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Mathematical Modeling of Epidemic Diseases; A Case Study of the COVID-19 Coronavirus

Reza Sameni

arXiv:2003.11371v4q-bio.PE

TL;DR

This paper examines how to model epidemic spread and how social measures relate to model parameters and outcomes. It extends a compartmental epidemic model and uses simulations to explore these relationships, while also applying filtering and smoothing to noisy case reports. The authors report that quarantine can delay and reduce infection peaks, though it is insufficient in the long term without continued measures.

  • Problem

    Assessing pandemic spread and regional nonpharmaceutical interventions matters for planning, but daily case reports can contain errors, inconsistencies, and delays.

    Source: §I · p. 1, RATE FROM NOISY REPORTS · p. 14

  • Method

    The study extends an SEIR compartmental model to examine epidemic propagation and uses an extended Kalman filter and smoother to process daily reports and estimate the exponential rate.

    Source: §I · p. 1, A. Endogenous trend prediction · p. 14

  • Results

    Quarantine delays and reduces infection population peaks but is insufficient in the long term; social distancing and other measures should continue after the initial quarantine.

    Source: §VIII · p. 16

  • Takeaways & Limitations

    The authors conclude that the no-infection fixed point is unstable and that an initial seed can trigger pandemic waves, while epidemic dynamics are not specific to coronavirus apart from its parameters.

    Source: §IX · p. 17

    Daily reported cases have errors, inconsistencies, and delays across weekdays, weekends, and holidays.

    Source: RATE FROM NOISY REPORTS · p. 14

Abstract

from arXiv · show

In this research, we study the propagation patterns of epidemic diseases such as the COVID-19 coronavirus, from a mathematical modeling perspective. The study is based on an extensions of the well-known susceptible-infected-recovered (SIR) family of compartmental models. It is shown how social measures such as distancing, regional lockdowns, quarantine and global public health vigilance, influence the model parameters, which can eventually change the mortality rates and active contaminated cases over time, in the real world. As with all mathematical models, the predictive ability of the model is limited by the accuracy of the available data and to the so-called \textit{level of abstraction} used for modeling the problem. In order to provide the broader audience of researchers a better understanding of spreading patterns of epidemic diseases, a short introduction on biological systems modeling is also presented and the Matlab source codes for the simulations are provided online.

I. INTRODUCTION

The introduction motivates epidemic modeling as a tool for understanding disease spread and supporting healthcare planning, then outlines modeling principles and a transition from stochastic infection spread to differential-equation models.

I. INTRODUCTION

  • The study extends an SEIR compartmental model and uses simulations to show how distancing, lockdowns, and public-health vigilance influence model parameters and epidemic outcomes.The authors caution that models are valid only under their assumptions and recommend using this work for planning and decision-making, not direct clinical guidance.Source: §I · p. 1

A. Mathematical modeling

  • Models support understanding, simulation, prediction, and potentially control, but selecting a parsimonious model requires balancing accuracy, complexity, and parameter identifiability.AIC, BIC, and MDL are cited as criteria for comparing models with different numbers of parameters and levels of accuracy.Source: A. Mathematical modeling · p. 2

B. From stochastic infection propagation models to ordinary differential (difference) equation modeling

  • When R0 > 1, early epidemic spread is exponential, but growth eventually saturates in a finite population; reproduction can vary with susceptibility, contact patterns, and countermeasures.The paper adopts a heuristic model-forming approach that it describes as less rigorous but equally accurate in large populations.Source: B. From stochastic infection… · pp. 2–3

C. Compartmental modeling

Compartmental models represent population quantities and their flows as state variables and rate equations. The paper introduces epidemiological assumptions and uses the SIR model to illustrate how infection and recovery rates shape persistence and spread.

