A generalized distributed delay model of COVID-19: An endemic model with immunity waning

Sarafa A. Iyaniwura; Rabiu Musa; Jude D. Kong; Sarafa A. Iyaniwura; Rabiu Musa; Jude D. Kong

doi:10.3934/mbe.2023249

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 5379-5412. doi: 10.3934/mbe.2023249

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Research article Special Issues

A generalized distributed delay model of COVID-19: An endemic model with immunity waning

1.
Department of Mathematics and Institute of Applied Mathematics (IAM), University of British Columbia, Vancouver, British Columbia, Canada
2.
Faculty of Mathematics, Technion Israel Institute of Technology, Haifa 32000, Israel
3.
Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada
4.
Africa-Canada Artificial Intelligence and Data Innovation Consortium (ACADIC), York University, Toronto, Ontario, Canada

Received: 30 October 2022 Revised: 13 December 2022 Accepted: 20 December 2022 Published: 12 January 2023

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has been spreading worldwide for over two years, with millions of reported cases and deaths. The deployment of mathematical modeling in the fight against COVID-19 has recorded tremendous success. However, most of these models target the epidemic phase of the disease. The development of safe and effective vaccines against SARS-CoV-2 brought hope of safe reopening of schools and businesses and return to pre-COVID normalcy, until mutant strains like the Delta and Omicron variants, which are more infectious, emerged. A few months into the pandemic, reports of the possibility of both vaccine- and infection-induced immunity waning emerged, thereby indicating that COVID-19 may be with us for longer than earlier thought. As a result, to better understand the dynamics of COVID-19, it is essential to study the disease with an endemic model. In this regard, we developed and analyzed an endemic model of COVID-19 that incorporates the waning of both vaccine- and infection-induced immunities using distributed delay equations. Our modeling framework assumes that the waning of both immunities occurs gradually over time at the population level. We derived a nonlinear ODE system from the distributed delay model and showed that the model could exhibit either a forward or backward bifurcation depending on the immunity waning rates. Having a backward bifurcation implies that $R_c < 1$ is not sufficient to guarantee disease eradication, and that the immunity waning rates are critical factors in eradicating COVID-19. Our numerical simulations show that vaccinating a high percentage of the population with a safe and moderately effective vaccine could help in eradicating COVID-19.

Keywords:

Citation: Sarafa A. Iyaniwura, Rabiu Musa, Jude D. Kong. A generalized distributed delay model of COVID-19: An endemic model with immunity waning[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5379-5412. doi: 10.3934/mbe.2023249

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Abstract

1. Introduction

The method of monitoring physiological condition by means of bioimpedance remains a novel technique for more than 50 years or so. The non-destructive method of bioimpedance measurement for fruits and vegetables is extensively used for various purposes such as to access health conditions [1],[2], effect of storage conditions [3], physiological changes including assessing the quality of fruits and vegetables [4]–[8]. On the other hand, investigations are also carried out to understand the science behind the characteristics of bioimpedance of plant tissues [9]–[17]

The necessity of invariability of bioimpedance parameters is extremely important in practical application of Cole model for discriminating fruits (or vegetables) from a reference fruit (or vegetable). Say for example someone wants to compare the Cole parameters for two different species of a given fruit. In such cases, the parameters can be compared if the parameters are not dependent on position (of a given fruit or sample). If the position (of measurement) dependent variability is high, one cannot distinguish two different samples under investigation. Thus, the measurement method should be robust enough to establish invariability of bioimpedance parameters which can be treated as reference parameters for comparison purpose.

Figure 1. This figure is not in a scale. (a) Schematic of biological tissue. Straight dashed horizontal lines (green) represent the ac current path while the curvy lines (red) represent dc current path in biological tissue. (b) Cole equivalent circuit of lump tissue. Here, R₀ and R_∞ are the high and low frequency resistances respectively. C is the fractional capacitance and α (0 < α ≤ 1) is the dispersion (nonideality factor) parameter. The equivalent circuit consists of single dispersion parameter α.

