An agent-based nested model integrating within-host and between-host mechanisms to predict an epidemic

Yuichi Tatsukawa; Md. Rajib Arefin; Kazuki Kuga; Jun Tanimoto

doi:10.1371/journal.pone.0295954

Abstract

The COVID-19 pandemic has remarkably heightened concerns regarding the prediction of communicable disease spread. This study introduces an innovative agent-based modeling approach. In this model, the quantification of human-to-human transmission aligns with the dynamic variations in the viral load within an individual, termed “within-host” and adheres to the susceptible–infected–recovered (SIR) process, referred to as “between-host.” Variations in the viral load over time affect the infectivity between individual agents. This model diverges from the traditional SIR model, which employs a constant transmission probability, by incorporating a dynamic, time-dependent transmission probability influenced by the viral load in a host agent. The proposed model retains the time-integrated transmission probability characteristic of the conventional SIR model. As observed in this model, the overall epidemic size remains consistent with the predictions of the standard SIR model. Nonetheless, compared to predictions based on the classical SIR process, notable differences existed in the peak number of the infected individuals and the timing of this peak. These nontrivial differences are induced by the direct correlation between the time-evolving transmission probability and the viral load within a host agent. The developed model can inform targeted intervention strategies and public health policies by providing detailed insights into disease spread dynamics, crucial for effectively managing epidemics.

Citation: Tatsukawa Y, Arefin MR, Kuga K, Tanimoto J (2023) An agent-based nested model integrating within-host and between-host mechanisms to predict an epidemic. PLoS ONE 18(12): e0295954. https://doi.org/10.1371/journal.pone.0295954

Editor: Irene Sendiña-Nadal, Universidad Rey Juan Carlos, SPAIN

Received: May 14, 2023; Accepted: December 1, 2023; Published: December 15, 2023

Copyright: © 2023 Tatsukawa et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting information files.

Funding: This study was partially supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (JSPS), Japan, KAKENHI (Grant No. JP 19KK0262, JP 20H02314 and JP 20K21062) awarded to Professor Tanimoto. We would like to express our gratitude to them. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

The COVID-19 pandemic presented challenges in forecasting the dynamics of communicable diseases. The seminal work of Kermack & McKendrick in 1927 marked the inception of quantitative predictions in epidemic spread [1], which was followed by numerous groundbreaking studies. In epidemiological research, COVID-19 has garnered widespread attention from both the scientific community and the general public. Media coverage has popularized the concept of the “basic reproduction number,” denoted as R₀. The basic reproduction number R₀ represents the average number of the secondary infections generated by a primary infection in a fully susceptible population during the infectious period [2–4]. It is a critical statistical index for indirectly evaluating the infectiousness of a communicable disease.

As established in existing research, infectious diseases transmit through physical contact [2,5,6]. A practical approach to modeling epidemic spread involves the utilization of complex network topology, where nodes symbolize individuals and links denote the interactions [5,7]. In this framework, following the SIR process, an infected individual transmits the disease to a susceptible neighbor with a transmission probability β_p. Fig 1 schematically explains the concept of transmission probability β_p, which comprises three components: (i) emitter body, (ii) exposed environment, and (iii) recipient body. The emitter body pertains to the viral load in an infectious agent, where an increase in virus production raises β_p. The exposed environment—varying from indoor to open-air and from densely to less densely populated areas—also significantly influences the trend of β_p.

Download:

Fig 1. Schematic representation of transmission probability.

Entire transmission process is threefold: (i) emitter body, (ii) exposed environment, and (iii) recipient body. The infectious emitter releases the virus in the environment—either indoor or outdoor. Compared to the opposite scenario, an indoor and densely populated environment is a favorable condition for virus transmission. The recipient absorbs the virus from the environment and may become infected. These three factors may combinedly determine the transmission probability.

https://doi.org/10.1371/journal.pone.0295954.g001

Regarding the third component, the recipient body, the likelihood of infection relies on the amount of virus inhaled from the emitter and the activity level of the biological defense system of the recipient. These factors collectively determine a composite probability for β_p. However, the time scales for these components vary significantly. The overall time scale for β_p is expected to be on the order of days, similar to the time frame for processes within the emitter’s body. In contrast, the time scales for the exposed environment and recipient body are considerably shorter, typically in minutes or seconds. The instantaneous turbulent effects caused by environmental factors and agent movement poses challenges for accurate quantification. Owing to this disparity in time scales, incorporating all three influences on β_p is complex. Nevertheless, if the latter two components are overlooked owing to their brief occurrence, β_p estimation primarily considers the contribution of the emitter body. Therefore, the transmission probability can be modeled by observing the viral load (represented as (t)) in a human body, resulting in a formulation of β_p as a function of time t and viral load V(t), expressed as β_p(t, V(t)). This study explores whether the time- and viral load-dependent transmissibility influences the epidemic outcomes compared to traditional models.

Most conventional SIR models operate on the assumption of a constant transmission rate, especially when employing the ordinary differential equations (ODE) approach, as reported in references [3,8,9], or a constant transmission probability in the context of multi-agent simulations (MAS), as indicated in [7,10–12]. Several studies including [13–16] have explored time-dependent transmission rates or probabilities, especially in networked models. These studies primarily focus on the impact of various interventions, such as mask usage [17,18], social distancing measures [19–21], and reduction of social activities, exemplified by the implementation of lockdowns [22]. These interventions are often not in place initially but are introduced after a specific period.

In the current study, a novel agent-based framework for epidemic spread is proposed, incorporating viral kinetics within the human body. This model is categorized into two components: the “within-host” mechanism, representing viral kinetics, and the “between-host” aspect, quantifying the dynamics of disease spread through human-to-human interactions within a network. Consequently, the within-host mechanism is integrated into the agent-based between-host dynamics, governed by the SIR process. In this model, the transmission probability during the infectious period is determined by the viral load in each infected agent.