C. Compartmental modeling

  • A compartmental model represents quantities as internally homogeneous compartments linked by flows, yielding state-space equations that relate dynamics, inputs, parameters, and noisy measurements.The graph representation can also support estimating and predicting compartment variables from noisy measurements with Kalman or extended Kalman filters.Source: C. Compartmental modeling · p. 3

D. Mathematical epidemiology

1) The susceptible-infected-recovered model:

  • When α > β, infection can spread; in the nonfatal, non-immunizing SIR model, illness can persist after the transient period.The model’s first fixed point is unstable, and its simulation without lifetime immunity shows a constant rate of illness after the transient period.Source: 1) The susceptible-infected-recovered… · p. 5, Figure 3 · p. 5

2) The fatal SIR model:

The conventional basic reproduction number is defined through the next-generation matrix, while an alternative definition incorporates the generation interval. The two definitions yield equivalent epidemic stability conditions in the models studied, though the alternative’s mathematical equivalence remains unproved.

E. The basic reproduction number (R0)

  • The next-generation-matrix definition sets R0 to the spectral radius of the negative next-generation matrix FV^-1.The matrix is formed by partitioning new-infection terms from other transitions.Source: E. The basic reproduction number (R0) · p. 6
  • The epidemic stability condition R0 < 1 is related to, but is not the same quantity as, the outbreak exponent or infection slope.For the SIR model, the stability condition also matches α < β.Source: E. The basic reproduction number (R0) · p. 6
  • The conventional definition discards the discretization time, so it lacks a direct analogy between discrete- and continuous-time outbreak behavior.The proposed alternative is motivated by this limitation and by an analogy between the two model types.Source: E. The basic reproduction number (R0) · p. 6
  • For an irreducible dynamic model, alternative reproduction number values below 1 (or λ1 < 0) correspond to stable epidemic conditions, while values above 1 (or λ1 > 0) correspond to unstable conditions.Here, λ1 is the real part of the dominant eigenvalue of the dynamic model’s Jacobian evaluated at the fixed point of interest; Δ is the generation-time unit (or discretization period).Source: E. The basic reproduction number (R0) · p. 7
  • Both definitions give the same stability condition in the models studied, but the mathematical proof of their equivalence remains future work.The alternative definition makes reproduction numbers from different generation-time units comparable.Source: E. The basic reproduction number (R0) · pp. 6–7

III. PROPOSED EPIDEMIC MODEL I

The proposed fatal SEIR model adds an asymptomatic exposed compartment and a death compartment to represent coronavirus spread and mortality. Its compartment proportions sum to one, and susceptible-exposed and susceptible-infected interactions determine exposure rates.

III. PROPOSED EPIDEMIC MODEL I

  • The model adds an exposed, asymptomatic stage because transmission can occur before clinical symptoms, and includes a separate compartment for deaths.The paper describes exposed people as particularly consequential during an incubation period.Source: §III · p. 7
  • The proposed fatal SEIR model has a unique recovery group, alongside susceptible, exposed, infected, recovered, and passed-away population fractions.The five fractions sum to one.Source: Figure 5 · p. 7, §III · p. 7
  • Exposure-rate terms s(t)e(t) and s(t)i(t) make exposure proportional to susceptible people interacting with exposed or infected people.Source: §III · p. 7
  • The model can be reduced in order by using the population-closure constraint to replace s(t) with 1 − e(t) − i(t) − r(t) − p(t).Source: §III · p. 7

A. Measurements model

The measurement model uses reported infections, recoveries, and deaths, while acknowledging that exposed cases are not directly measured. The model also makes simplifying demographic and geographic assumptions and relates its transition parameters to clinical factors and interventions.

A. Measurements model

  • Reported infections, recoveries, and deaths are modeled as measurements with noise, while the exposed state is not directly measurable from available public data.Noise sources include uncertain population totals, misreporting, death misclassification, and unknown or misclassified cases; exposed cases may be estimated under certain conditions.Source: A. Measurements model · p. 8
  • The measurement model initially assumes reported recoveries include symptomatic and asymptomatic cases, but a later version measures only symptomatic recoveries.Source: A. Measurements model · p. 8

B. Model assumptions and level of abstraction

  • The model treats variables as continuous and omits births, natural deaths, sex, age, and vaccination.These exclusions constrain its scope; it also ignores geopolitical differences while fitting parameters to country-level data.Source: B. Model assumptions and level of… · p. 8

