations of stochastic memorydependent disease outbreaks to generate the training and testing data.
Here, we first analyze the influence of η on the estimation of R_{0} using the Markovian theory, and design two tasks to evaluate the applicability of the theory in epidemic forecasting and prevention evaluation of memorydependent spreading. For comparison, we also generate the corresponding results from the nonMarkovian theory in the two tasks.
Estimation of basic reproduction number
Estimating basic reproduction number R_{0} is crucial for determining the ultimate prevalence of disease spreading and for assessing the effectiveness of various disease containment measures^{45,53,54}. When using the Markovian theory to fit the earlystage transmission of a memorydependent process, a key parameter that can affect the estimation of R_{0} is the ratio η. To develop an analysis, recall the basic principle for estimating R_{0}: disease spreading dynamics can be viewed as a combination of two parallel processes: infection and removal. In particular, the infection process is the reproduction of the disease within each generation, where each infected individual generates an average of R_{0} newly infected individuals in the subsequent generation after a mean time period T_{gen}. In the removal process, infected individuals are removed from the spreading chain, where each generation takes an average time T_{rem} to be removed. For a Markovian type of dynamics with constant γ and μ, the equality T_{gen} = T_{rem} holds. Consequently, during the Markovian fitting step, the average number of new infections upon the removal of a single infected individual is taken as the value of R_{0}. For memorydependent spreading, if the equality T_{gen} = T_{rem} holds, the memorydependent spreading curves will possess an approximate memoryless feature so that R_{0} can be still be estimated by counting the number of new infections at the time when the current generation of infections is removed, as shown in Fig. 3a. However, for T_{gen} < T_{rem}, more than one generation is produced while the current generation is removed, R_{0} estimated by the Markovian theory will represent an overestimate, as shown in Fig. 3b. For T_{gen} > T_{rem}, less than one generation is created during T_{rem}, the Markovian theory will give an underestimate of R_{0}, as shown in Fig. 3c.
Figure 3d shows the Markovian fitting for memorydependent spreading simulation curves, which exhibit identical values of R_{0} under T_{gen} = T_{rem} (red curves), T_{gen} < T_{rem} (blue curves), and T_{gen} > T_{rem} (green curves). When the equality T_{gen} = T_{rem} holds, the Markovian theory with fitted parameters generates accurate predictions of the future evolution (red × symbols). For T_{gen} < T_{rem}, the outbreak in the initial stage is accelerated, resulting in an overestimation by the Markovian theory (blue + symbols). For T_{gen} > T_{rem}, the initial outbreaks are decelerated, leading to an underestimation by the Markovian approach (green Δ symbols).
The above qualitative insights lead to a semiempirical relationship between the Markovianestimated basic reproduction number \({\hat{R}}_{0}\) and its actual value R_{0} as:
where a is a positive coefficient (see “Method” for a detailed derivation). The value of a is a crucial and constant parameter in Eq. (15), and it needs to be determined by fitting it to the data. Once this constant a is obtained, the actual value of R_{0} can be derived by adjusting the estimated \({\hat{R}}_{0}\) based on Eq. (15), and more accurate steady state can be calculated by using Eq. (7).
Equation (15) implies the relationship \(\ln (\ln {R}_{0}/\ln {\hat{R}}_{0})=a\ln \eta\). We use Weibull time distributions \({\psi }_{\inf }(\tau )\) and ψ_{rem}(τ) constrained by certain average infection and removal times: \({T}_{\inf }=5\), T_{rem} = 7; \({T}_{\inf }=7\), T_{rem} = 7; \({T}_{\inf }=5\), T_{rem} = 5; \({T}_{\inf }=7\), T_{rem} = 5; and \({T}_{\inf }=6\), T_{rem} = 6 for memorydependent Monte Carlo simulations. Figure 3e shows the linear relationship between \(\ln (\ln {R}_{0}/\ln {\hat{R}}_{0})\) and \(\ln \eta\), providing support for our qualitative analysis of the Markovian estimation. The estimation of R_{0} also depends on the ratio η and is relatively insensitive to the specific values of T_{gen}, \({T}_{\inf }\), or T_{rem} for the same form of time distribution. The results in the inset of Fig. 3e further confirm that the estimated \({\hat{R}}_{0}\) approaches 1 when T_{gen} is much larger than T_{rem} and tends to +∞ when T_{gen} is much smaller than T_{rem}. By fitting the available data, we have determined the value of a to be 1.59. After obtaining the value of a, we can develop our webbased application for rectifying R_{0} and epidemic forecasting^{50}. We also plot additional experimental results for lognormal and gamma distributions (lognormal, \({T}_{\inf }=5\), T_{rem} = 7; and gamma, \({T}_{\inf }=5\), T_{rem} = 7) in Supplementary Note 7. Referring to Supplementary Fig. S2, various forms of time distributions may affect the Markovian estimation of R_{0}. However, their influence appears less compared to the impact of the value of η. Hence, it can be concluded that the distribution form seldom affects the estimation of R_{0}.