The nested models have garnered significant interest in recent years. For an in-depth review, readers may consult the article by Martcheva et al. [23]. This research provides a concise overview of relevant studies. Pereira et al. [24] developed a nested model linking within-host dynamics to a time-discretized between-host dynamics, specifically tailored for tuberculosis, thereby limiting its application to general SIR-type dynamics. Feng et al. [25] employed a mean-field approach to integrate within-host viral dynamics with a susceptible–infected (SI) epidemic model for HIV. Steinmeyer et al. [26] established a model featuring viral load dynamics (within-host), which connects to specific between-host dynamics by estimating the viral emission and absorption between emitters and recipients. However, their model does not incorporate host-contact network structures, a critical component in epidemic modeling, as the degree of connectivity among individuals significantly influences disease spread [26]. For example, epidemic propagation in a scale-free network [27] tends to be hierarchical, initially impacting highly connected individuals before reaching those with fewer connections [26,28,29]. A related methodology is detailed in Ref. [30].

The novelty of our model lies in its integration of within-host viral dynamics with a network-restricted (scale-free) SIR epidemic model. This model establishes a power-law relationship between transmission probability and the viral load of an infected agent. The present focus is on examining the final epidemic size, peak number of infections, and the timing of these peaks under this framework and comparing these outcomes with those predicted by conventional models.

This study is structured as follows: In the Methods section, the nested model is described, combining the ODE-based within-host viral dynamics with agent-based between-host disease spreading dynamics. The Results section presents findings from numerical experiments pertaining to the nested model. These results are thoroughly analyzed in the subsequent Discussion section. Finally, the conclusions of this study are summarized at the end.

Methods

Fig 2 provides a schematic depiction of the current model. The within-host component, structured as an ODE-type target cell-exposed-infected-viral load (TEIV) process, quantifies viral dynamics in each agent. Conversely, the between-host aspect, modeled using the MAS approach and adhering to the SIR process, chronicles sequential stochastic events on a complex network. The study employs either a complete graph, representing a well-mixed situation, or a Barabási–Albert scale-free graph (BA-SF) [27] with a population size of N = 10⁴. In the BA-SF network, a small number of nodes, known as hubs, have significantly higher degrees than other nodes. Specifically, we assume a BA-SF network with an average degree of 8 (⟨𝑘⟩ = 8). The framework includes two temporal dimensions. Within each host agent, the infection age denotes the period during which the viral load fluctuates. This duration ceases when the host is no longer infectious, marked by a substantially reduced viral load. In the between-host context, the focus shifts to monitoring various agents classified as susceptible, infected with a specific infection age, or recovered. The infection age correlates with a particular viral load, influencing the likelihood of infecting susceptible neighbors. The process concludes when no infected agents remain. To derive the average values for each quantity, we conducted 10³ realizations, each commencing with five randomly positioned infected hosts.

Download:

Fig 2. A schematic explanation of the nested model.

Viral kinetics inside the host body is formulated by a deterministic ordinary differential equation-type target cell-exposed cell-infected-viral load process. The between-host part, following the SIR process, is implemented by an agent-based stochastic simulation model on a network. The simulation starts with randomly assigning five initially infected agents on the network. Each infected agent may infect susceptible neighbors with a time-varying transmission probability β_p(t), which depends on the viral load inside the host.

https://doi.org/10.1371/journal.pone.0295954.g002

Within-host model

In this study, we adopt the TEIV process outlined by Baccam et al. [31], which was empirically determined using subjects voluntarily infected with influenza A. The formulation of this process is as follows: (1) where T(t), 𝐸(t), 𝐼(t), and V(t) represent the number of target, exposed, and infected cells and viral load, respectively, at time t. Experimentally identified parameters and initial conditions are stated as follows [31]: infection rate, β_V = 1.466 × 10⁻⁵) [(TCID₅₀/mL)⁻¹ day⁻¹]; transition rate, γ_V = 3.274 [day⁻¹]; production rate, p = 5.826 × 10⁻² [(TCID₅₀/mL)⁻¹ day⁻¹]; cell death rate, δ = 3.934 [day⁻¹]; virus extraction rate, c = 9.575 [day⁻¹]; T(0) = 4 × 10⁸ [cells]; and V(0) = 12.41 [TCID₅₀/mL]. Note that the median tissue culture infectious dose is abbreviated as TCID₅₀ and defined as the dilution of a virus required to infect 50% of a given cell culture [32]. The outcomes depicted in Fig 3(a) and 3(b) were obtained using Eq (1). Consequently, the viral load (panel (a)) was deterministically quantified for a duration of seven days following the infection of a focal agent.

Download:

Fig 3. Deterministic dynamics of the within-host mechanism.

(a) temporal evolution of the viral load; (b) time series of the number of target cells, exposed cells, and infected cells.

https://doi.org/10.1371/journal.pone.0295954.g003

Disease spreading process: Between-host model

For the SIR process, an agent-based model was constructed. Each agent is situated on a node of the underlying network, either a complete graph or a BA-SF network, with links between nodes signifying physical contact. At the start of a simulation episode, all agents are classified as susceptible, except for a few initially infected agents (herein, I₀ = 5), randomly distributed within the network. In this agent-based simulation, the transition from susceptible (S) to infected (I) status is considered stochastic.

(2)

In this model, if an infectious agent has a susceptible neighbor, there exists a probability, denoted as β_p, that the neighbor will become infected Eq (2). We assume that each infected agent, on average, recovers after a period of seven days. Therefore, the reciprocal of the recovery rate, γ⁻¹ = 7 days, i.e., γ = 1/7 given that the recovery time follows an exponential distribution. This infected period is referred to as T_inf (= 7 days). Referring to the typical basic reproduction number for seasonal influenza, we set R₀ = 2.5. Consequently, the transmission probability (per link) in the MAS corresponding to R₀ is expressed as follows: (3)

For comparative analysis, we also utilize a conventional SIR model adapted to the MAS approach. This model employs a time-constant transmission probability throughout the infectious period, here defined as seven days. Practically, for the complete graph, we assumed for the BA-SF network, .