DownLoad: Full-Size Img PowerPoint

Cole model is widely applied for living biological tissue [15],[18]–[21]. In general, lump tissue is represented by Cole equivalent circuit and single dispersion model is accepted for its simplicity [7],[14] (Figure 1). Among the various method of bioimpedance measurements, direct measurement (i.e. using LCR meter) with two-electrode configuration is the most generic one due to its high degree of reliability [5],[20]. The electrical current paths inside a tissue is schematically shown in Figure 1 (a). It is intuitive from this figure, that if impedance is measured for the same tissue at other position, keeping the distance between the electrode invariant, one should obtain the same results (within the limits of experimental error for living tissue). On the other hand, if the distance between the electrodes is not kept same for measuring impedance at two different positions, the variability in measured parameters are clearly observed [14],[15],[22]. Thus, in order to obtain the comparable results among the samples, it is quite obvious to keep the distance between the electrode invariant. The reasons of choosing apple and potato are as follows. First of all, apple is a fruit whereas potato is a vegetable (tuber) and both of them are extensively used as food materials and therefore have importance in agriculture sector. Further, apple grows in open atmosphere whereas potato grows under soil. Not only these, the tissue of apple will have a natural tendency towards ripen (or to be eaten) so that seed can be dispersed. On the other hand, potato tissue will have tendency to fight against external environment to sustain so as to produce the next generation potato. The porosity (fraction of empty space) of apple is highly different than that of potato.

In this work, a specific electrode-pair is designed and is used to measure bioimpedance of apple and potato tissues. It is needless to mention that, though apple is a fruit, potato is a vegetable and providing us two different kind of living tissue to be investigated under this study. Applying single dispersion Cole model, impedance parameters are extracted by curve fitting. It is established here that the tissue of apple and potato can be specified in terms of bioimpedance parameters which in turn can be used to discriminate similar tissue of different species with 98% confidence. Further, with 99% confidence, it is evidently shown here that the position of measurement of bioimpedance of a given fruit or vegetable is not a factor of variability. Thus, our process of measurement can directly be applied in assessing quality (or condition) of fruits or vegetables. In addition to this, our method can be applied to monitor the ripening of fruits similar to citrus fruit like orange [8].

2. Single dispersion Cole bioimpedance model

Single dispersion Cole bioimpedance model is a widely accepted model is extensively used for various kinds of biological tissue [5]–[8],[10]–[21]. The single dispersion Cole equivalent circuit for lump biological tissue is shown in Figure 1(b). The equivalent circuit consists of a constant phase element (CPE) comprising of a fractional capacitor (C) with a parallel resistance (R0 - R∞) and a series resistance R∞. The CPE maintains the phase invariancy by the dispersion parameter α. In the equivalent circuit, R0 is the dc resistance (or low frequency resistance of the tissue); R∞ is the ac resistance of tissue at high frequency. The impedance of single dispersion equivalent circuit (Cole equivalent circuit) is usually expressed as

$Z = R_{\infty} + \frac{(R_{0} - R_{\infty})}{1 + {(jωτ)}^{α}}$ (1)

where, $j = \sqrt{- 1}$ . The symbol τ in Eq. (1) is known as the characteristics time constant of the tissue and is expressed as

$τ = {[(R_{0} - R_{\infty}) C]}^{1 / α}$ (2)

Separation of Z into real and imaginary parts is not straight forward due to the non-integer value of α (0 < α ≤1), which turns the Eq. (1) into the category of fractional calculus. Nevertheless, analytically it is possible to express the impedance Z in terms of real and imaginary part using $j^{α} = (j \sin \frac{απ}{2} + \cos \frac{απ}{2})$ and after simplifying we obtain

$Z = X + jY$ (3)

where,

$X = R_{\infty} + \frac{(R_{0} - R_{\infty}) (1 + ω^{α} τ^{α} \cos \frac{απ}{2})}{{(1 + ω^{α} τ^{α} \cos \frac{απ}{2})}^{2} + {(ω^{α} τ^{α} \sin \frac{απ}{2})}^{2}}$ and

$Y = - \frac{(R_{0} - R_{\infty}) (ω^{α} τ^{α} \sin \frac{απ}{2})}{{(1 + ω^{α} τ^{α} \cos \frac{απ}{2})}^{2} + {(ω^{α} τ^{α} \sin \frac{απ}{2})}^{2}}$ .

