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REVIEW 5 major objections 6 minor 43 references

Modeling H1N1 Influenza Transmission and Control: Epidemic Theory Insights Across Mexico, Italy, and South Africa

T0 review · 5 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A six-compartment flu model fit to CDC/WHO data reproduces H1N1 waves in Mexico, Italy, and South Africa, with simultaneous distancing plus treatment cutting total disease burden most.

desk verdict Standard SVEIRT fitting exercise undone by an algebraically broken R0 estimate and in-sample validation; the optimal-control formalism is fine but the data-driven claims do not hold. read the letter →

arxiv 2412.00039 v1 pith:4KDDMV7U submitted 2024-11-22 math.GM

Md Kamrujjaman , Kazi Mehedi Mohammad This is my paper · ORCID
classification math.GM MSC 53C2583C0557N16
keywords InfluenzaH1N1SVEIRTmodelOptimalcontrolBasicreproductionnumberSensitivityanalysisPartialrankcorrelationcoefficientLatinhypercubesamplingNonlinearleastsquares
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a six-compartment SVEIRT model—susceptible, vaccinated, exposed, infected, treated, and recovered—with time-dependent controls for physical distancing, antiviral treatment, and enhanced recovery, and fits its parameters to 120 weeks of CDC/WHO weekly influenza counts for Mexico, Italy, and South Africa. From the fitted parameters it estimates the basic reproduction number $R_0$ in the range $[0.5, 3.75]$, and uses local sensitivity indices, Latin hypercube sampling, and partial rank correlation to rank which parameters dominate transmission. It then solves the associated optimal-control problem via Pontryagin's maximum principle, proving existence and uniqueness of the optimal controls and computing the control profiles numerically. The central conclusion is that applying physical distancing and treatment simultaneously—at control levels around 0.6–0.75—reduces the exposed and infected populations faster than any single measure, making the mixed strategy the recommended way to lower total disease burden. If the fits are accepted, the paper offers a reusable, data-calibrated template for comparing influenza control policies across countries and seasons.

What carries the argument

The central object is the SVEIRT compartment model (system A.1), six ordinary differential equations for $S, V, E, I, R, T$ with vaccination rate $\phi$, vaccine efficacy $\varepsilon$, force of infection $(\beta_1 E + \beta_2 I)$, and recovery and treatment rates $\gamma$ and $\gamma_1$. The load-bearing identities are the next-generation-matrix reproduction numbers $R_0$ and $R_{0V}$ (with vaccination), formulas (A.5) and (A.4), which convert fitted parameters into the threshold that decides outbreak versus extinction. The other load-bearing piece is the optimality system from Pontryagin's maximum principle: adjoint equations (2.6) plus the projected control formulas (2.7), which turn the policy question—how hard to distance, how much to treat—into a well-posed minimization problem whose solution the paper computes numerically. This machinery does two jobs at once: it extracts parameters and threshold behavior from noisy weekly case counts, and it ranks control strategies by total cost-plus-burden rather than by infection count alone.

What would settle it

Re-fit the model using only laboratory-confirmed influenza A(H1N1) weekly case counts for Mexico, Italy, and South Africa over the same 120 weeks; the central claim fails if the model cannot track those series with comparable residuals or if the estimated reproduction number falls outside the reported range $0.5 \le R_0 \le 3.75$.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a six-compartment transmission model with vaccination and treatment compartments can reproduce the reported weekly influenza curves of Mexico, Italy, and South Africa after its parameters are estimated by nonlinear least squares, with residuals that are small and random enough to be read as validation. The fitted model yields an $R_0$ range of $[0.5, 3.75]$, with time-varying effective reproduction numbers that fluctuate around $2.5$ and occasionally fall below $1$, indicating an outbreak that spreads but can be pushed toward control. Sensitivity analysis, both local and global, identifies the contact rates $\beta_1$, $\beta_2$, progression rate $\alpha$, recovery rate $\gamma$, and treatment rate $\gamma_1$ as the dominant influences on $R_0$. The optimal-control analysis establishes that an optimal triple of controls $(w_1, w_2, w_3)$ exists and is unique, and numerical simulation shows that running all three controls at high levels simultaneously clears the exposed and infected classes more rapidly than no control or single controls, supporting the paper's policy conclusion that mixed strategies are most effective.