Bridging between within-host and between-host

The present model illustrates the dependence of the transmission probability of an agent i, denoted by , on its viral load V(t) (as in Eq (1)). However, the exact kinetic relationship between viral load and transmission probability for epidemics like seasonal influenza or COVID-19 remains largely undefined [33]. Handel et al.[34] comprehensively reviewed this issue. Park et al. [35] endeavored to estimate this relationship using experimental data derived from horses rather than humans. Saad-Roy et al. [36] suggested that the transmission rate of influenza roughly correlates with the power of the viral load. Extensive data analysis by He et al.[37] on the COVID-19, Larremore et al.[38] to hypothesize three potential functional relationships between the viral load and transmission rate: (i) β(t) is proportional to log[V(t)], (ii) β(t) is linearly proportional to V(t), and (iii) β(t) is saturated with a certain fixed value if V(t) exceeds a threshold. Note that time t refers to the age of infection inside the host. Considering these studies, we postulated a power-law relationship between the transmission probability and viral load as follows: (4) (5) where V_i(t) denotes the viral load of agent i at time t, obeying to V(t) in Eq (1), and V_max indicates its peak value in the time evolution, as depicted in Fig 3(a). V_i(t) in Eq (5) depicts the normalized virus level of agent i. β_max and α are considered positive parameters. Fig 4(a) displays that α controls the power-law tendency. In particular, α = 1 recovers the linear relationship.

Download:

Fig 4.

(a) Power-law relationship between transmission probability β_p(t) and normalized viral titer depicted by the relationship given in Eqs (4) and (5). Parameter α > 1(α < 1) renders the curve convex (concave). System ranges from up to (1, β_max) with the monotonically increasing tendencies depending on α. (b) viral load V(t) (green dashed line) and the transmission probability β_p(t) for various selection of α, resulting from Eq (4). Here, we selected β_max = 0.3 and γ = 1/7 to generate panel (b). Note that the total area under each curve in panel (b) is the same and equal to that of the time-constant transmission probability. Note that panel (b) yields information about the generation time distribution, i.e., the time interval between infections in infector-infected pairs.

https://doi.org/10.1371/journal.pone.0295954.g004

The parameter set (β_max, α) is uniquely determined to satisfy , i.e., the area underneath each curve, corresponding to various choices of α, is equal and consistent with the case for the time-constant transmission probability (Fig 4(b)). This specific condition is one of the most vital assumptions in the current framework, physically indicating that the time integral of the infectiousness is preserved, regardless of the conventional SIR model (a time constant β_p by Eq (3)) or the present model (time variable depending on V_i(t). Based on this assumption, whenever either β_max or α is considered a constant control parameter, the remaining one (i.e., either α or β_max) will be automatically adjusted. Note that as α → 0, β_max approximates to a time constant β_p (by Eq (3)) that recovers the conventional SIR model. Fig 4(b) further elucidates how disease transmission speed varies with different α values. For instance, a curve for α > 1 (pink curve) features a higher peak, indicating a concentration of probability distribution around the mean, which translates to an increased likelihood of transmission between the second and fourth days of the infection period. Conversely, α < 1 corresponds to a lower peak curve (blue curve), suggesting a reduced transmission probability during the infectivity period.

This observation is intricately connected to the concept of generation time [39,40], which calculates the interval between infections in infector–infected pairs. Therefore, the model implies that the generation time for α > 1 is shorter compared to α < 1, indicating faster transmissions in the former scenario. In both cases, the average generation time is estimated to be slightly less than three days. Thus, Fig 4(b) offers insight into the generation time distribution of our model.

Results

Fig 5 presents outcomes for scenarios involving BA-SF networks. Our primary focus is on the final epidemic size (FES), determined by the count of recovered individuals at equilibrium (the point when no infected individuals remain) for each simulation episode. Additionally, we evaluate the peak number of infected individuals and the timing of this peak relative to β_max, with values of β_max ranging from 0 to 5.0. Panels (a), (b), and (c) in Fig 5 illustrate all 10³ realizations for FES, peak infected, and peak time, respectively. Panel (d) compares the ensemble average of FES in our model to that in the conventional SIR model (depicted as a blue solid line) in a BA-SF network. Panel (e) contrasts the average peak infected number in the nested model (dashed lines) against the conventional model (solid lines).

Download:

Fig 5.

(a), (b), and (c) respectively display all 10³ realizations for final epidemic size (FES), peak infected, and its appearing time for the nested model (on Barabási–Albert scale-free network) as a function of β_max. (d) Ensemble average of FES (red-dashed line derived from panel (a)) along β_max, which is compared with the result obtained from the conventional SIR model (horizontal blue solid line). The assembled averages of the peak infected (orange solid line derived from panel (b)) and the peak time (green solid line derived from panel (c)) are finally compared with the conventional model (solid lines) in panel (e).

https://doi.org/10.1371/journal.pone.0295954.g005

Fig 6 displays the time-series data for susceptible (S), infected (I), and recovered (R) populations across 10³ realizations on the BA-SF network, as observed in panels (a-*). The time series for the fraction of infected individuals is also plotted to highlight the average peak infected level and average peak time in panels (b-*). The top row illustrates results from the conventional SIR model (panels (*-i)), whereas the middle and bottom rows exhibit outcomes from the nested model for lower (β_max = 0.1) and higher (β_max = 5.0) β_max values, respectively. Notably, results in panels (b-ii) and (b-iii) indicate that a higher β_max results in a shorter epidemic duration and a more pronounced infection peak. Corresponding findings for a well-mixed setting, which exhibit similar trends (S1 Fig).