Thus, the modulus of Z becomes

$| Z | = \sqrt{{(R_{\infty} + \frac{(R_{0} - R_{\infty}) (1 + ω^{α} τ^{α} \cos \frac{απ}{2})}{{(1 + ω^{α} τ^{α} \cos \frac{απ}{2})}^{2} + {(ω^{α} τ^{α} \sin \frac{απ}{2})}^{2}})}^{2} + {(\frac{(R_{0} - R_{\infty}) (ω^{α} τ^{α} \sin \frac{απ}{2})}{{(1 + ω^{α} τ^{α} \cos \frac{απ}{2})}^{2} + {(ω^{α} τ^{α} \sin \frac{απ}{2})}^{2}})}^{2}}$ (4)

Experimentally, Eq. (4) |Z| is obtained as a function of frequency f (= ω/2π). The experimental data is then fitted (non-linear curve fitting) and values of R₀, R_∞, C, α are obtained. From these parameters, the relaxation time τ is calculated using Eq. (2).

3. Specially designed electrode-pair

In general, the physiological parameters are non-repetitive and standard deviation is large. For example, if at identical condition, the cell size is measured for a given tissue, the variation of more than 20% is not uncommon while the grain size of a metallic conductor is measured, the result is found to be within the tolerance level (about 5% or so). The variation in cell size would therefore lead to a large variation in the tissue related parameters. Thus, it is extremely important to specify the tissue by means of Cole model parameters as well as relaxation time. On the other hand, it is essential to keep the distance (or gap) between the electrode-pair invariant. Intuitively, this is due to the fact that the Cole parameters (except α) are highly dependent on the gap between the electrodes. Hence, to discriminate tissues from one another, it is essential to obtain the invariability of impedance irrespective of measuring position (of tissue) of fruits or vegetable.

An electrode-pair is designed and fabricated for the purpose of bioimpedance measurement of fruit or vegetable. A schematic of the fabricated electrode-pair is shown in Figure 2. The depth of penetration and diameter of the electrodes are 1.5 cm and 0.28 mm respectively and these are maintained throughout this work unless otherwise stated. The electrodes are separated by plastic material (of negligibly low leakage current). The plastic holder of the electrode-pair is made by cutting a disposable syringe. The gap between the electrode is kept at 3 mm. For electrode material, stain-less steel is chosen for its inertness of reacting with any other materials and retain sufficient strength at low wire diameter.

Figure 2. Schematic of electrode-pair. Typical pictures of accessing tissue of apple and potato using electrode-pair are shown in the inset of the Figure.

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4. Experiments

Table 1. Sample matrix under study.

Sample	Species/Description	Designation of Measurement
Apple-1	Apple-1 and Apple-2 are of same species (Green Granny Smith) and both are purchased from same local shop. Apple-1 and Apple-2 can be considered to be biologically identical.	A₁₁, A₁₂, A₁₃, A₁₄, A₁₅
Apple-2		A₂₁, A₂₂, A₂₃, A₂₄, A₂₅
Apple-3	Apple-3 (Red Washington) is of different species that of Apple-1 and Apple-2. Apple-3 is biologically different from Apple-1 and Apple-2.	A₃₁, A₃₂, A₃₃, A₃₄, A₃₅
Potato-1	Potato-1 and Potato-2 are of same species (Kufri Chandramukhi) and both are purchased from same local shop. Potato-1 and Potato-2 can be considered to be biologically identical.	P₁₁, P₁₂, P₁₃, P₁₄, P₁₅
Potato-2		P₂₁, P₂₂, P₂₃, P₂₄, P₂₅
Potato-3	Potato-3 (Kufri Jyoti) is of different species that of Potato-1 and Potato-2. Potato-3 is biologically different from Potato-1 and Potato-2.	P₃₁, P₃₂, P₃₃, P₃₄, P₃₅

Remarks: In this table, A_mn (or P_mn) represents impedance measurement of Apple-m (or Potato-m) at position n.