C. Model parameters

  • The incubation-to-symptom rate κ can be set to 0.43 per day under the example’s 14-day quarantine assumption and 99.75% infection drop-out target.The authors note that this example simplifies away age, individual immune-system features, and disease severity; screening can increase κ.Source: C. Model parameters · p. 8
  • The contagion factors αi and αe represent transmission from symptomatic and exposed people, respectively, with αe expected to be much greater under ordinary pre-lockdown conditions.αi is related to viral contagiousness and social factors such as hygiene, population density, and human interaction.Source: C. Model parameters · p. 8
  • The reinfection rate γ represents return from recovery to susceptibility, but the paper says COVID-19 immunity was too uncertain to characterize at the time.Preliminary research suggested short-term immunity of up to four months.Source: C. Model parameters · pp. 8–9
  • The recovery rate β can be approximated from daily recoveries relative to infected cases and depends partly on healthcare infrastructure; the mortality rate μ is similarly approximated from daily deaths.The exposed-case recovery rate ρ is not directly measurable through observations and requires lab-based experiments.Source: C. Model parameters · p. 9

D. Fixed-point analysis

The disease-free equilibrium is unstable to a small exposure, and outbreak growth depends on the dominant eigenvalue. The analysis links stability conditions to countermeasures and draws practical conclusions about case-count comparisons, seed size, and estimating exposed cases.

D. Fixed-point analysis

  • The only fixed point is the disease-free case, but a minor exposure makes it unstable and drives the system toward further infection.Source: D. Fixed-point analysis · p. 9

E. Model analysis during outbreak

  • When infection remains a small population fraction, policy performance should be compared using net case tolls rather than tolls normalized by total population.Source: E. Model analysis during outbreak · p. 9
  • The linearized system is stable exactly when λ1 < 0, a condition equivalent to R0 < 1.Source: E. Model analysis during outbreak · p. 10
  • Stability is sufficient when αi ≈ 0 and κ + ρ > αe; the paper associates these conditions with distancing and lockdown, including isolation of asymptomatic cases.The authors also give αi = αe = 0 as a sufficient condition, while noting that αe = 0 is difficult to achieve in practice.Source: E. Model analysis during outbreak · p. 10
  • When λ1 > 0, exposed and infected cases grow exponentially, continuing toward infection of a significant share of the population without temporary lockdowns, distancing, and quarantine.This applies while the linear approximation holds; growth eventually reaches a point where it no longer holds.Source: E. Model analysis during outbreak · p. 10
  • Under countermeasures, λ1 changes and can serve as a score for tracking how well distancing and quarantine are performing.Source: E. Model analysis during outbreak · p. 10
  • Initial seed size is not the most critical management parameter: regions with smaller seeds may still have higher infection and death tolls depending on infection rates.Source: E. Model analysis during outbreak · pp. 10–11
  • During early exponential growth, when the infected share remains small, exposed cases can be approximated from infected cases.Source: E. Model analysis during outbreak · p. 11

F. Repeated waves of epidemic

The model allows infected-population peaks to recur over time. Their amplitude and spacing depend on model parameters, and the paper notes a historical example of multiple pandemic waves.

F. Repeated waves of epidemic

IV. PROPOSED EPIDEMIC MODEL II

Model II extends the fatal SEIR model by separating recoveries from exposure and infection. It makes infection recoveries observable separately, while retaining Model I’s outbreak properties.

IV. PROPOSED EPIDEMIC MODEL II

  • Model II separates recovery after exposure from recovery after infection; healthcare units can observe infected cases, deaths, and recoveries from infection.Recovery without symptoms may require broad public screening, which the authors describe as impractical for a large population.Source: §IV · p. 11, Figure 6 · p. 12
  • Under the low-infection assumption, Model II omits the susceptible variable and writes the remaining five state variables as a linear system.Source: §IV · pp. 11–12
  • Model II has the same first three eigenvalues as Model I, so its outbreak properties, stability conditions, and reproduction number are unchanged.Source: §IV · p. 12

V. EPIDEMIC TREND ESTIMATION, MODEL OBSERVABILITY

The section establishes when epidemic-model states, including unmeasured exposed cases, can be estimated from observations, then describes control and parameter-estimation approaches. It distinguishes offline fitting with fixed parameters from joint state-and-parameter tracking when parameters vary over time.