Download:

Fig 6. Temporal evolution of the fractions S, I, and R for 10³ realizations on the Barabási–Albert scale-free network.

(a-*) time series of S, I, and R; (b-*) fraction of infection (I) to illustrate the average peak infected and the average peak time. Insets in panel (a-*) display the average over 10³ realizations for the fractions S, I, and R. Panels (*-i) exhibit the case of the conventional SIR model, whereas panels (*-ii) and (*-iii) demonstrate the instances of the nested model corresponding to a lower (β_max = 0.1) and higher (β_max = 5.0)β_max.

https://doi.org/10.1371/journal.pone.0295954.g006

Discussion

A key observation of the nested model is that it yields the same FES as the conventional SIR model (Fig 5(d)). However, marked differences emerge concerning the peak level and its timing (refer to Fig 5(e)) between the nested (dashed lines) and conventional (solid lines) models. Simulations indicate that both the peak infected level and its occurrence time are higher in the nested model compared to the conventional SIR model. This pattern is evident in the well-mixed population scenario on a complete graph (S2 Fig). Notably, a larger β_max produces a more pronounced peak of , although with a shorter active duration, characterized by a higher peak and narrower curve width (illustrated by a red line in Fig 4(b)). This result signifies that considering a time-varying transmission probability results in an earlier and higher peak infected level compared to predictions by the conventional SIR model, especially when the characteristic time-variable transmission probability features a narrow and high peak.

As observed from Fig 6, a consistent trend regarding the infected peak is observed in both BA-SF and complete graph scenarios. Nonetheless, the peak times in the nested model are delayed relative to the conventional model, attributable to the heterogeneity effect of the BA-SF compared to the complete graph (as depicted by comparing panel (b-i) with panels (b-ii) and (b-iii) in Fig 6). To validate the robustness of our findings, the simulations were extended to other network types such as small-world [41], random regular, and Erdős–Rényi (ER) random networks [42,43], (S3 Fig). Additionally, various initial conditions were examined (S4 Fig). In each scenario, the FES in both nested and time-constant epidemic models were consistent. Therefore, the phenomenon of identical FES in both conventional and nested models can be considered general, based on the assumption that the time-integrated transmission probability is similar in both frameworks.

In our analysis thus far, we assumed a constant β_max for each agent. However, let us consider a scenario where β_max varies among agents. In this context, we explore the dynamics wherein, during each infection event (i.e., virus transfer) on a scale-free network, the recipient’s β_max is equal to the β_max of the emitter, adjusted by a mutation bias, expressed as , where Δ ∈ [−0.3, 0.3] represents a mutation bias following a uniform distribution.

Under this framework, our simulations demonstrate that the overall β_max value in the population, starting from an initial value (e.g., 1.0 for each of the initially infected agents; I₀ = 5), fluctuates in both upward and downward directions over time. However, during a prolonged epidemic period, β_max tends to settle at a lower value (S5 Fig). We observed that this variable β_max scenario does not significantly impact the FES, peak infected level, or peak time, in comparison to the fixed β_max scenario (Fig 5).

Conclusions

In this research, we developed a nested within- and between-host model where a global SIR process is driven by an agent-based model on a complex network. This model features a time-varying transmission probability, dependent on the viral load of each infected agent, as determined by the viral kinetics (TEIV process) within the host.

When the time-integrated infectiousness of an infected individual aligns with the conventional framework (i.e., time-constant transmissibility), the FES remains consistent between the nested model and the standard SIR process. However, the nested model exhibits a higher peak infected level, which appears later than that predicted by the conventional SIR model. This deviation results from the time-varying transmission probability characterized by a higher peak and shorter active period, particularly notable in networks with heterogeneous degree distributions. This observation remains unaltered under various initial simulation conditions (S4 Fig). The robustness of the model was further validated across different network topologies, including small-world, random regular, and ER random networks (S3 Fig). Thus, the findings from this model can be regarded as general, provided the baseline assumption of consistent time-integrated transmission probability with the time-constant probability during the infection period is maintained.

The primary limitation of this study includes its assumption of uniform within-host dynamics, potentially oversimplifying the diversity of real-world scenarios. Future research could address this by incorporating more varied within-host dynamics and exploring alternate disease models. Despite this, the research significantly contributes a novel framework bridging individual viral kinetics with epidemic spread, providing practical implications for public health policy and intervention strategies.

Supporting information

S1 Fig. Comparison between the nested and conventional SIR models for the well-mixed setting.

https://doi.org/10.1371/journal.pone.0295954.s001

(TIF)

S2 Fig. Illustrations of final epidemic size, peak infection, and time to attain the peak infection for the well-mixed setting.

https://doi.org/10.1371/journal.pone.0295954.s002

(TIF)

S3 Fig. Final epidemic size and the peak infection time on three different networks.

https://doi.org/10.1371/journal.pone.0295954.s003

(TIF)

S4 Fig. Final epidemic size versus β_max graph on the scale-free network.

https://doi.org/10.1371/journal.pone.0295954.s004

(TIF)

S5 Fig. Evolution of β_max in the time direction.

https://doi.org/10.1371/journal.pone.0295954.s005

(TIF)