| Show Table

DownLoad: CSV

The sample matrix is shown Table 1. In order to remove any parasitic effect that may arise from electrodes and probes, the LCR meter (Model: 8101G, Make: Gw Instek) is calibrated before taking any measurement data. Each time electrodes are cleaned by medicated sprit and dried properly before inserting into fruit or vegetable. By inserting the electrode-pair at desired position of the sample, impedance is measured in the frequency range of 1 Hz to 1 MHz with amplitude of 1 V (peak to peak). All measurements are conducted at identical condition. In order to avoid any physical damage in the tissue due to high electric field, small signal is chosen for measurement. The impedance is found to decrease with the increase in frequency and becomes sufficiently flat around the frequency of 1 MHz and thus restricts the choice of upper limit on the frequency of interest. For each sample, five measurements are done at five different positions and each measurement contains four hundred data points. Typical measurement data are plotted as a function of frequency and is shown in Figure 3.

Figure 3. Measured tissue impedance as a function of frequency. (a) apple, (c) potato. Fitted Cole plots (b) apple, (d) potato.

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5. Results and discussions

Experimentally |Z| is obtained as a function of frequency for different samples and for different position of a given sample as mentioned in Table 1. The non-linear curve fitting is performed by commercial MS Excel 2016 package. As a common practice, in the curve fitting process, the value of R₀ and R_∞ is considered to be the measured values R₀ and R_∞. Hence in this work, the measured impedances at 1 Hz and 1 MHz are considered to be R₀ and R_∞ respectively. The value of α and C are estimated by fitting the experimental curve with the Eq. (4). The characteristic time constant τ is then estimated using Eq. (2) for each set of data.

From the experimental plots of |Z| as a function of frequency (see Figure 3) it is not possible to distinguish the species of a given fruit or vegetable and this is because the plots of different species almost overlap on each other when plotted simultaneously in a single graph. On the other hand, the important parameters related to equivalent circuit of these plots may differ significantly.

The fitted Cole plots are shown in Figure 3(b) and Figure 3(d) for apple and potato respectively. Since, the sample Apple-1 and Apple-2 are from same species, the Cole plots of these do not differ and follow one another for the entire span of frequency range. While the Cole plot for the sample Apple-3 slightly differ from the other two, especially at the low frequency range. This is probably due to the fact that the species of sample Apple-3 is different from that of sample Apple-1 or Apple-2. Exactly similar observation is found for potato samples (Figure 3. (d)). From the fitted Cole plots, it is noticed that the distinguishable capability of bioimpedance method is better at the low frequency range for apple as well for potato. This fact suggests that the bioimpedance study of apple and potato can be limited to 3 kHz. By reducing the frequency range while keeping the number data point same, we can obtain better experimental results.

5.1. ANOVA on apple and potato samples

In this study, all ANOVA are performed using commercial MS Excel 2016 package. The Cole parameters for apple samples are shown in Table 2. ANOVA on τ is performed considering position and sample as two factors. The first factor is considered to understand whether position of measurement is a factor or not. On the other hand, the second factor is considered to understand the discriminating capability of the bioimpedance measurement.

To test these hypotheses, two apples of same species (Green Granny Smith in this case) are taken from a local shop and thus the biological differences between the apples are minimized. Then for each apple, impedance is measured at five different positions. The measured and calculated data is shown in Table 2. Result of ANOVA for tissue of apple is shown in Table 3. Interestingly it is found that none of the factors are significant (since, Fcritical > F-value) at the confidence level as high as 99% when ANOVA is performed among the sample Apple-1 and Apple-2. This means that one the Cole parameters i.e. the relaxation time are indistinguishable as far position is concerned. The two samples (Apple-1/Apple-2) are also found to be indistinguishable. Or in other words, the tissues of Apple-1 and Apple-2 are statistically same. Thus, with respect to our electrode system, the relaxation time is invariable and it is does depend on the position of measurement and remains same for biologically same samples.

On the other hand, though the position is found be indistinguishable in Apple-1 and Apple-3 (or Apple-2 and Apple-3), they are found to be disguisable (Fcritical < F-value) when ANOVA is performed for these two pair of samples. Thus, again as before, the position of measurement is not found to be a factor. However, in the present two cases, the sample is found to be a factor and is discriminated by respective relaxation time. This means the tissues of Apple-1 and Apple-3 are different. This fact proves that the relaxation time of apple tissue can be used to discriminate different species of apple.