Epidemic forecasting
As suggested in Fig. 1d, we evaluate the efficacy of Markovian theory for epidemic forecasting. We use the early stage of Monte Carlo simulation data to fit parameters under both Markovian and nonMarkovian hypotheses and then to predict future disease outbreaks. The remaining simulation data are leveraged to evaluate the accuracy of the Markovian and nonMarkovian forecasting results. Regardless of the forms of time distributions in the memorydependent Monte Carlo simulations (Weibull, lognormal, or gamma), the nonMarkovian model fits the training data in a consistent manner, i.e., by assuming the actual time distributions as unknown and treating them as Weibull time distributions.
Figure 4a–c shows the evolution of the spreading dynamics from three types of memorydependent Monte Carlo simulations with Weibull infection and removal distributions, where the shape parameters \({\alpha }_{\inf }\) and α_{rem} are selected according to \(\ln {\alpha }_{\inf }=0.3,\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}=1.2\) (Fig. 4a), \(\ln {\alpha }_{\inf }=0.45,\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}=0.45\) (Fig. 4b), and \(\ln {\alpha }_{\inf }=1.2,\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}=0.3\) (Fig. 4c), for \({T}_{\inf }=5\) and T_{rem} = 7. For the Weibull distributions, we have \({\alpha }_{\inf }\, < \,{\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\), \({\alpha }_{\inf }={\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\) and \({\alpha }_{\inf }\, > \,{\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\), corresponding to T_{gen} < T_{rem}, T_{gen} = T_{rem}, and T_{gen} > T_{rem}, respectively. We compare the simulated cumulative infected fractions to those predicted by the Markovian and nonMarkovian theories. In general, the nonMarkovian theory provides more accurate predictions than the Markovian theory. For the specific parameter setting \(\ln {\alpha }_{\inf }=0.45,\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}=0.45\) (i.e., T_{gen} = T_{rem}), both theories yield a high accuracy.
The accuracy can be assessed through the forecasting error ε^{+} that evaluates whether a theory overestimates or underestimates the steadystate cumulative infection, i.e., quantifying the extent of deviation between the results obtained from Markovian or nonMarkovian theories and those derived from Monte Carlo simulations (see “Method” for detailed definition of ε^{+}). A plus value of ε^{+} means overestimation while minus value indicates underestimation. We evaluate the accuracy measure ε^{+} in the parameter plane of \(\ln {\alpha }_{\inf }\) and \(\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\), ranging from −0.3 to 1.2. Figure 4d, e shows that the Markovian accuracy is sensitive to parameter changes: underestimated if \({\alpha }_{\inf }\) is greater than α_{rem} (T_{gen} > T_{rem}), overestimated when \({\alpha }_{\inf }\) is smaller than α_{rem} (T_{gen} < T_{rem}), and a high forecasting accuracy is achieved only for \({\alpha }_{\inf }={\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\) (T_{gen} = T_{rem}). In contrast, the nonMarkovian theory yields highly accurate results in the whole parameter plane, with only a slight underestimation for \({\alpha }_{\inf }\gg {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\). This can be primarily attributed to the growing challenge of fitting simulation data, where the absolute derivatives of R_{0} with respect to parameters become extremely high, making the fitting process much sensitive to the parameters^{55,56}. This occurs when \({\alpha }_{\inf }\) is greater than α_{rem} or T_{gen} is much larger than T_{rem}, causing the Monte Carlo curves to become more sensitive to the parameters and potentially deviate from the accurate theoretical curves (see Supplementary Fig. S3 and Supplementary Note 8 for details).
Using the five scenarios specified in Fig. 2j for memorydependent Monte Carlo simulations, we obtain the relationship between ε^{+} and \(\ln \eta\), as shown in Fig. 4f. It can be seen that, in the Markovian framework, an overestimation arises for T_{gen} < T_{rem}, and an underestimation occurs for T_{gen} > T_{rem}. Only when T_{gen} = T_{rem} is an accurate estimate achieved. In general, the nonMarkovian theory provides much more accurate forecasting than the Markovian theory, especially when T_{gen} and T_{rem} are not equal. The results further illustrate that the specific values of T_{gen}, \({T}_{\inf }\), or T_{rem} have little impact on forecasting accuracy, and the impact of the time distribution forms is much lower compared to the influence exerted by the η value.