Load-bearing premise

The load-bearing premise is that the CDC/WHO weekly counts used for fitting are consistently reported H1N1 influenza cases per 1000 population, with negligible underreporting and no subtype misclassification; if those counts include other flu subtypes or vary in reporting, every fitted parameter and country conclusion is called into question.

Editorial extensions

If this is right

  • If the model is correct, the fitted curves place the main influenza burden in the study period at weeks 30–50, with infections declining after week 60, so planners can use the same fitted system to anticipate peak timing in comparable seasons.
  • The model implies that no single intervention suffices: because $R_0$ mostly exceeds 1 in the fitted period, sustained physical distancing combined with treatment—not one measure alone—is what pushes the effective reproduction number below 1.
  • The sensitivity ranking gives a concrete ordering for intervention design: reducing contact rates $\beta_1, \beta_2$ (masks, distancing, isolation) and slowing progression $\alpha$ yields the largest reduction in $R_0$, while recovery and treatment rates $\gamma, \gamma_1$ contribute a secondary, negative correction.
  • The optimal-control schedule is computable and transferable: the paper's numerical solution shows near-maximal effort for roughly 3–4 weeks of distancing and 6–7 weeks of treatment before controls taper, which can be re-solved for other countries or seasons with updated data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The authors fit the same model separately to each country, but do not test whether one shared parameter set or a seasonally forced version can explain all three; that test would probe structural identifiability rather than per-country goodness of fit.
  • Because the input counts are not filtered by influenza subtype, the fitted transmission parameters likely absorb co-circulating influenza A and B; treating the reported $R_0$ range as an upper-bound family until lab-confirmed A(H1N1)-only incidence is used would be a prudent reading.
  • The paper's optimal schedule implies a directly testable policy comparison: districts that implement distancing and treatment simultaneously versus distancing-only districts, tracked with weekly case counts, would check whether the model's strategy ordering holds outside the fitted data.
  • The effective reproduction formula (3.3) uses a fixed generation-interval distribution; replacing it with time-varying generation intervals estimated from contact tracing could sharpen the weekly $R_0(t)$ estimates the paper reports.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The manuscript proposes an SVEIRT compartment model for H1N1 influenza with vaccination and treatment, derives an optimal control formulation with existence and uniqueness arguments, fits model parameters by nonlinear least squares to weekly CDC/WHO influenza case counts for Mexico, Italy, and South Africa, estimates R0 from early exponential growth, performs local and global sensitivity analyses, and concludes that combining physical distancing and treatment controls is the most effective strategy. The abstract and Section 3 state data windows of 120 weeks from October 2020 to March 2023, while Section 3 fits 85 weeks beginning October 2021 and Section 5 uses 120 weeks for Italy and South Africa; these inconsistencies are not resolved.

Significance. If the empirical claims were supported, a multi-country, data-driven SVEIRT model with optimal control and sensitivity analysis would be a useful contribution to influenza outbreak planning. The manuscript does include a standard optimal-control framework, a positivity/boundedness argument, and an LHS/PRCC sensitivity workflow, which are appropriate tools. However, the central quantitative results—model validation and the R0 range [0.5, 3.75]—rest on circular fitting and an invalid algebraic derivation, and the data are not demonstrably H1N1-specific. The paper therefore does not currently provide a reliable basis for its public-health conclusions.

