Open problems for the nonlocal integro-differential COVID-19 model of Liu–Wang–Zhang–Yau: parameter identification from outbreak data, turning-point uniqueness, and asymptotic theory beyond Weibull delay kernels

The COVID-19 transmission model of Jijun Liu, Liyan Wang, Qiang Zhang and Shing-Tung Yau (Applied Mathematical Modelling 89 (2021) 1965–1982; online 2020) reduces the epidemic in a closed region to a linear Volterra integro-differential equation — an evolution equation whose rate of change at time $t$ depends on the entire prior history through convolution integrals — for the number $I_s(t)$ of daily existing infected people, with Weibull-distributed incubation, cure and death delays, infection rate $\beta$, isolation ratio $\kappa(t)$, recovery rates and imported cases $s(t)$. The paper rigorously proved eventual vanishing of $I_s(t)$ and $I_s'(t)$ under limited imported cases with a small infection rate or strong isolation, and existence of at least one turning point, and it matched the officially issued Wuhan and Italy confirmed-case series using parameters chosen by trial and error. Four issues were explicitly left open: (i) the nonlinear inverse problem of identifying the unknown parameters and input sources — including the unobservable initial value $I_s^0$ — from officially issued data; (ii) uniqueness of the turning point and monotone decline after the peak, which the authors state they cannot assert; (iii) extension of the rigorous asymptotic analysis from Weibull delays to other delay densities used numerically, such as the modified Gaussian; and (iv) validation of the Italy predictions beyond April 20, 2020, and of the inferred trajectory of the unobserved existing infected population, against actual later data. A follow-up by three of the same authors (Applied Mathematics Letters 125 (2022) 107783) added vaccination and proved the vanishing theorem for Gamma-distributed delays, but reiterated that parameter identification is still an open problem and left the other components untouched. The problem of completing the mathematical theory and data identification of this model therefore remains open.

Background

Classical mathematical epidemiology describes disease spread through coupled systems of ordinary differential equations for population compartments: the susceptible, infectious and removed classes of the SIR model, with an added exposed (infected but not yet infectious) class in SEIR, and their variants. For COVID-19, two features motivated delay-type extensions of these compartment models: the incubation period between infection and diagnosis, and the arrival of imported cases from outside a closed region. This model, due to Jijun Liu, Liyan Wang, Qiang Zhang and Shing-Tung Yau, takes a different route: it tracks a single aggregated quantity, the number $I_s(t)$ of people currently existing in the infected state at time $t$ (infected but not yet cured or dead), and derives a closed governing equation for it.

The derivation treats the incubation period, the cure time after diagnosis, and the time to death as independent random variables with probability density functions. In the analytical part these are Weibull densities $f^k_\lambda(t)=\frac{k}{\lambda}(t/\lambda)^{k-1}e^{-(t/\lambda)^k}$ for $t\ge 0$, which specialize to the exponential distribution for $k=1$ and the Rayleigh distribution for $k=2$. Delay effects then enter through convolution integrals. For example, the newly confirmed cases $j(t)$ form the output of a cascade: a fraction $\kappa(t)\in(0,1)$ of the newly infected people at time $t$ — internal infections $i(t)=\beta(1-\kappa(t))I_s(t)$ plus imported cases $s(t)$, where $\beta$ is the infection rate and $\kappa(t)$ the isolation ratio — is isolated, passes through incubation with density $h_s(\tau;\tau_1)$, and is then confirmed; cured and dead cases are modeled similarly, with hospital recovery rate $\kappa_2$, self-healing rate $\kappa_4$ of exposed patients, and densities with mean periods $\tau_1,\dots,\tau_5$ for the five delay mechanisms. Eliminating all auxiliary components yields, for $I_s(t)$, a linear Volterra integro-differential equation (equation (2.21) of the paper) — an equation in which the rate of change of the unknown at time $t$ depends on its entire history through convolution-type integral terms — with initial value $I_s(0)=I_s^0$. For fixed parameters and inputs this forward problem is well posed — it has a unique solution depending continuously on the data — and all other epidemic series are recovered from $I_s$. A crucial observability gap drives one of the open issues: officially issued data contain the daily confirmed cases $j(t)$, the cumulative count $J(t)=\int_0^t j(\tau)d\tau$, and the hospital cured and dead cases $c_1(t),d_1(t)$, but never $I_s(t)$ itself, and the initial value $I_s^0$ is unknown.