To establish the relevance of these results to realworld diseases, we obtain the distributions of \({\psi }_{\inf }(\tau )\) and ψ_{rem}(τ) for four known infectious diseases, including COVID19, SARS, H1N1 influenza, and smallpox, using the information in refs. ^{2,33,34,35,36,37,38,39,40}. We then calculate the corresponding values of ε^{+} and \(\ln \eta\) based on the Markovian and nonMarkovian approaches. As demonstrated in Fig. 4f, the positions of the four diseases in the (\(\ln \eta\), ε^{+}) plane are consistent with the results of our estimations. Because the data were from the reports of laboratoryconfirmed cases incorporating the effects of the quarantine and distancing from susceptible individuals after the confirmation of the diagnosis, T_{gen} of the four diseases are all smaller than the corresponding values of T_{rem}, leading to some overestimation for the Markovian forecasting results.
Evaluation of vaccination strategies
In the development and application of a theory for disease spreading, assessing the effects of different vaccination strategies is an important task. Here we consider five prioritization strategies for vaccine distribution^{5}: individuals under 20 years (denoted as m = 1), adults between 20 and 49 years (m = 2), adults above 20 years (m = 3), adults above 60 years (m = 4), and all age groups (m = 5), and implement these strategies in Monte Carlo simulations (see “Method” for the detailed procedure of vaccination in Monte Carlo simulations). Figure 5a shows the results of epidemic evolution with these vaccination strategies, where the shape parameters are chosen according to \(\ln {\alpha }_{\inf }=0.45\) and \(\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}}=0.45\) (T_{gen} = T_{rem}). Figure 5b–c shows the results from the Markovian and nonMarkovian theories, respectively, with the corresponding fitted parameters for the vaccination strategies (see “Method” for the detailed procedure of vaccination in theoretical calculations). These results indicate that the Markovian and nonMarkovian theories yield the correct epidemic evolution and future outbreaks under different vaccination scenarios, when T_{gen} = T_{rem}.
To characterize the effectiveness of different vaccination strategies in blocking disease transmission, we introduce a vaccination effectiveness vector, δ, whose mth element quantifies the cumulative infected fraction with the mth vaccination strategy in the steady state: \({\delta }_{m}={\tilde{c}}_{m}\), for m = 0, …, 5 (subscript m = 0 indicates the results without vaccination). Figure 5d shows that the δ vectors from the Monte Carlo simulation, Markovian and nonMarkovian theories from Fig. 5a–c, respectively.
We further introduce a metric, the socalled prevention evaluation error ε*, that gauges the ability of the Markovian and nonMarkovian theories to estimate the total effectiveness of vaccination, i.e., measuring the disparity between the results calculated by the Markovian or nonMarkovian theories and those obtained through Monte Carlo simulations considering various vaccination strategies (see “Method” for the detailed definition of ε*). Figure 5e, f shows the average values of ε^{*} of the two theories in the simulation parameter plane using 100 independent realizations, which are similar to those in Fig. 4d–e, indicating that the error mainly comes from the R_{0} estimation. In general, the Markovian theory performs well only in the diagonal area of the parameter plane where \({\alpha }_{\inf }={\alpha }_{{{{{{{{\rm{rem}}}}}}}}}\), as shown in Fig. 5e, and the nonMarkovian theory outperforms the Markovian counterpart in most cases, as shown in Fig. 5f.
Meanwhile, we assess the ability of both the Markovian and nonMarkovian theories to detect the optimal vaccination strategy. We also define a quantity, optimization failure probability \(\hat{\varepsilon }\), to quantify the probability of a theory failing to identify the optimal strategy, i.e., that leads to the lowest cumulative infection among these strategies (see “Method” for the detailed definition of \(\hat{\varepsilon }\)). Figure 5g–h illustrates the results of \(\hat{\varepsilon }\) for the two theories within the parameter plane \((\ln {\alpha }_{\inf },\ln {\alpha }_{{{{{{{{\rm{rem}}}}}}}}})\). While the nonMarkovian theory still demonstrates superior performance, the Markovian approach proves capable of identifying the optimal strategy across a larger parameter space compared to the Markovian results depicted in Fig. 5e.
We obtain the relationships between ε* and \(\ln \eta\), as well as between \(\hat{\varepsilon }\) and \(\ln \eta\), as shown in Fig. 5i–j with the same five timedistribution scenarios as in Fig. 2j. In all cases of Fig. 5i, ε* reaches a minimum for \(\ln \eta =0\) and increases as \(\ln \eta\) deviates from zero. The results from the five scenarios further illustrates that the specific values of T_{gen}, \({T}_{\inf }\), or T_{rem} play little role in the errors in vaccination evaluation, and the effect of time distribution forms is also limited. Figure 5i also includes the values of ε* for the realworld infectious diseases COVID19, SARS, H1N1 influenza, and smallpox, which are consistent with those from the nonMarkovian and Markovian theories. Regarding the results depicted in Fig. 5j, it is observed that the nonMarkovian theories consistently outperform the Markovian counterparts. On the other hand, within a wide range of \(\ln \eta\) values around 0, the Markovian theories successfully identify the optimal vaccination strategy among various commonly employed ones. When the value of \(\ln \eta\) deviates from 0, Markovian theories become ineffective in determining the optimal strategy. (Note that on the left side of Fig. 5j, we only present the failures of Markovian theories to identify the optimal strategy in the Monte Carlo simulations with lognormal distribution. This is primarily due to the fact that the parameters associated with the Weibull and gamma distributions fall outside the acceptable range when we keep \({T}_{\inf }\) and T_{rem} fixed to modify \(\ln \eta\) to a very low value.) Furthermore, we demonstrate that even when employing Markovian approaches, the optimal vaccination strategy can still be determined among these strategies considered for the four distinct real diseases.