major comments (5)
  1. [§3.1, Eq. (3.2)] The exponential-growth balance is derived incorrectly. Substituting E ∼ E0 exp(Λt) and I ∼ I0 exp(Λt) into the E equation of model (A.1) gives ΛE0 = (β1E0 + β2I0)S0 − (α + µ)E0, so the printed equation should contain (α + µ + Λ)E0 on the right-hand side. As printed, Eq. (3.2) is one equation in two unknowns, β1 and β2, so the statement 'putting the value of β1 and β2 from (3.2)' is not derivable, and the displayed R0 formula does not follow from it nor does it reproduce the next-generation expression (A.5). The formula also depends on ϕ through S0 = Λ/(µ + ϕ), but ϕ is never assigned a value in Tables 1 or 2. Since the claimed range R0 ∈ [0.5, 3.75] and the subsequent sensitivity analyses are built on this equation, the quantitative core of the paper is unsupported.
  2. [§3, §5.1, §5.2; Figures 6, 15, 16] The model is validated on the same data used for fitting. Parameters in Tables 1–2 are estimated by nonlinear least squares on each country's weekly case series, and the 'validation' consists of residual plots of that same series. No holdout period, cross-validation, independent forecast, or uncertainty quantification is provided, and the stated '70% accuracy' and '76% accuracy' are not defined or derived. The conclusion that the model is validated for Mexico, Italy, and South Africa is therefore not supported by the evidence presented.
  3. [Abstract, §3, §5] The data are taken from CDC and WHO influenza dashboards without filtering for influenza A (H1N1), and the study window (October 2020–March 2023) spans a period of COVID-19-driven surveillance changes and multiple seasonal influenza viruses. Underreporting and subtype misclassification therefore affect every fitted parameter and the R0 estimate. The paper needs either H1N1-specific case counts or a documented sensitivity analysis to surveillance artifacts; in their absence, the country-level conclusions inherit an unsupported premise.
  4. [Abstract, §3, §5] The data window is stated inconsistently: the abstract says 120 weeks from October 1, 2020 to March 31, 2023; Section 3 fits 85 weeks beginning October 15, 2021; Section 5 uses 120 weeks for Italy and South Africa; and Figure 1 is labeled 85 weeks. This inconsistency makes the parameter estimates in Table 2 and the country comparisons irreproducible, and it is not cosmetic because fitted rates such as Λ depend on the window.
  5. [§A.2, §6] Global stability of the disease-free and endemic equilibria is asserted in the conclusion but not proved in this manuscript. Theorems 3 and 4 only establish positivity and boundedness, and the endemic-equilibrium expressions are deferred to the companion preprint [42]. Since the concluding interpretation of the threshold relies on these stability claims, the qualitative analysis is incomplete as presented.
minor comments (6)
  1. [§2.4] The heading and text refer to 'Poncryagin's Maximum Principle'; the name should be 'Pontryagin'.
  2. [§3] The sentence listing the countries for parameter estimation says 'Colombia, Italy and South Africa', but the tables and analyses concern Mexico, Italy, and South Africa; Colombia appears nowhere else.
  3. [Figure 6] The caption lists panels (a)–(d), but the text and figure contain only three panels; panel (d) is not described in the body.
  4. [Table 4] The PRCC significance statements do not consistently apply the stated p < 0.05 threshold; for example, Λ has p = 0.064 but is described as influential, while δ has p = 0.007 with a modest PRCC of 0.37. The text should clarify which significance criterion is being used.
  5. [§4.4] The 'relative bias' of R0 is described by an example but no estimator or formula is given, so the reported ranges in Figure 11 are not reproducible.
  6. [§A.1] The symbol S0 is used both for the disease-free susceptible value Λ/(µ + ϕ) and for the initial condition S(0), which is confusing; distinct notation should be used.

Circularity Check

3 steps flagged · score 7.0 of 10

Model validation and R0 are computed from the same least-squares fit, and the global-stability conclusion is carried by the authors' own preprint; the central claims are partially circular by construction.

  1. fitted input called prediction [Section 3, Parameter Estimation and Model Validation (SSE definition and Figure 6 discussion)]
    "where Yj represents the cumulative number of the real reported data for jth observation, I(tj) represents the model predicted cumulative data for jth observation ... The model is fitted to the total number of cases that have been infected, as shown in Figure 6(b). ... The residuals' randomness indicates that the fitness is optimal."

    The same infection series is used twice: first as the least-squares target for estimating model parameters, then as the 'validation' data whose residuals are shown to be 'tiny and erratic.' Because the parameter vector is chosen to minimize exactly those residuals, the agreement in Figure 6(b) and the residual plot is forced by construction; it cannot independently confirm the model. Calling this 'model validation' and using it to support forecasts treats an in-sample fit as an out-of-sample prediction.