Table 2. Cole parameters obtained from experiment and curve fitting for apple.

Sample	R₀ (kΩ)	R_∞ (kΩ)	α	C (nF sec^α−1)	τ (µsec)	Regression coefficient
	12.93	0.64	0.719	76.65	63	0.9989
	14.21	0.74	0.703	88.30	70	0.9987
Apple-1	13.58	0.77	0.691	99.99	65	0.9983
	13.75	0.65	0.705	91.55	72	0.9989
	12.12	0.63	0.692	103.83	60	0.9983
	12.83	0.61	0.694	98.25	63	0.9980
	13.67	0.70	0.708	86.55	69	0.9985
Apple-2	13.90	0.72	0.696	103.47	76	0.9985
	12.47	0.67	0.693	112.05	70	0.9983
	10.97	0.70	0.694	113.46	59	0.9984
	11.85	0.83	0.745	62.35	57	0.9993
	10.53	0.77	0.747	64.20	52	0.9991
Apple-3	11.35	0.84	0.754	55.15	51	0.9994
	13.75	0.84	0.767	47.47	65	0.9997
	11.74	0.84	0.751	56.65	53	0.9993

Remarks: Apple-1&Apple-2 are of same species and are taken from same lot. While Apple-3 is different from the other two.

| Show Table

DownLoad: CSV

Table 3. Summary of ANOVA on τ for apple.

Sample	Factor	P-value	F-value	F_critical	Distinguishability	Level of Confidence
Apple-1 & Apple-2	Position	0.119	3.627	15.977	No	99%
Apple-1 & Apple-2	Sample	0.594	0.334	21.197	No	99%
Apple-1 & Apple-3	Position	0.146	3.153	10.899	No	98%
Apple-1 & Apple-3	Sample	0.017	15.384	14.039	Yes	98%

| Show Table

DownLoad: CSV

Table 4. Cole parameters obtained from experiment and curve fitting for potato.

Sample	R₀ (kΩ)	R_∞ (kΩ)	α	C (nF sec^α−1)	τ (µsec)	Regression coefficient
	5.17	0.14	0.6400	341	48	0.9888
	5.48	0.16	0.7000	204	41	0.9956
Potato-1	5.58	0.13	0.6638	246	47	0.9920
	5.65	0.12	0.6505	325	60	0.9922
	5.09	0.11	0.6797	243	51	0.9951
	5.14	0.12	0.6275	445	60	0.9897
	5.43	0.13	0.6532	286	48	0.9918
Potato-2	6.66	0.16	0.6077	450	68	0.9890
	4.70	0.12	0.6415	363	47	0.9891
	5.67	0.12	0.5979	580	68	0.9865
	5.52	0.12	0.6360	420	70	0.9934
	5.57	0.15	0.6099	610	68	0.9906
Potato-3	6.70	0.14	0.6202	432	78	0.9912
	5.86	0.11	0.6362	393	69	0.9916
	5.46	0.13	0.6436	349	57	0.9924

Remarks: Potato -1 & Potato -2 are of same species and are taken from same lot, while Potato -3 is different from the other two.

| Show Table

DownLoad: CSV

Table 5. Summary of ANOVA on τ for potato.

Sample	Factor	P-value	F-value	F_critical	Distinguishability	Level of Confidence
Potato-1 & Potato-2	Position	0.6046	0.7537	15.977	No	99%
Potato-1 & Potato-2	Sample	0.2124	2.1975	21.198	No	99%
Potato-1& Potato-3	Position	0.6201	0.7218	10.899	No	98%
Potato-1& Potato-3	Sample	0.0182	14.856	14.040	Yes	98%

| Show Table

DownLoad: CSV

The various Cole parameters for potato samples are tabulated in Table 4. Similar to apple samples, ANOVA is performed on τ for potato samples and the result of the ANOVA is shown in Table 5. Based on the ANOVA, we have obtained similar conclusion as in apples. That is position of measurement is not a factor and different tissue of potato samples can be distinguished based on the relaxation time.

In order to gain confidence on our discrimination method described above, the entire experiment is repeated by taking another two specimens of each type using the same electrode-pair. When data is analyzed similar outcome is resulted. Thus, our method is found to be repetitive and robust.