Conducting accurate evaluations in prevention serves as the sufficient condition of the successful identification of the optimal strategy. In comparison to the prevention evaluation errors ε^{*} of Markovian theories, the optimization failure probability \(\hat{\varepsilon }\) exhibits a wider range of \(\ln \eta\) values that result in the lowest value. The lack of mathematical continuity among these strategies is the primary reason for this. It indicates that there is no smooth transition or mathematical relationship connecting these strategies, resulting in the rank of the strategies not changing promptly when the value of \(\ln \eta\) deviates from 0. Therefore, only large errors from the Markovian theories can result in the failure to detect the optimal strategy. Based on this analysis, the extent to which \(\ln \eta\) deviates from 0, leading to the failure of Markovian theories, as well as whether such failure will occur, depends on the selection of the tested strategies.
Discussion
The COVID19 pandemic has emphasized the importance of investigating disease transmission in human society through modeling. Empirical observations have consistently demonstrated strong memory effects in realworld transmission phenomena. The initial transient stage of an epidemic is critical for data collection, prediction, and articulation of control strategies, but an accurate nonMarkovian model presents difficulties. In contrast, a Markovian model offers great advantages in parameter estimation, computation, and analyses. Uncovering the conditions under which Markovian modeling is suitable for transient epidemic dynamics is necessary.
We have developed a comprehensive mathematical framework for both Markovian and nonMarkovian compartmentalized SIR disease transmissions in an agestratified population, which allows us to identify two types of equivalence between Markovian and nonMarkovian dynamics: in the steady state and transient phase of the epidemic. Our theoretical analysis reveals that, in the steady state, nonMarkovian (memorydependent) transmissions are always equivalent to the Markovian (memoryless) dynamics. However, transientstate equivalence is approximate and holds when the average generation and removal times match each other. In particular, when the average generation time is approximately equal to the average removal time, the disease transmission and removal of an infected individual exhibit a memoryless correlation, thereby minimizing the memory effects of the dynamic process. This results in highly accurate results from the Markovian theory that captures the characteristics of memorydependent transmission based solely on the early epidemic curves. Our analysis also suggests that the Markovian accuracy is mainly determined by the value of generationtoremoval time ratio in disease transmission, where a largerthanone (smallerthanone) ratio can lead to underestimation (overestimation) of the basic reproduction number and epidemic forecasting, as well as the errors in the evaluation of control or prevention measures. The estimation accuracy primarily depends on this ratio, but not greatly affected by the specific values at the various times associated with the epidemic; although distribution forms might affect accuracy to some extent, their influence is much less compared to the impact of the ratio. This property exhibits substantial practical importance, because the average generation and removal times can be readily assessed based on sparse data collected from the transient phase of the epidemic, but to estimate their distributions with only sparse data is infeasible. These results provide deeper quantitative insights into the influence of memory effects on epidemic transmissions, leading to a better understanding of the connection and interplay between Markovian and nonMarkovian dynamics.
There were previous studies of the equivalence between Markovian and nonMarkovian transmission in the SIS model^{18,21,22}. However, these studies addressed the steadystate equivalence rather than the transientstate equivalence. To our knowledge, our work is the first to investigate the transientstate equivalence of the SIR model. In addition, previous studies mainly examined the impact of the average generation time on the transmission dynamics, such as how the shape of the generation time distribution affects the estimation of R_{0}^{53} or the use of serial time distributions in estimating R_{0} during an epidemic^{45}. There was a gap in the literature regarding how generation times affect the accuracy of different models. Our paper fills this gap by providing a criterion for using Markovian frameworks to model memorydependent transmission based on the relationship between the average generation and removal times.
From an application perspective, our study suggests that the impact of the time distribution forms on Markovian estimation accuracy is limited, making it easier to select models between Markovian and nonMarkovian dynamics in the initial outbreak of an epidemic based primarily on the generationtoremoval time ratio. This insight is especially useful since detailed time distribution forms are often harder to detect than their corresponding mean values. In addition, we note that in previous studies, it was observed that in various scenarios, serial intervals, albeit with larger variances, are anticipated to possess a consistent mean value with the average generation time and are more straightforward to measure^{2,45,46,47,48,49}. Given the practical difficulties in observing the generation time, our finding of minimal impact from the distribution forms suggests that the average serial interval can be utilized as a substitute of the average generation time to determine the applicability of the Markovian theories for modeling purposes without compromising accuracy, although numerous studies have indicated that replacing the generation time distribution with the serial interval distribution may affect the analysis of transmission dynamics^{44,45,57}. Meanwhile, based on Eq. (15), if we determine the ratio of generationtoremoval time, the estimated R_{0} obtained through the Markovian approach can be adjusted to approximate the true value. And our webbased application showcases the demonstration of rectifying R_{0} and epidemic forecasting^{50}.