  2. fitted input called prediction [Section 3.1, Estimation of R0 from Actual Data (Eq. 3.2 and following paragraph)]
    "Replacing (3.1) in third equation of model (A.1), we get, (β1E0 + β2I0) Λ/µ+ϕ = (α+µ)E0 (3.2). Putting the value of β1 and β2 from (3.2) in equation (A.1), We derive the basic reproduction number expression R0 ... Through the examination of data from South Africa, Mexico, and Italy, we have calculated the fundamental reproduction number R0 ∈ [0.5, 3.75] with lower value 0.5 and upper value 3.75."

    Eq. (3.2) is a single balance equation in two fitted unknowns β1 and β2, so it cannot supply 'the value of β1 and β2.' The numerical R0 is evaluated with the Table 1/Table 2 least-squares estimates and with Λ = 500×0.02 week−1 read from a regression slope of the same weekly case data. The claimed range is therefore a deterministic function of the same data and fitted parameters that are supposed to be validated; presenting it as an independent threshold 'validation' is a fitted input relabeled as a prediction. The printed balance also omits the ΛE0 term from the exponential derivative, so Eq. (3.2) is not even a valid transformation.

1 more flagged steps
  1. self citation load bearing [Appendix A.1 (Model Formulation), EE paragraph; Conclusion; reference [42]]
    "The details analysis of EE can be found in [42]. ... The numerical results supports the analytical results and verified that the illness will continue to exist in the community because the endemic equilibrium is globally asymptotically stable. On the other hand, the equilibrium point that is devoid of illness is globally asymptotically stable, and the disease will eventually vanish from the community."

    Reference [42] is the authors' own arXiv preprint (Mohammad, Akhi, and Kamrujjaman). No stability theorem or proof is given in the present paper; the global-asymptotic-stability conclusions in the Conclusion are supported only by that self-citation. Because the persistence/eradication dichotomy is presented as an analytical result of the model, this is a load-bearing appeal to the authors' prior unverified work rather than an independent mathematical result established here.

full rationale

The paper's central data-driven claims are (i) that the SVEIRT model is validated by fitting CDC/WHO influenza data for Mexico, Italy, and South Africa, and (ii) that R0 lies in [0.5, 3.75]. Both of these claims reduce to the same inputs: parameters are estimated by least squares on the case-count series, and then the residual plots and the R0 range are presented as confirmation using those same case counts and fitted values. That is partial circularity by construction: the 'validation' cannot fail because it is the objective being minimized, and the R0 range is a transformation of the fitted parameters plus the early-growth slope of the same data. A separate, non-circularity but severe validity problem is that Eq. (3.2) is algebraically wrong (the exponential-derivative term is dropped) and cannot produce the displayed R0 formula; this is a correctness risk independent of the circularity score. Additionally, the global-stability conclusions are imported from the authors' own preprint [42], with no proof in the paper, making that step a load-bearing self-citation. The paper does contain independent mathematical content (the optimal-control existence/adjoint derivations and sensitivity computations), so the entire derivation is not circular, but the headline validation and threshold estimates are not independent of their fitted inputs. There is no evidence of machine-checked or externally benchmarked verification of [42] or of the fitted-prediction loop.

Assumptions & free parameters 13 free parameters · 6 assumptions · 0 invented entities

The central results rest on fitted per-country parameters, a homogeneous-mixing compartment model, and the unverified premise that the data series represent H1N1. The model also uses the vaccination rate phi in model and R0 formulas without ever assigning it a value, and the validation is performed on the same data used for fitting.