For Weibull delays the paper proves two results under a growth condition on the imported cases, $\int_0^\infty s(t)e^{t/\lambda_0}dt\le s^*$, where $\lambda_0$ depends on the densities. Theorem 3.1: if $\beta\kappa_0$ is small enough, with $\kappa_0=\max_{t\ge 0}(1-\kappa(t))$ — that is, a sufficiently small infection rate or sufficiently strong isolation — then $I_s(t)$, $c(t)$ and $d(t)$ tend to zero as $t\to\infty$, with explicit exponential decay for $k=1,2$ obtained by Grönwall-type estimates (standard integral inequalities that bound a function in terms of its own time-integral) and estimates on iterated kernels (the repeated self-convolutions of the delay kernel that build up the Volterra solution series). Theorem 3.3: under the same conditions $I_s'(t)\to 0$ and $I_s'(t)$ has at least one zero, that is, the epidemic has a turning point (a peak of the existing-infection count). Remark 3.4, however, states that the authors cannot assert that $I_s'(t)$ has only one zero point on $(0,\infty)$, so the infected count need not decrease monotonically after the peak — attributed to the weak restrictions on the external input $s(t)$ — and rebounds are in fact produced numerically for oscillating imported cases. Numerically, the model with Weibull, Rayleigh, and modified Gaussian kernels $f^*(t)=c^*e^{-(t-\tau^*)^2/\sigma^2}$ ($t\ge 0$) reproduces the officially issued Wuhan (January 23 to March 18, 2020) and Italy (March 10 to April 20, 2020) confirmed-case series, but only with $I_s^0$ and all parameters chosen by trial and error. The paper formulates the resulting data problem explicitly: determining the unknown parameters from observed data is a nonlinear inverse problem left as future work. It also notes, in Section 2, that while the model and the numerical simulations are the same for other density functions, the corresponding theoretical analysis needs to be carried out separately. A companion literature strand (the work of Yu Chen, Jin Cheng, Yu Jiang and Keji Liu) studies a related time-delay system of ordinary differential equations with external source, Applicable Analysis 101(1) (2022) 146–157 (online first 2020), together with its parameter identification, Journal of Inverse and Ill-Posed Problems 28(2) (2020) 243–250; that system is not the convolution model studied here, and its identification theory does not transfer.

Problem Statement

The source-faithful open problem is the completion of the mathematical theory and of the data identification of this specific nonlocal transmission model — the linear Volterra integro-differential equation (2.21) of the cited paper governing the daily existing infected population $I_s(t)$, with initial value $I_s(0)=I_s^0$ and driven by the imported cases $s(t)$ and by the initial injection $s_0(t)$ (the equivalent continuous input over a short initial interval $[0,\epsilon_0]$ with total mass $\int_0^{\epsilon_0}s_0(t)\,dt=I_s^0$, which encodes the initially infected population) — in four linked components:

1. Parameter and input identification (the nonlinear inverse problem). From the officially issued observations — daily confirmed cases $j(t)$, cumulative confirmed cases $J(t)$, hospital cured cases $c_1(t)$ and hospital dead cases $d_1(t)$ — determine the unknown quantities of the governed system: the initial value $I_s^0$, the infection rate $\beta$, the isolation ratio $\kappa(t)$, the recovery rates $\kappa_2$ and $\kappa_4$, the mean delay periods $\tau_1,\dots,\tau_5$ (and the density family with its shape parameters), and the imported-case input $s(t)$. This includes establishing identifiability — whether distinct parameter sets can produce indistinguishable observed series, given that $I_s(t)$ itself is never observed and the observables are convolved functionals of it — together with a stable reconstruction method, thereby replacing the trial-and-error parameter specification used in the paper's Wuhan and Italy case studies.

2. Turning-point uniqueness and monotone decline. Determine conditions on the imported cases $s(t)$, the isolation ratio $\kappa(t)$ and the infection rate $\beta$ under which the turning point of $I_s(t)$ is unique and $I_s(t)$ decreases monotonically after the peak; or, conversely, characterize or explicitly construct rebound scenarios (multiple zeros of $I_s'(t)$) showing when externally driven resurgences occur. The paper proves only existence of a zero of $I_s'(t)$ (Theorem 3.3) and states in Remark 3.4 that uniqueness cannot be asserted.