A. Observability during outbreak (low fraction of infection)

  • For linearized outbreak models, observability-matrix rank n means all state variables are estimable from outputs in finite time.Numerical checks report no linearly dependent columns for arbitrary model-parameter choices.Source: A. Observability during outbreak (low… · p. 12

B. Observability in the general case

  • When exposed or infected cases exceed several percent of the population, the changing susceptible fraction requires the original nonlinear models.The general-case observability test is checked numerically for arbitrary parameter values and the later simulations.Source: B. Observability in the general case · p. 12
  • Exposed cases can be estimated from other observations when model parameters are known or accurately estimated.The result is stated for both the general and linear-approximated cases.Source: B. Observability in the general case · p. 13

C. Epidemic control and model controllability

  • Models I and II have no external inputs, so interventions such as distancing, quarantine, medication, and vaccination are represented as changes to model parameters.Without parameter changes, the models have no control mechanism; the section instead considers internal stability.Source: C. Epidemic control and model… · p. 13

A. Constrained least squares parameter estimation

  • Constrained weighted least squares fits fixed parameters by minimizing a quadratic model-measurement error, with physical bounds and optional noise-covariance weighting.Averaging over all time samples makes this formulation an offline fitting procedure.Source: A. Constrained least squares parameter… · p. 13
  • For time-varying parameters, an adaptive least-squares version updates estimates sample by sample, while an alternative approach jointly estimates states and parameters.The alternative is described as more flexible and does not require nonlinear least-squares solvers.Source: A. Constrained least squares parameter… · p. 13

B. An extended Kalman filter for joint parameter and variable estimation

  • An extended Kalman filter can track the augmented state and time-varying parameters, while its covariance estimates provide confidence intervals for infected and exposed counts.Under the assumed process- and measurement-noise statistics, those intervals can inform strategists about estimate accuracy and lockdown or quarantine timing.Source: B. An extended Kalman filter for joint… · p. 14

VII. ESTIMATION PANDEMIC TREND AND REPRODUCTION

The paper addresses noisy, delayed daily case reports by modeling exponential epidemic growth and estimating its rate over time. It uses Kalman filtering and smoothing, with NPI information incorporated through a learned mapping to that rate.

RATE FROM NOISY REPORTS

  • An extended Kalman filter and smoother are proposed to predict or filter daily reports while estimating the reproduction rate under NPIs, despite reporting errors and delays.The reported-case problem includes weekday, weekend, and holiday inconsistencies; the paper notes analysts often monitor three- or seven-day averages.Source: RATE FROM NOISY REPORTS · p. 14

A. Endogenous trend prediction

  • With no NPI input, the fixed-interval extended Kalman smoother smooths noisy daily reports and estimates the exponential rate during training; the filter predicts cases without new reports.The filter and smoother can also estimate missing reporting dates, and their covariance matrices support confidence intervals.Source: A. Endogenous trend prediction · p. 14

B. Endogenous prediction

  • For NPI-informed forecasts, regression models such as LASSO or LSTM learn a mapping from regional NPI indexes to the exponential rate, which is then used in the filter.The training target is the residual associated with the exogenous NPI input and autoregressive noise; forecasts include confidence intervals.Source: RATE FROM NOISY REPORTS · p. 14, B. Endogenous prediction · p. 15

C. Alternative methods for exponential rate estimation

The section presents alternative ways to estimate epidemic growth and simulations showing how immunity, quarantine, and healthcare saturation shape modeled outbreaks. Its conclusion emphasizes instability around the no-infection fixed point and identifies real-data fitting and further predictions as future work.