5.2. Physiological difference between apple and potato

It is intuitive from the Figure 1(a), that the values of the Cole parameters are highly dependent on (a) cell size, (b) porosity, (c) constituents etc. In order to assess the cell size, extensive analysis is performed on the cellular images obtained using a confocal microscope (Olympus CX41). The average cell size of apple and potato (of typical species used in bioimpedance measurement) are estimated to be (10–12) × 10³ µm² and (18–20) × 10³ µm² respectively. The measured cell areas agree well with the similar measurement presented in published references [23],[24]. Thus, the cell size of potato is about 1.8 times larger than that of apple.

Figure 4. Images cellular structure of apple (top-left and top-right) and potato (bottom-left and bottom-right) obtained by confocal microscope (without any staining). The images are taken at the same scale for comparison purpose. Typical dc current paths are shown by dark detouring lines. The dc current paths are found to be longer in apple tissue in comparison to that of potato tissue.

DownLoad: Full-Size Img PowerPoint

On the other hand, the porosity of the apple and potato are in the range of 0.15–0.22 and 0.03–0.08 respectively [25]. Hence, apple tissue contains much more extracellular space than that of the potato. The estimated cell size and porosity in the tissues under investigation can be used to explain the capacitance obtained experimentally. The capacitance of potato tissue is found larger than that of apple tissue (see Table II and Table IV). In the cellular structure, two nearby cells and extracellular space between cells behave like the plates and dielectric material respectively of a parallel plate capacitor. As the capacitance of a parallel plate capacitor is proportional to the plate area and is inversely proportional to the separation between the plates (the extracellular space), the capacitance (C) is found larger in potato tissue as its cell size is larger and extracellular space (porosity) is smaller.

The dc resistance on the other hand depends on the length the dc current path. Longer the dc current path, larger is the dc resistance R0. Typical dc current paths are drawn manually (based on the principle that dc current flows through the extracellular space) and are shown in Figure 4. It is clear from this figure that the detouring of dc current path of apple tissue is greater than that of potato tissue and as a result apple shows larger R0 which is indeed found experimentally (see Table 2 and Table 4). Thus, the orientation of cells in the tissue highly affects the dc resistance. More the misorientation among the cells, longer is the dc current path and yields larger dc resistance. Another factor that may affect the dc resistance is the resistivity of the constituents of the extracellular space in tissues and such investigation is beyond the scope of this work. Therefore, the comparison of Cole parameters of the tissues of Apple and Potato from the biophysical point of view is tentative.

6. Conclusions

An experimental method is established here to distinguish different species of fruits or vegetable by means of measuring Cole parameters. Cole parameters are extracted for apple and potato tissue by specially designed electrode-pair using direct measurement technique. Among the various Cole parameters, the relaxation time τ is found to be most promising tissue characterizing parameter, since it contains all other Cole parameters of a given tissue. With statistical relevance, it is shown that tissues of different species of apple as well as potato can be distinguished by our experimental method. Our method can have wide application to distinguish unhealthy (or treated) fruits or vegetables from healthy one. The method is not costly and therefore this method has potential application in accessing plant tissue.