Our study highlights the critical importance of accurately quantifying R_{0} for achieving precise epidemic forecasting and prevention evaluation. A previous work^{54} revealed that the value of R_{0} depends on three key components: the duration of the infectious period (e.g., ψ_{rem}(τ)), the probability of infection resulting from a single contact between an infected individual and a susceptible one (e.g., \({\psi }_{\inf }(\tau )\)), and the number of new susceptible individuals contacted per unit of time. However, given the practical limitations inherent in obtaining all three components, numerous methods have been developed for estimating R_{0}. Although our work presents a specific approach, which fits the parameters of exponential or nonexponential time distribution by using the initial outbreak curves, it is not the only one available. For example, when contact patterns are unknown, R_{0} can be estimated by fitting the growth rate g and the generation time distribution ψ_{gen}(τ), and then applying them in the EulerLotka equation^{45,53,54}. However, since the focus of our work is on epidemic forecasting and evaluation of prevention measure where the data of contact patterns are given, R_{0} can be directly calculated once \({\psi }_{\inf }(\tau )\) and ψ_{rem}(τ) are fitted, without requiring the fitting of any additional quantity such as growth rate g. The estimation of R_{0} can also be achieved by using data in the steady state, such as the final size of an epidemic or equilibrium conditions^{54}. However, this method is not suitable for the transient phase where only earlystage curves are available. Utilizing the approach delineated in this paper is practically more appropriate for estimating R_{0}.
While our study focused on transmission within the SIR framework, extension to SEIR or SIS models is feasible. While we emphasized the significance of the transientstate equivalence in disease transmission, transient dynamics are more relevant or even more crucial than the steady state in nonlinear dynamical systems^{58}. For example, in ecological systems, transient dynamics play a vital role in empirical observations and are therefore a key force driving natural evolution^{59,60,61,62,63,64}. In neural dynamics, transient changes in neural activity can mediate synaptic plasticity, a crucial mechanism for learning and memory^{65,66,67}. Therefore, the identification of suitable conditions for choosing between Markovian and nonMarkovian dynamics may not be limited to epidemic dynamics alone and may serve as a valuable reference for other fields as well.
Taken together, our study establishes an approximate equivalence between Markovian and nonMarkovian dynamics in the transient state, assuming that time distributions follow Weibull forms (see Supplementary Note 4 for details). While the applicability of our findings to most synthetic and empirical distributions has been analyzed qualitatively, a quantitative analysis requires further studies. For extreme cases with nonWeibull distributions, the transmission should be evaluated using other specific methods. While we have provided a qualitative analysis of the mechanism underlying why time distribution forms have minimal impacts on the errors of Markovian estimations, a more rigorous theoretical analysis is needed and requires further exploration. Meanwhile, in the spreading process, there are other memory effects beyond the “temporal” memory considered in our study, for example, the transmission capacity can be influenced by the previous transmission path^{42}, which is referred to as “spatial” memory effect. However, our study specifically focuses on analyzing the impact of “temporal” memory effects. To understand how “spatial” memory effects affect the accuracy of the Markovian approach, further research is needed. In addition, due to the complexity of the nonlinear transmission, our study has produced a semiempirical relationship to estimate the overestimation and underestimation of Markovian methods. Further research is required to develop a rigorous formula that can accurately predict these effects. Additionally, the nonMarkovian SIR model utilized in this study shares similarities with the Hawkes or HawkesN process^{41,68} (see Supplementary Note 9 for details). These processes are commonly employed for the statistical modeling of events in diverse fields, where random events exhibit selfexciting behavior. We anticipate that our research will offer fresh perspectives and valuable insights to advance the understanding of related studies in this area.
Method
Monte Carlo simulation
In the simulation, we classify N individuals into n subgroups based on the age distribution p. The index of the subgroup to which an individual belongs is denoted by l (where 1 ≤ l ≤ n), and the index of the individual within the subgroup is denoted by u (where 1 ≤ u ≤ p_{l}N). The state of the individual u in the age group l is represented by X_{lu}, which includes the states S (susceptible), I (infected), W (recovered), and D (dead), where W and D both represent R (removed), while I and R both represent C (cumulative infected). For each individual, we also record the absolute time of infection and removal using two variables: \({t}_{lu}^{\inf }\) and \({t}_{lu}^{{{{{{{{\rm{rem}}}}}}}}}\), respectively. The absolute time of the system is denoted by t, and we implement the total spreading simulation step by step using a finite time step Δt as follows:

i.