C. Alternative methods for exponential rate estimation

  • The alternative growth-rate methods include nonlinear least squares over N-day windows, linear regression on log new cases, and an empirical geometric mean of case ratios.The proposed extended Kalman smoother is also illustrated on US daily reports since the 100th case; NPIs are not included in that example.Source: C. Alternative methods for exponential… · p. 15, Figure 7 · p. 15

VIII. SIMULATED RESULTS

  • During exponential outbreak growth, i(t)/e(t) remains approximately constant, consistent with the derived relationship, when i(t) ≪ s(t) and before the model’s nonlinear phase.This scenario assumes γ = 0, representing lifetime immunity.Source: §VIII · p. 15, Figure 8 · p. 16
  • With loss of immunity γ = 0.001, the model produces recurrent decaying epidemic peaks over time, unlike the lifetime-immunity case with γ = 0.The recurrent pattern is compared in the text to the 1918 Spanish flu.Source: §VIII · p. 15, Figure 9 · p. 16
  • An insufficient one-month quarantine reduces exposed and infected peaks but does not significantly change the five-month mortality toll, because residual exposed cases can restart spread.The authors conclude that distancing and other measures should continue long after quarantine to drive contaminated cases to zero.Source: §VIII · p. 16
  • At the modeled healthcare breakpoint i(t) = 0.1, the death rate significantly increases as the system reaches saturation.The example uses β0 = 0.1, µ0 = 0.02, βs = 0.01, and µs = 0.2; the authors note the real breakpoint is expected far below this infection rate.Source: §VIII · pp. 16–17, Figure 11 · p. 17

IX. CONCLUSION AND FUTURE WORK

  • The conclusion states that the model is unstable around the no-infection fixed point and that a single initial infection can trigger pandemic waves; real-data fitting and predictions under quarantine or vaccination remain future work.The authors say the dynamics are not specific to coronavirus beyond its parameters.Source: §IX · p. 17

APPENDIX

The appendix gives the Jacobian for the time-varying compartmental model and outlines an extended Kalman filter for jointly tracking compartments and model parameters. It relates the filter implementation to the discrete nature of reported epidemic measurements.

A. Compartmental Model Jacobians

  • The appendix augments the state with epidemic compartments and time-varying parameters, then gives the Jacobian’s non-zero entries for the compact state-space model.The augmented vector includes e(t), i(t), r(t), p(t), αi(t), αe(t), κ(t), β(t), ρ(t), µ(t), and γ(t).Source: A. Compartmental Model Jacobians · pp. 17–18

B. The extended Kalman filter algorithm for simultaneous parameter and variable tracking

  • Because reported epidemic data are discrete while the compartmental model is continuous, the appendix identifies continuous-discrete filtering or model discretization as the two EKF implementation options.The appendix details the discrete EKF approach in Algorithm 2.Source: B. The extended Kalman filter… · p. 18
  • Algorithm 2 takes noisy regular epidemic reports and initial conditions, and outputs estimates of the compartment variables and model parameters.It includes hard constraints on variable and parameter ranges and performance monitoring.Source: B. The extended Kalman filter… · p. 18

C. The extended Kalman filter algorithm for trend tracking from noisy data

The section formulates trend tracking from daily noisy reports as a discrete state-space model and summarizes an extended Kalman filter for estimating epidemic states and growth rate. It states that the algorithm is asymptotically stable and can track the state vector under exponential growth.

C. The extended Kalman filter algorithm for trend tracking from noisy data

  • For pandemic trend smoothing and forecasting, observations yk = y(k∆) are modeled in discrete form with state equation sk+1 = f(sk, wk; h(p(t))) and observation equation yk = [1 0]sk + nk.Source: C. The extended Kalman filter… · p. 18
  • The proposed EKF is stated to be asymptotically stable, allowing it to track the state vector even when new cases follow exponential growth.The section notes that the observation equation is linear and the state equation has a Jacobian with respect to state variables.Source: C. The extended Kalman filter… · p. 18
  • Algorithm 3 takes daily epidemic reports and noise and initialization inputs to produce smoothed and forecasted estimates of the pandemic trend and exponential rate.The algorithm is titled as smoothing and forecasting the pandemic trend and its exponential rate.Source: C. The extended Kalman filter… · p. 18
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