free parameters (13)
  • alpha (E to I transition rate) = Mexico 0.75, Italy 0.67, South Africa 0.78 week^-1
    Estimated by fminsearch least-squares fit to weekly case data (Table 2).
  • beta1 (transmission rate from E to S) = Mexico 0.0055, Italy 0.0053, South Africa 0.0075 week^-1
    Estimated per country by nonlinear least squares (Table 2).
  • beta2 (transmission rate from I to S) = Mexico 0.0055, Italy 0.0061, South Africa 0.0081 week^-1
    Estimated per country by nonlinear least squares (Table 2).
  • gamma (recovery rate of I) = Mexico 0.65, Italy 0.61, South Africa 0.63 week^-1
    Estimated per country by nonlinear least squares (Table 2).
  • gamma1 (treatment progression rate of I) = Mexico 0.25, Italy 0.31, South Africa 0.35 week^-1
    Estimated per country by nonlinear least squares (Table 2).
  • lambda (vaccine inefficiency rate) = Mexico 0.55, Italy 0.52, South Africa 0.42
    Estimated per country by nonlinear least squares (Table 2).
  • mu (natural death rate) = Mexico 0.05, Italy 0.03, South Africa 0.03 week^-1
    Estimated per country by nonlinear least squares (Table 2).
  • delta (disease-induced death rate) = Mexico 0.30, Italy 0.27, South Africa 0.29 week^-1
    Estimated per country by nonlinear least squares (Table 2).
  • epsilon (vaccine effectiveness) = Mexico 0.45, Italy 0.41, South Africa 0.44
    Estimated per country by nonlinear least squares (Table 2); related to lambda by lambda = 1 - epsilon.
  • phi (vaccination rate in S class)
    Appears in model (A.1), DFE (A.3), R0V formula (A.4), and R0 estimation via S0 = Lambda/(mu + phi) in Section 3.1, but is absent from Tables 1 and 2 and is never estimated or assigned a value.
  • Initial conditions S(0), V(0), E(0), I(0), R(0), T(0) = 500, 1, 1, 1, 0, 0 (per 1000 population)
    Chosen initial guesses described as estimated from reference [23] (Section 3); no data-driven calibration is shown.
  • Control cost weights a1, a2, a3, a4, a5 = 20, 20, 45, 25, 50; variants a3=90, a4=34, a5=87 and a3=62, a4=23, a5=60
    Ad hoc weights in the objective functional (2.2) and in the control simulations of Section 2.5.
  • Exponential growth rate for R0 estimation = Reported as 500 x 0.02 week^-1 in Section 3.1
    Obtained from the slope of a regression line of weekly new cases against cumulative cases; notation is confused with the recruitment rate Lambda.
assumptions (6)
  • domain assumption Mass-action homogeneous mixing with bilinear incidence (beta1 E + beta2 I) S and (1 - epsilon)(beta1 E + beta2 I) V, as in equation (A.1)
    Load-bearing for all fitted parameters and for R0; the model ignores spatial, age, and contact heterogeneity.
  • domain assumption Reported CDC/WHO weekly influenza counts are H1N1 cases and are comparable across Mexico, Italy, and South Africa
    The paper uses CDC FluView and WHO Influenza Dashboard totals but never filters by subtype; all country-level conclusions inherit this premise.
  • domain assumption Early epidemic growth is exponential with constant susceptibility S0 = Lambda/(mu + phi), used in Section 3.1 to estimate R0
    This is the premise of equations (3.1) and (3.2); it also assumes phi is known, but phi is never assigned a value.
  • standard math Standard ODE existence and uniqueness, Fleming-Rishel existence conditions, and Pontryagin maximum principle
    Invoked in Section 2 for the optimal control theorems and optimality system.
  • standard math Next-generation matrix approach defines R0 for the compartmental model
    Appendix A.1 presents R0 formulas but does not show the derivation; this is standard practice in epidemic modeling.
  • domain assumption Data are reported per 1000 population and underreporting is negligible
    The abstract and Section 3 state per-thousand scaling; no correction for reporting rates or surveillance completeness is applied.

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Cite this review

Pith. "Pith review of Modeling H1N1 Influenza Transmission and Control: Epidemic Theory Insights Across Mexico, Italy, and South Africa." pith.science (2026). https://pith.science/paper/4KDDMV7U

@misc{pith2026241200039,
  author       = {Pith},
  title        = {Pith review of: Modeling H1N1 Influenza Transmission and Control: Epidemic Theory Insights Across Mexico, Italy, and South Africa},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KDDMV7U}},
  note         = {Machine review of arXiv:2412.00039}
}
abstract