References

1. Kermack WO, McKendrick AG. 1927 A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721. ().
- View Article
- Google Scholar
2. Anderson RM, May RM. 1991 Infectious Diseases of Humans. Oxford University Press.
3. Martcheva M. 2015 An Introduction to Mathematical Epidemiology. Springer: New York.
4. van den Driessche P. 2017 Reproduction numbers of infectious disease models. Infect. Dis. Modell. 2, 288–303. (). pmid:29928743
- View Article
- PubMed/NCBI
- Google Scholar
5. Kiss IZ, Miller J, Simon PL. 2017 Mathematics of Epidemics on Networks. 1st edn. Springer International Publishing.
6. Alvarez-Zuzek LG, Di Muro MA, Halvin S, Braunstein LA. 2010 Dynamic vaccination in partially overlapped multiplex network. Phys. Rev. E 99, 1–11. (). pmid:30780375
- View Article
- PubMed/NCBI
- Google Scholar
7. Pastor-Satorras R, Castellano C, Van Mieghem P, Vespignani A. 2015 Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979. ().
- View Article
- Google Scholar
8. Tanimoto J. 2018 Evolutionary Game Theory with Sociophysics: Analysis of Traffic Flow and Epidemics. Springer: Singapore.
9. Brauer F, Castillo-Chavez C, Feng Z. 2019 Mathematical Models in Epidemiology. Springer: New York, NY New York.
10. Moreno Y, Pastor-Satorras R, Vespignani A. 2002 Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B 26, 521–529. ().
- View Article
- Google Scholar
11. Pastor-Satorras R, Vespignani A. 2001 Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203. (). pmid:11290142
- View Article
- PubMed/NCBI
- Google Scholar
12. Keeling MJ, Eames KTD. 2005 Networks and epidemic models. J. R. Soc. Interface 2, 295–307. (). pmid:16849187
- View Article
- PubMed/NCBI
- Google Scholar
13. Al-Radhawi MA, Sadeghi M, Sontag ED. 2022 Long-term regulation of prolonged epidemic outbreaks in large populations via adaptive control: A singular perturbation approach. IEEE Control Syst. Lett. 6, 578–583. ().
- View Article
- Google Scholar
14. Chaturvedi D, Chakravarty U. 2021 Predictive analysis of COVID-19 eradication with vaccination in India, Brazil, and U.S.A. Infect. Genet. Evol. 92, 104834. (). pmid:33798755
- View Article
- PubMed/NCBI
- Google Scholar
15. Tetteh JNA, Nguyen VK, Hernandez-Vargas EA. 2021 Network models to evaluate vaccine strategies towards herd immunity in COVID-19. J. Theor. Biol. 531, 110894. (). pmid:34508758
- View Article
- PubMed/NCBI
- Google Scholar
16. Otunuga OM. 2021 Estimation of epidemiological parameters for COVID-19 cases using a stochastic SEIRS epidemic model with vital dynamics. Results Phys. 28, 104664. (). pmid:34395184
- View Article
- PubMed/NCBI
- Google Scholar
17. Kabir KMA, Risa T, Tanimoto J. 2021 Prosocial behavior of wearing a mask during an epidemic: an evolutionary explanation. Sci. Rep. 11, 12621. (). pmid:34135413
- View Article
- PubMed/NCBI
- Google Scholar
18. Li W, Zhou J, Lu JA. 2020 The effect of behavior of wearing masks on epidemic dynamics. Nonlinear Dyn. 101, 1995–2001. (). pmid:32836804
- View Article
- PubMed/NCBI
- Google Scholar
19. Gosak M, Kraemer MUG, Nax HH, Perc M, Pradelski BSR. 2021 Endogenous social distancing and its underappreciated impact on the epidemic curve. Sci. Rep. 11, 3093. (). pmid:33542416
- View Article
- PubMed/NCBI
- Google Scholar
20. Matrajt L, Leung T. 2020 Evaluating the effectiveness of social distancing interventions to delay or flatten the epidemic curve of coronavirus disease. Emerg. Infect. Dis. 26, 1740–1748. (). pmid:32343222
- View Article
- PubMed/NCBI
- Google Scholar
21. Maharaj S, Kleczkowski A. 2012 Controlling epidemic spread by social distancing: do it well or not at all. BMC Public Health 12, 679. (). pmid:22905965
- View Article
- PubMed/NCBI
- Google Scholar
22. Schlosser F, Maier BF, Jack O, Hinrichs D, Zachariae A, Brockmann D. 2020 COVID-19 lockdown induces disease-mitigating structural changes in mobility networks. Proc. Natl. Acad. Sci. U. S. A. 117, 32883–32890. (). pmid:33273120
- View Article
- PubMed/NCBI
- Google Scholar
23. Martcheva M, Tuncer N, St Mary C. 2015 Coupling within-host and between-host infectious diseases models. Biomath 4,1–12. ().
- View Article
- Google Scholar
24. Pereira RS, Bauch CT, Penna TJP, Espíndola AL. 2021 A nested model for tuberculosis: combining within-host and between-host processes in a single framework. Int. J. Mod. Phys. C 32, 2150167. ().
- View Article
- Google Scholar
25. Feng Z, Velasco-Hernandez J, Tapia-Santos B, Conceição M, Leite A, Feng Z, et al. 2011 A model for coupling within-host and between-host dynamics in an infectious disease. Nonlinear Dyn. 683, 401–411. ().
- View Article
- Google Scholar
26. Steinmeyer SH, Wilke CO, Pepin KM. 2010 Methods of modelling viral disease dynamics across the within- and between-host scales: the impact of virus dose on host population immunity. Philos. Trans. R. Soc. Lond. B Biol. Sci. 365, 1931–1941. (). pmid:20478888
- View Article
- PubMed/NCBI
- Google Scholar
27. Barabási AL, Albert R. 1999 Emergence of scaling in random networks. Science 286, 509–512. (). pmid:10521342
- View Article
- PubMed/NCBI
- Google Scholar
28. Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A. 2005 Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288. (). pmid:15862595
- View Article
- PubMed/NCBI
- Google Scholar
29. Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A. 2004 Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92, 178701. (). pmid:15169200
- View Article
- PubMed/NCBI
- Google Scholar
30. Lukens S et al. 2014 A large-scale immuno-epidemiological simulation of influenza A epidemics. BMC Public Health 14, 1–15. ().
- View Article
- Google Scholar
31. Baccam P, Beauchemin C, Macken CA, Hayden FG, Perelson AS. 2006 Kinetics of influenza A virus infection in humans. J. Virol. 80, 7590–7599. (). pmid:16840338
- View Article
- PubMed/NCBI
- Google Scholar
32. Lei C, Yang J, Hu J, Sun X. 2021 On the calculation of TCID50 for quantitation of virus infectivity. Virol. Sin. 2020 36, 141–144. (). pmid:32458296
- View Article
- PubMed/NCBI
- Google Scholar
33. Marc A, Kerioui M, Blanquart F, Bertrand J, Mitjà O, Corbacho-Monné M, et al. 2021 Quantifying the relationship between sars-cov-2 viral load and infectiousness. eLife 10. (). pmid:34569939
- View Article
- PubMed/NCBI
- Google Scholar
34. Handel A, Rohani P. 2015 Crossing the scale from within-host infection dynamics to between-host transmission fitness: a discussion of current assumptions and knowledge. Philos. Trans. R. Soc. Lond. B Biol. Sci. 370. (). pmid:26150668
- View Article
- PubMed/NCBI
- Google Scholar
35. Park AW, Daly JM, Lewis NS, Smith DJ, Wood JLN, Grenfell BT. 2009 Quantifying the impact of immune escape on transmission dynamics of influenza. Science 326, 726–728. (). pmid:19900931
- View Article
- PubMed/NCBI
- Google Scholar
36. Saad-Roy CM, McDermott AB, Grenfell BT. 2019 Dynamic perspectives on the search for a universal influenza vaccine. J. Infect. Dis. 219 Supplement 1, S46–S56. (). pmid:30715467
- View Article
- PubMed/NCBI
- Google Scholar
37. He X et al. 2020 Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med. 26, 672–675. () pmid:32296168
- View Article
- PubMed/NCBI
- Google Scholar
38. Larremore DB, Wilder B, Lester E, Shehata S, Burke JM, Hay JA, et al. 2021 Test sensitivity is secondary to frequency and turnaround time for COVID-19 screening. Sci. Adv. 7. (). pmid:33219112
- View Article
- PubMed/NCBI
- Google Scholar
39. Hart WS, Miller E, Andrews NJ, Waight P, Maini PK, Funk S, et al. 2022 Generation time of the alpha and delta SARS-CoV-2 variants: an epidemiological analysis. Lancet Infect. Dis. 22, 603–610. pmid:35176230
- View Article
- PubMed/NCBI
- Google Scholar
40. Carrat F, Vergu E, Ferguson NM, Lemaitre M, Cauchemez S, Leach S, et al. 2008 Time lines of infection and disease in human influenza: a review of volunteer challenge studies. Am. J. Epidemiol. 167, 775–785. pmid:18230677
- View Article
- PubMed/NCBI
- Google Scholar
41. Watts DJ, Strogatz SH. 1998 Collective dynamics of ‘small-world’ networks. Nature 393, 440–442. (). pmid:9623998
- View Article
- PubMed/NCBI
- Google Scholar
42. Bollobás B. 1998 Random graphs. In Modern Graph Theory (215–252). Springer: New York, NY.
43. Holme P, Saramäki J. 2012 Temporal networks. Phys. Rep. 519, 97–125. ().
- View Article
- Google Scholar