Initialization: set t = 0, X_{lu} for every individual is set to S.

ii.
Set infection seeds: choose a set of individuals as the infection seeds and the corresponding X_{lu} are set to I, the corresponding \({t}_{lu}^{\inf }\) are set to 0, and \({t}_{lu}^{{{{{{{{\rm{rem}}}}}}}}}\) are set to a random value following the removal time distribution ψ_{rem}(τ).

iii.
Infection in one time step: calculate the infection rate, \({\hat{\omega }}_{lu}^{\inf }(t)\), of infected individual u in age group l during the current time step by
$${\hat{\omega }}_{lu}^{\inf }(t)=1{{{\Psi }}}_{\inf }(t{t}_{lu}^{\inf }+{{\Delta }}t)/{{{\Psi }}}_{\inf }(t{t}_{lu}^{\inf }).$$
The probability \({\bar{\omega }}_{l}^{\inf }(t)\) of each susceptible individual in age group l being infected can be calculated by
where \({{{{{{{{\mathcal{I}}}}}}}}}_{m}(t)\) is the index set of the infected individuals in age group m at time t. The number of the susceptible individuals being infected in age group l follows a binomial distribution \(B({s}_{l}^{\star }(t){{{{{p}}}}}_{l}N,{\bar{\omega }}_{l}^{\inf }(t))\), where \({s}_{l}^{\star }(t)\) denotes the fraction of susceptible individuals in age group l at time t. Then generate a random number N_{l}(t) following this binomial distribution and randomly select N_{l}(t) susceptible individuals in age group l to set them as I state. The corresponding \({t}_{lu}^{\inf }\) of the new infected individuals are set to the current t and \({t}_{lu}^{{{{{{{{\rm{rem}}}}}}}}}\) are set to τ_{rem} + t, where the random τ_{rem} follow the removal time distribution ψ_{rem}(τ).

iv.
Removal in one time step: check if \({t}_{lu}^{{{{{{{{\rm{rem}}}}}}}}}\) of each infected individual is during the current time step. If this condition is satisfied, set their state to D with the probability σ_{l} and to W otherwise, where σ_{l} is the infection fatality rate of age group l^{69}. Then let t ← t + Δt.

v.
Repeat the process iii) and iv), until no individual with I index exists.
Time distributions
In the numerical calculations or Monte Carlo simulations, we employ three types of time distributions, i.e., Weibull, lognormal, and gamma, to describe the nonMarkovian or memorydependent transmission process.
For Weibull time distribution, it follows:
where α and β denote the shape and scale parameters, respectively.
The lognormal time distribution is defined as follows:
The gamma time distribution is expressed as follows:
where Γ(⋅) denotes gamma function, while α and β represent the shape and scale parameters, respectively.
Additionally, the survival function could be calculated by:
while the hazard function could be derived from:
For time distribution, survival function, hazard function and the corresponding parameters, incorporating subscripts “\(\inf\)” and “rem” signifies those related to infection and removal, respectively. For example, \({\psi }_{\inf }(\tau )\) represents the time distribution of infection, while \({\alpha }_{\inf }\) denotes its shape parameter.
Derivation of semiempirical estimation of basic reproduction number
Intuitively, the period of T_{rem} can accommodate T_{rem}/T_{gen} = 1/η time intervals of length T_{gen}, corresponding to the result of 1/η generations of infections. This can lead to an exponential increase in the number of infections during T_{rem}. This intuition suggests a relationship between the fitted basic reproduction number \({\hat{R}}_{0}\) and the actual R_{0}, which can be expressed as an exponential function:
where f(⋅) is a monotonically increasing function that satisfies three conditions. First, f(1) = 1, indicating that \({\hat{R}}_{0}\) can be accurately estimated when T_{gen} = T_{rem}. Second, f(0) = 0, meaning that if T_{rem} is an extremely small fraction of T_{gen}, the transmission will take a long time to reach the steady state, causing the curve to be flat in the initial stage and potentially causing the Markovian fitting to produce the estimate \({\hat{R}}_{0}=1\). Third, f( + ∞) = +∞, implying that if T_{rem} is extremely large compared to T_{gen}, the transmission will quickly reach the steady state, causing the Markovian fitting to give an extremely large estimate of \({\hat{R}}_{0}\). Because the actual transmission process involves many complicated nonlinear relationships, identifying the specific form of the function f(⋅) is a challenging task. We thus assume
where a is an unknown positive coefficient. This leads to Eq. (15).