3. Asymptotic theory beyond Weibull delays. Extend the rigorous asymptotic analysis — the vanishing of $I_s(t)$, $c(t)$ and $d(t)$ (with decay estimates) under the growth condition on $s(t)$ and the smallness condition on $\beta\max(1-\kappa)$ — from the Weibull density family to other delay densities used numerically, in particular the modified Gaussian $f^*(t)=c^*e^{-(t-\tau^*)^2/\sigma^2}$, or to an explicitly stated class of admissible kernels.

4. Predictive validation against actual data. Compare the model's Italy predictions — the confirmed-case series $j(t)$ and $J(t)$ beyond April 20, 2020 produced with the paper's Rayleigh-distribution parameters ($\beta=0.475$, $\kappa(t)\equiv 0.951$, $\kappa_2=0.9$, $\kappa_4=0.6$, mean periods $\tau_1=18$, $\tau_2=11$, $\tau_3=15$, $\tau_4=18$, $\tau_5=10$, $I_s(0)=110000$ at March 10, 2020), and the inferred trajectory of the unobserved $I_s(t)$ through June 17, 2020 — with the actual later Italian data, as the paper itself requested, stating what the comparison establishes about the model's validity.

The common objective is to turn the model from a trial-parameter forecasting illustration into an identified and qualitatively understood dynamical system: components 1 and 4 connect the model to observed data, while components 2 and 3 complete its qualitative and asymptotic theory for the same governed system. Components 1–3 are open mathematical problems; component 4 is a bounded data-comparison task, performable from public data.

References

1. Jijun Liu, Liyan Wang, Qiang Zhang, Shing-Tung Yau, 'The dynamical model for COVID-19 with asymptotic analysis and numerical implementations', Applied Mathematical Modelling 89 (2021) 1965–1982, DOI: 10.1016/j.apm.2020.07.057, https://doi.org/10.1016/j.apm.2020.07.057; open-access copy: PMC7414781, https://pmc.ncbi.nlm.nih.gov/articles/PMC7414781/
2. Liyan Wang, Qiang Zhang, Jijun Liu, 'On the dynamical model for COVID-19 with vaccination and time-delay effects: A model analysis supported by Yangzhou epidemic in 2021', Applied Mathematics Letters 125 (2022) 107783, DOI: 10.1016/j.aml.2021.107783, https://doi.org/10.1016/j.aml.2021.107783; open-access copy: PMC8570462, https://pmc.ncbi.nlm.nih.gov/articles/PMC8570462/
3. Yu Chen, Jin Cheng, Yu Jiang, Keji Liu, 'A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification', Journal of Inverse and Ill-Posed Problems 28(2) (2020) 243–250, DOI: 10.1515/jiip-2020-0010, https://doi.org/10.1515/jiip-2020-0010
4. Yu Chen, Jin Cheng, Yu Jiang, Keji Liu, 'A time delay dynamic system with external source for the local outbreak of 2019-NCoV', Applicable Analysis 101(1) (2022) 146–157 (online first February 2020), DOI: 10.1080/00036811.2020.1732357, https://doi.org/10.1080/00036811.2020.1732357
5. Efrén M. Benavides, María Ordobás Gavín, Raúl Mallaina García, Sara de Miguel García, Maira Ortíz Pinto, Ramón Doménech Gimenez, Ana Gandarillas Grande, 'COVID-19 dynamics in Madrid (Spain): A new convolutional model to find out the missing information during the first three waves', PLOS ONE 17(12) (2022) e0279080, DOI: 10.1371/journal.pone.0279080, https://doi.org/10.1371/journal.pone.0279080
6. Xinhe Zhu, Yuanyou Shi, Yongmin Zhong, 'An EKF prediction of COVID-19 propagation under vaccinations and viral variants', Mathematics and Computers in Simulation 231 (2025) 221–238, DOI: 10.1016/j.matcom.2024.12.012, https://doi.org/10.1016/j.matcom.2024.12.012
7. Zhigang Liu, Tiejun Zhou, 'Analysis of global stability of an epidemic model with asymptomatic infected persons', Advances in Continuous and Discrete Models 2025 (2025), DOI: 10.1186/s13662-025-03945-5, https://doi.org/10.1186/s13662-025-03945-5