[ref1] 1. Kermack WO, McKendrick AG. 1927 A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721. ().
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Anderson RM, May RM. 1991 Infectious Diseases of Humans. Oxford University Press.

[ref3] 3. Martcheva M. 2015 An Introduction to Mathematical Epidemiology. Springer: New York.

[ref4] 4. van den Driessche P. 2017 Reproduction numbers of infectious disease models. Infect. Dis. Modell. 2, 288–303. (). pmid:29928743
View Article
PubMed/NCBI
Google Scholar

[7] View Article

[8] PubMed/NCBI

[9] Google Scholar

[ref5] 5. Kiss IZ, Miller J, Simon PL. 2017 Mathematics of Epidemics on Networks. 1st edn. Springer International Publishing.

[ref6] 6. Alvarez-Zuzek LG, Di Muro MA, Halvin S, Braunstein LA. 2010 Dynamic vaccination in partially overlapped multiplex network. Phys. Rev. E 99, 1–11. (). pmid:30780375
View Article
PubMed/NCBI
Google Scholar

[12] View Article

[13] PubMed/NCBI

[14] Google Scholar

[ref7] 7. Pastor-Satorras R, Castellano C, Van Mieghem P, Vespignani A. 2015 Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979. ().
View Article
Google Scholar

[16] View Article

[17] Google Scholar

[ref8] 8. Tanimoto J. 2018 Evolutionary Game Theory with Sociophysics: Analysis of Traffic Flow and Epidemics. Springer: Singapore.

[ref9] 9. Brauer F, Castillo-Chavez C, Feng Z. 2019 Mathematical Models in Epidemiology. Springer: New York, NY New York.

[ref10] 10. Moreno Y, Pastor-Satorras R, Vespignani A. 2002 Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B 26, 521–529. ().
View Article
Google Scholar

[21] View Article

[22] Google Scholar

[ref11] 11. Pastor-Satorras R, Vespignani A. 2001 Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203. (). pmid:11290142
View Article
PubMed/NCBI
Google Scholar

[24] View Article

[25] PubMed/NCBI

[26] Google Scholar

[ref12] 12. Keeling MJ, Eames KTD. 2005 Networks and epidemic models. J. R. Soc. Interface 2, 295–307. (). pmid:16849187
View Article
PubMed/NCBI
Google Scholar

[28] View Article

[29] PubMed/NCBI

[30] Google Scholar

[ref13] 13. Al-Radhawi MA, Sadeghi M, Sontag ED. 2022 Long-term regulation of prolonged epidemic outbreaks in large populations via adaptive control: A singular perturbation approach. IEEE Control Syst. Lett. 6, 578–583. ().
View Article
Google Scholar