Definition of errors
The difference ε between nonMarkovian and the corresponding Markovian results calculated from Eqs. ((13), (14)) is defined as:
where the pairs (s^{‡}, s^{†}), (i^{‡}, i^{†}) and (r^{‡}, r^{†}) correspond to the nonMarkovian and Markovian susceptible, infected and removed curves, respectively. The 2norm \(\parallel \!\!\cdot {\parallel }_{2,{{\mathbb{T}}}_{\theta }^{{{{\ddagger}}} }}\) on time duration \({{\mathbb{T}}}_{\theta }^{{{{\ddagger}}} }\) ensures that ε measures the “distance” between nonMarkovian and the Markovian transient states. It is not appropriate to set time duration as the total transmission period because the cumulative infected fraction approaches the steadystate value asymptotically, making it difficult to determine the exact time point of the steady state. To address this issue, we choose \({{\mathbb{T}}}_{\theta }^{{{{\ddagger}}} }\) as \([0,{t}_{\theta }^{{{{\ddagger}}} }]\), where \({t}_{\theta }^{{{{\ddagger}}} }\) is the time when the nonMarkovian cumulative infected fraction c^{‡}(t) reaches the θ percentile point within the range that spans from its initial value to its steadystate value. For instance, let ć^{‡} denote the cumulative infected fraction in nonMarkovian calculation at time 0, i.e., ć^{‡} = c^{‡}(0), while \({\tilde{c}}^{{{{\ddagger}}} }\) represents this fraction at the steady state, i.e., \({\tilde{c}}^{{{{\ddagger}}} }=\mathop{\lim }\nolimits_{t\to +\infty }{c}^{{{{\ddagger}}} }(t)\). \({t}_{\theta }^{{{{\ddagger}}} }\) is determined to satisfy the following equation:
Therefore, θ determines the time period during which we measure the “distance” ε between nonMarkovian and the Markovian transient states. The value of θ in Fig. 2j is selected as 50 (see Supplementary Figs. S4, S5 and Supplementary Note 10 for more selection of θ and detailed analysis).
The forecasting error ε^{+} that evaluates whether a theory overestimates or underestimates the steadystate cumulative infected fraction is defined as:
where \({\tilde{t}}^{\star }\) denotes the time when the Monte Carlo stochastic simulation reaches the steady state when no infection occurs in the population, \({c}^{\lozenge}({\tilde{t}}^{\star })\) and \({c}^{\star }({\tilde{t}}^{\star })\) are the cumulative infected fractions from theory and simulation, respectively. A positive value of ε^{+} indicates overestimation, whereas a negative value indicates underestimation.
The prevention evaluation error ε*, which gauges the ability of the Markovian and nonMarkovian theories to estimate the total effectiveness of vaccination, is defined as:
where δ^{⋆} is the result from Monte Carlo simulation, δ^{◇} represents the result from theoretical calculation and ∥ ⋅ ∥_{2} is the 2norm of a vector.
The optimization failure probability \(\hat{\varepsilon }\), which measures the probability that a theory fails to identify the optimal vaccination strategy, is defined as:
where
\({\delta }_{(l)}^{\star }\) and \({\delta }_{(l)}^{\lozenge}\) represent the vectors, δ^{⋆} and δ^{◇}, for the lth experiment, respectively, and z denotes the total number of experiments (in this paper, z is set to 100). Consequently, \(\hat{\varepsilon }\) quantifies the fraction of experiments in which a theory fails to identify the optimal vaccination strategy, and serves as a measure of the probability of failure in optimizing the vaccination strategy.
Selection of training data
In our study, we selected the curves of all states that occurred prior to the time point at which the cumulative infected fraction reached a specific percentile ζ situated between the initial and steadystate cumulative infected fractions, as the training data. For example, let ć^{⋆} denote the cumulative infected fraction at time 0, while \({\tilde{c}}^{\star }\) represents this fraction at the steady state. The selection of the maximum time point of the time duration \({{\mathbb{T}}}_{\zeta }^{\star }\) for training data, denoted as \({t}_{\zeta }^{\star }\), is determined to meet the following equation:
Therefore, ζ determines the time duration, i.e., \({{\mathbb{T}}}_{\zeta }^{\star }=[0,{t}_{\zeta }^{\star }]\), of the training data, which could vary over different Monte Carlo simulations. In our study, we consistently choose ζ to be 20. Note that choosing a specific constant time period as the training data may not be appropriate, as it can result in an overabundance of data points for fitting due to some instances of fast transmission already having reached the steady state, while some instances of slow transmission may not have spread out yet, leading to invalid training data.
Fitting method
Because the removal process is independent of the infection one, we divide the fitting method into two parts: removal parameter fitting and infection parameter fitting. Specifically, we use c^{⋆}(t) and r^{⋆}(t) to denote the cumulative infected and removed fractions of a Monte Carlo simulation at time t. These two types of data are substituted into the Eq. (3) to fit the parameters of ψ_{rem}(τ). To explain further, we can define a loss function:
where \({t}_{\zeta }^{\star }\) represents the maximum time point of the training data, \({\hat{r}}^{\star }(t)\) denotes the removed fraction calculated by cumulative infected fraction, i.e.,
In this equation, Ψ_{rem}(τ) is the survival function of a specific removal time distribution ψ_{rem}(τ), such as Weibull, lognormal, or gamma distributions for the nonMarkovian framework, and exponential distribution for the Markovian framework. By minimizing the loss function L_{rem}, we can determine the optimal parameters for removal time distribution ψ_{rem}(τ). This can be accomplished using the LBFGSB optimization algorithm, which is wellsuited for minimizing the loss function^{70}.