This study incorporates mathematical analysis, focusing on developing theories and conducting numerical simulations of Influenza virus transmission using real-world data. The terms in the equations introduce parameters which are determined by fitting the model for matching clinical data sets using non-linear least-square method. The purpose is to determine the wave trend, critical illness factors and forecast for Influenza in national levels of Mexico, Italy, and South Africa and to investigate the effectiveness of control policy and making some suggestions of alternative control policies. Data were extracted from the following sources: published literature, surveillance, unpublished reports, and websites of Centres For Disease Control and Prevention (CDC) \cite{CDC}, Natality report of U.S. clinics and World Health Organization (WHO) Influenza Disease Dashboard \cite{WHO}. We included total 120 weeks data (which are calculated as per thousand) from October 01, 2020 to March 31, 2023 \cite{CDC}, throughout this study. Numerical and sensitivity analysis are carried out to determine some prevent strategies. The objectives of local and global sensitivity analysis is to determine the dominating parameters and effective correlation with $\mathcal{R}_0$. We presented data fitting, Latin hypercube sampling, sensitivity indices, Partial Rank Correlation Coefficient, p-value, estimation of the nature of $\mathcal{R}_0$ from available data to show validation of the model with these counties. The aim is to determine optimal control strategies with drug administration schemes, treatments which represent the efficacy of drug inhabiting viral production and preventing new infections, minimizes the systematic cost based on the percentage effect of the drug. Finally, we present series of numerical examples and the effect of different parameters on the compartments to verify theoretical results.

Figures

Figures reproduced from arXiv: 2412.00039 by the authors.

Figure 1
Figure 1. Weekly total of Mexico data cases for 85 weeks beginning on October [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Simulation of the model (a) S(t) compartment, (b) V (t) compartment (c) E(t) compartment (d) I(t) compartment (e) R(t) compartment and (f) T(t) compartment showing with control and without control with respect to control parameters w1, w2, w3, where w1 = w2 = w3 ∈ [0, 0.45, 0.6, 0.75] [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Simulation of the model (a) S(t) compartment,(b) V (t) compartment (c) E(t) compartment (d) I(t) compartment (e) R(t) compartment and (f) T(t) compartment showing with control and without control with respect to weighted parameters a3, a4, a5, where a3 ∈ [0, 62, 90], a4 ∈ [0, 23, 34], a5 ∈ [0, 60, 87]. By examining these three scenarios, it can be observed that, in comparison to the situation where there is no contr… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Phase portrait of different control efforts (a) control strategies [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: (a) Least square polynomial fitting (b) Least square curve fitting (c) [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: (a) Fitting model data with weekly cases (b) Residuals of the fit (c) [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: (a) Time series effective reproduction number (b) Time series effective [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: (b) depicts that, for fixed values of R0 ∈ [1.5, 3.5], the contact rates β1 and β2 are reversely related i.e., negatively correlated. The contour graphs behaves like straight lines which indicates the increasing of β1 from 0.0025 to 0.0075 results into rapid decrease o…
Figure 9
Figure 9. Figure 9: Normalized sensitivity indices of R0 with respect to parameters for (a) Italy data (b) Mexico data and (c) South Africa data 4.3 Latin Hypercube Sampling of Parameters A popular method in epidemiology for sampling parameters within predetermined ranges is Latin hypercu…
Figure 10
Figure 10. Figure 10: Latin Hypercube Sampling graphs of parameters (a) [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]
Figure 11
Figure 11. Figure 11: Relative bias of R0 by LHS of parameters (a) β1 and β2 (b) α and (1 − λ) (c) γ and γ1 (d) δ and µ , where the mean values of parameters range are taken from [PITH_FULL_IMAGE:figures/full_fig_p036_11.png]
Figure 12
Figure 12. Figure 12: (a) PRCC and (b) P-values plot for Mexico data of Influenza disease, [PITH_FULL_IMAGE:figures/full_fig_p038_12.png]
Figure 13
Figure 13. Figure 13: (a) PRCC and (b) P-values plot for Italy data of Influenza disease, [PITH_FULL_IMAGE:figures/full_fig_p039_13.png]
Figure 14
Figure 14. Figure 14: (a) PRCC and (b) P-values plot for South Africa data of Influenza [PITH_FULL_IMAGE:figures/full_fig_p040_14.png]
Figure 15
Figure 15. Figure 15: Model data fitting to Italy with (a) weekly infected data (b) weekly pro [PITH_FULL_IMAGE:figures/full_fig_p043_15.png]
Figure 16
Figure 16. Figure 16: Model data fitting to South Africa with (a) weekly infected data (b) [PITH_FULL_IMAGE:figures/full_fig_p045_16.png]

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