[32] View Article

[33] Google Scholar

[ref14] 14. Chaturvedi D, Chakravarty U. 2021 Predictive analysis of COVID-19 eradication with vaccination in India, Brazil, and U.S.A. Infect. Genet. Evol. 92, 104834. (). pmid:33798755
View Article
PubMed/NCBI
Google Scholar

[35] View Article

[36] PubMed/NCBI

[37] Google Scholar

[ref15] 15. Tetteh JNA, Nguyen VK, Hernandez-Vargas EA. 2021 Network models to evaluate vaccine strategies towards herd immunity in COVID-19. J. Theor. Biol. 531, 110894. (). pmid:34508758
View Article
PubMed/NCBI
Google Scholar

[39] View Article

[40] PubMed/NCBI

[41] Google Scholar

[ref16] 16. Otunuga OM. 2021 Estimation of epidemiological parameters for COVID-19 cases using a stochastic SEIRS epidemic model with vital dynamics. Results Phys. 28, 104664. (). pmid:34395184
View Article
PubMed/NCBI
Google Scholar

[43] View Article

[44] PubMed/NCBI

[45] Google Scholar

[ref17] 17. Kabir KMA, Risa T, Tanimoto J. 2021 Prosocial behavior of wearing a mask during an epidemic: an evolutionary explanation. Sci. Rep. 11, 12621. (). pmid:34135413
View Article
PubMed/NCBI
Google Scholar

[47] View Article

[48] PubMed/NCBI

[49] Google Scholar

[ref18] 18. Li W, Zhou J, Lu JA. 2020 The effect of behavior of wearing masks on epidemic dynamics. Nonlinear Dyn. 101, 1995–2001. (). pmid:32836804
View Article
PubMed/NCBI
Google Scholar

[51] View Article

[52] PubMed/NCBI

[53] Google Scholar

[ref19] 19. Gosak M, Kraemer MUG, Nax HH, Perc M, Pradelski BSR. 2021 Endogenous social distancing and its underappreciated impact on the epidemic curve. Sci. Rep. 11, 3093. (). pmid:33542416
View Article
PubMed/NCBI
Google Scholar

[55] View Article

[56] PubMed/NCBI

[57] Google Scholar

[ref20] 20. Matrajt L, Leung T. 2020 Evaluating the effectiveness of social distancing interventions to delay or flatten the epidemic curve of coronavirus disease. Emerg. Infect. Dis. 26, 1740–1748. (). pmid:32343222
View Article
PubMed/NCBI
Google Scholar

[59] View Article

[60] PubMed/NCBI

[61] Google Scholar

[ref21] 21. Maharaj S, Kleczkowski A. 2012 Controlling epidemic spread by social distancing: do it well or not at all. BMC Public Health 12, 679. (). pmid:22905965
View Article
PubMed/NCBI
Google Scholar

[63] View Article

[64] PubMed/NCBI

[65] Google Scholar

[ref22] 22. Schlosser F, Maier BF, Jack O, Hinrichs D, Zachariae A, Brockmann D. 2020 COVID-19 lockdown induces disease-mitigating structural changes in mobility networks. Proc. Natl. Acad. Sci. U. S. A. 117, 32883–32890. (). pmid:33273120
View Article
PubMed/NCBI
Google Scholar

[67] View Article

[68] PubMed/NCBI

[69] Google Scholar

[ref23] 23. Martcheva M, Tuncer N, St Mary C. 2015 Coupling within-host and between-host infectious diseases models. Biomath 4,1–12. ().
View Article
Google Scholar

[71] View Article

[72] Google Scholar

[ref24] 24. Pereira RS, Bauch CT, Penna TJP, Espíndola AL. 2021 A nested model for tuberculosis: combining within-host and between-host processes in a single framework. Int. J. Mod. Phys. C 32, 2150167. ().
View Article
Google Scholar

[74] View Article

[75] Google Scholar

[ref25] 25. Feng Z, Velasco-Hernandez J, Tapia-Santos B, Conceição M, Leite A, Feng Z, et al. 2011 A model for coupling within-host and between-host dynamics in an infectious disease. Nonlinear Dyn. 683, 401–411. ().
View Article
Google Scholar

[77] View Article

[78] Google Scholar

[ref26] 26. Steinmeyer SH, Wilke CO, Pepin KM. 2010 Methods of modelling viral disease dynamics across the within- and between-host scales: the impact of virus dose on host population immunity. Philos. Trans. R. Soc. Lond. B Biol. Sci. 365, 1931–1941. (). pmid:20478888
View Article
PubMed/NCBI
Google Scholar

[80] View Article

[81] PubMed/NCBI

[82] Google Scholar

[ref27] 27. Barabási AL, Albert R. 1999 Emergence of scaling in random networks. Science 286, 509–512. (). pmid:10521342
View Article
PubMed/NCBI
Google Scholar

[84] View Article

[85] PubMed/NCBI

[86] Google Scholar

[ref28] 28. Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A. 2005 Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288. (). pmid:15862595
View Article
PubMed/NCBI
Google Scholar

[88] View Article

[89] PubMed/NCBI

[90] Google Scholar

[ref29] 29. Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A. 2004 Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92, 178701. (). pmid:15169200
View Article
PubMed/NCBI
Google Scholar

[92] View Article

[93] PubMed/NCBI

[94] Google Scholar

[ref30] 30. Lukens S et al. 2014 A large-scale immuno-epidemiological simulation of influenza A epidemics. BMC Public Health 14, 1–15. ().
View Article
Google Scholar

[96] View Article

[97] Google Scholar

[ref31] 31. Baccam P, Beauchemin C, Macken CA, Hayden FG, Perelson AS. 2006 Kinetics of influenza A virus infection in humans. J. Virol. 80, 7590–7599. (). pmid:16840338
View Article
PubMed/NCBI
Google Scholar