Likewise, we use \({c}_{l}^{\star }(t)\) to denotes the cumulative infected fraction of age group l of a Monte Carlo simulation at time t and \({s}_{l}^{\star }(t)=1{c}_{l}^{\star }(t)\) represents the corresponding susceptible fraction (note that cumulative infected fraction is precisely defined as the sum of the infected and removed fractions, which is accurately expressed as \({c}_{l}^{\star }(t)={i}_{l}^{\star }(t)+{r}_{l}^{\star }(t)=1{s}_{l}^{\star }(t)\), leading to \({s}_{l}^{\star }(t)=1{c}_{l}^{\star }(t)\)). After obtaining the removal parameters, these two types of data are put into Eq. (1) to fit the infection time distribution parameters. In details, we define a loss function \({L}_{\inf }\), which is given by:
where \({\hat{c}}_{l}^{\star }(t)\) represents the cumulative infected fraction of age group l at time t calculated by the cumulative infected fractions of simulation data by using the equation:
In this equation, \({\omega }_{\inf }(\tau )\) is the hazard function of a specific infection time distribution \({\psi }_{\inf }(\tau )\), such as Weibull, lognormal, or gamma distributions for the nonMarkovian framework, and exponential distribution for the Markovian framework. And Ψ_{rem}(τ) is the survival function of the removal time distribution ψ_{rem}(τ), which is already obtained by fitting. To determine the optimal parameters for the infection time distribution \({\psi }_{\inf }(\tau )\), we minimize the loss function \({L}_{\inf }\) using the LBFGSB optimization algorithm^{70}.
Vaccination method
We assume that the individuals will build enough immune protection from the disease κ days after vaccination with the probability ρ, where ρ is the vaccine efficacy. In Monte Carlo simulations, if a susceptible individual gets vaccinated at time t_{vac}, he/she will be marked with the probability ρ. When the absolute time reaches t_{vac} + κ, if this individual has not been infected, he/she will be set to a state called protected state, indicating that this individual is protected from the disease and will never be infected.
In theoretical calculation, when a fraction of individuals in age group l get vaccinated with the detailed vaccination fraction υ_{l}, vaccination time t_{vac} and the fraction of susceptible individuals, which have not been vaccinated by time t, is recorded (denoted as s_{l,*}(t_{vac})). When the absolute time reaches t_{vac} + κ, the corresponding value of s_{l}(t_{vac} + κ) will be set as \({s}_{l}({t}_{{{{{{{{\rm{vac}}}}}}}}}+\kappa )[1\frac{\rho {\upsilon }_{l}}{{s}_{l,*}({t}_{{{{{{{{\rm{vac}}}}}}}}})}]\).
Data availability
All relevant data are available at https://github.com/fengmi9312/ValidityofMarkovianforMemory/tree/main/FigureData.
Code availability
The webbased application can be visited at https://cns.hkbu.edu.hk/toolbox/ValidityofMarkovianforMemory/main.html. The GitHub repository, which includes the source code for all the figure results, the webbased application, and an additional Python application, can be accessed at https://github.com/fengmi9312/ValidityofMarkovianforMemory.git.
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Acknowledgements
This work was partially supported by the Hong Kong Baptist University (HKBU) Strategic Development Fund; the Research Grants Council of Hong Kong (Grant No. C200522Y), the National Natural Science Foundation of China (Grant No. 12275229), and the Hong Kong Chinese Medicine Development Fund (Grant No. 22B2/049A) to L.T.; the Research Grants Council of Hong Kong (Grant No. GRF12201421) to C.S.Z. This research was conducted using the resources of the HighPerformance Computing Cluster Centre at HKBU, which receives funding from the Hong Kong Research Grant Council and the HKBU. Y.C.L. was supported by the Office of Naval Research through Grant No. N000142112323.
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M.F., L.T. and C.S.Z. designed research; M.F. performed research; L.T. and C.S.Z. contributed analytic tools; M.F., L.T. and C.S.Z. analyzed data; M.F., L.T., Y.C.L. and C.S.Z. discussed the results and wrote the paper.
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Feng, M., Tian, L., Lai, YC. et al. Validity of Markovian modeling for transient memorydependent epidemic dynamics. Commun Phys 7, 86 (2024). https://doi.org/10.1038/s4200502401578w
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DOI: https://doi.org/10.1038/s4200502401578w
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