[99] View Article

[100] PubMed/NCBI

[101] Google Scholar

[ref32] 32. Lei C, Yang J, Hu J, Sun X. 2021 On the calculation of TCID50 for quantitation of virus infectivity. Virol. Sin. 2020 36, 141–144. (). pmid:32458296
View Article
PubMed/NCBI
Google Scholar

[103] View Article

[104] PubMed/NCBI

[105] Google Scholar

[ref33] 33. Marc A, Kerioui M, Blanquart F, Bertrand J, Mitjà O, Corbacho-Monné M, et al. 2021 Quantifying the relationship between sars-cov-2 viral load and infectiousness. eLife 10. (). pmid:34569939
View Article
PubMed/NCBI
Google Scholar

[107] View Article

[108] PubMed/NCBI

[109] Google Scholar

[ref34] 34. Handel A, Rohani P. 2015 Crossing the scale from within-host infection dynamics to between-host transmission fitness: a discussion of current assumptions and knowledge. Philos. Trans. R. Soc. Lond. B Biol. Sci. 370. (). pmid:26150668
View Article
PubMed/NCBI
Google Scholar

[111] View Article

[112] PubMed/NCBI

[113] Google Scholar

[ref35] 35. Park AW, Daly JM, Lewis NS, Smith DJ, Wood JLN, Grenfell BT. 2009 Quantifying the impact of immune escape on transmission dynamics of influenza. Science 326, 726–728. (). pmid:19900931
View Article
PubMed/NCBI
Google Scholar

[115] View Article

[116] PubMed/NCBI

[117] Google Scholar

[ref36] 36. Saad-Roy CM, McDermott AB, Grenfell BT. 2019 Dynamic perspectives on the search for a universal influenza vaccine. J. Infect. Dis. 219 Supplement 1, S46–S56. (). pmid:30715467
View Article
PubMed/NCBI
Google Scholar

[119] View Article

[120] PubMed/NCBI

[121] Google Scholar

[ref37] 37. He X et al. 2020 Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med. 26, 672–675. () pmid:32296168
View Article
PubMed/NCBI
Google Scholar

[123] View Article

[124] PubMed/NCBI

[125] Google Scholar

[ref38] 38. Larremore DB, Wilder B, Lester E, Shehata S, Burke JM, Hay JA, et al. 2021 Test sensitivity is secondary to frequency and turnaround time for COVID-19 screening. Sci. Adv. 7. (). pmid:33219112
View Article
PubMed/NCBI
Google Scholar

[127] View Article

[128] PubMed/NCBI

[129] Google Scholar

[ref39] 39. Hart WS, Miller E, Andrews NJ, Waight P, Maini PK, Funk S, et al. 2022 Generation time of the alpha and delta SARS-CoV-2 variants: an epidemiological analysis. Lancet Infect. Dis. 22, 603–610. pmid:35176230
View Article
PubMed/NCBI
Google Scholar

[131] View Article

[132] PubMed/NCBI

[133] Google Scholar

[ref40] 40. Carrat F, Vergu E, Ferguson NM, Lemaitre M, Cauchemez S, Leach S, et al. 2008 Time lines of infection and disease in human influenza: a review of volunteer challenge studies. Am. J. Epidemiol. 167, 775–785. pmid:18230677
View Article
PubMed/NCBI
Google Scholar

[135] View Article

[136] PubMed/NCBI

[137] Google Scholar

[ref41] 41. Watts DJ, Strogatz SH. 1998 Collective dynamics of ‘small-world’ networks. Nature 393, 440–442. (). pmid:9623998
View Article
PubMed/NCBI
Google Scholar

[139] View Article

[140] PubMed/NCBI

[141] Google Scholar

[ref42] 42. Bollobás B. 1998 Random graphs. In Modern Graph Theory (215–252). Springer: New York, NY.

[ref43] 43. Holme P, Saramäki J. 2012 Temporal networks. Phys. Rep. 519, 97–125. ().
View Article
Google Scholar

[144] View Article

[145] Google Scholar

An agent-based nested model integrating within-host and between-host mechanisms to predict an epidemic

An agent-based nested model integrating within-host and between-host mechanisms to predict an epidemic

Figures

Abstract

Introduction

Methods

Within-host model

Disease spreading process: Between-host model

Bridging between within-host and between-host

Results

Discussion

Conclusions

Supporting information

S1 Fig. Comparison between the nested and conventional SIR models for the well-mixed setting.

S2 Fig. Illustrations of final epidemic size, peak infection, and time to attain the peak infection for the well-mixed setting.

S3 Fig. Final epidemic size and the peak infection time on three different networks.

S4 Fig. Final epidemic size versus β_max graph on the scale-free network.

S5 Fig. Evolution of β_max in the time direction.

S1 File.

S2 File.

S3 File.

S4 File.

S5 File.

S6 File.

S7 File.

S8 File.

S9 File.

S10 File.

S11 File.

S12 File.

S13 File.

S14 File.

S15 File.

S16 File.

References

Figures

Abstract

Introduction

Methods

Within-host model

Disease spreading process: Between-host model

Bridging between within-host and between-host

Results

Discussion

Conclusions

Supporting information

S1 Fig. Comparison between the nested and conventional SIR models for the well-mixed setting.

S2 Fig. Illustrations of final epidemic size, peak infection, and time to attain the peak infection for the well-mixed setting.

S3 Fig. Final epidemic size and the peak infection time on three different networks.

S4 Fig. Final epidemic size versus βmax graph on the scale-free network.

S5 Fig. Evolution of βmax in the time direction.

References

S4 Fig. Final epidemic size versus β_max graph on the scale-free network.

S5 Fig. Evolution of β_max in the time direction.