References

[1]	World Health Organization (WHO), WHO's COVID-19 response timeline, 2022. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/interactive-timeline/#category-Information.
[2]	World Health Organization (WHO), WHO Coronavirus (COVID-19) Dashboard, 2022. Available from: https://covid19.who.int/.
[3]	Y. Guo, Q. Cao, Z. Hong, Y. Tan, S. Chen, H. Jin, et al., The origin, transmission and clinical therapies on coronavirus disease 2019 (covid-19) outbreak–an update on the status, Mil. Med. Res., 7 (2020), 1–10. https://doi.org/10.1186/s40779-020-00240-0 doi: 10.1186/s40779-020-00240-0
[4]	R. Karia, I. Gupta, H. Khandait, A. Yadav, A. Yadav, Covid-19 and its modes of transmission, SN Compr. Clin. Med., 2 (2020), 1798–1801. https://doi.org/10.1007/s42399-020-00498-4 doi: 10.1007/s42399-020-00498-4
[5]	S. W. Ong, Y. K. Tan, P. Y. Chia, T. Lee, O. T. Ng, M. S. Wong, et al., Air, surface environmental, and personal protective equipment contamination by severe acute respiratory syndrome coronavirus 2 (sars-cov-2) from a symptomatic patient, Jama, 323 (2020), 1610–1612. https://doi.org/10.1001/jama.2020.3227 doi: 10.1001/jama.2020.3227
[6]	National Center for Immunization, Science brief: Sars-cov-2 and surface (fomite) transmission for indoor community environments, in CDC COVID-19 Science Briefs [Internet], Centers for Disease Control and Prevention (US), 2021.
[7]	World Health Organization (WHO), Scientific Brief: Transmission of SARS-CoV-2: implications for infection prevention precautions, (accessed August 3, 2021). Available from: https://www.who.int/news-room/commentaries/detail/transmission-of-sars-cov-2-implications-for-infection-prevention-precautions#: :text=Transmission.
[8]	S. Mallapaty, Why does the coronavirus spread so easily between people?, Nature, 579 (2020), 183–184.
[9]	Z. Abdelrahman, M. Li, X. Wang, Comparative review of sars-cov-2, sars-cov, mers-cov, and influenza a respiratory viruses, Front. Immunol., (2020), 2309. https://doi.org/10.3389/fimmu.2020.552909
[10]	World Health Organization (WHO), Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV), (accessed June 15, 2021). Available from: https://www.who.int/news/item/.
[11]	World Health Organization (WHO), WHO Director-General's opening remarks at the media briefing on COVID-19. 2020, (accessed June 15, 2021). Available from: https://www.who.int/director-general/speeches/detail/who-director-general-s-opening-remarks-at-the-media-briefing-on-covid-19—11-march-2020.
[12]	N. Lurie, M. Saville, R. Hatchett, J. Halton, Developing covid-19 vaccines at pandemic speed, N. Engl. J. Med., 382 (2020), 1969–1973. https://doi.org/10.1056/NEJMp2005630 doi: 10.1056/NEJMp2005630
[13]	G. Forni, A. Mantovani, Covid-19 vaccines: where we stand and challenges ahead, Cell Death Differ., 28 (2021), 626–639. https://doi.org/10.1038/s41418-020-00720-9 doi: 10.1038/s41418-020-00720-9
[14]	J. K. Jackson, Global economic effects of covid-19, in Technical report, Congressional Research Service, 2021. https://doi.org/10.1016/S2214-109X(21)00289-8
[15]	P. K. Ozili, T. Arun, Spillover of covid-19: impact on the global economy, SSRN Electron. J., 2020. https://doi.org/10.2139/ssrn.3562570
[16]	N. Fernandes, Economic effects of coronavirus outbreak (covid-19) on the world economy, in IESE Business School Working Paper, 2020. https://dx.doi.org/10.2139/ssrn.3557504
[17]	US FDA, Fda takes key action in fight against covid-19 by issuing emergency use authorization for first covid-19 vaccine, 2020. Available from: https://www.fda.gov/news-events/press-announcements/fda-takes-key-action-fight-against-covid-19-issuing-emergency-use-authorization-first-covid-19.
[18]	M. G. Thompson, J. L. Burgess, A. L. Naleway, H. L. Tyner, S. K. Yoon, J. Meece, et al., Interim estimates of vaccine effectiveness of bnt162b2 and mrna-1273 covid-19 vaccines in preventing sars-cov-2 infection among health care personnel, first responders, and other essential and frontline workers—eight us locations, december 2020–march 2021, Morb. Mortal. Wkly. Rep., 70 (2021), 495. https://doi.org/10.15585/mmwr.mm7013e3 doi: 10.15585/mmwr.mm7013e3
[19]	T. Pilishvili, K. E. Fleming-Dutra, J. L. Farrar, R. Gierke, N. M. Mohr, D. A. Talan, et al., Interim estimates of vaccine effectiveness of pfizer-biontech and moderna covid-19 vaccines among health care personnel—33 us sites, january–march 2021, Morb. Mortal. Wkly. Rep., 70 (2021), 753. https://doi.org/10.15585/mmwr.mm7020e2 doi: 10.15585/mmwr.mm7020e2
[20]	W. H. Self, M. W. Tenforde, J. P. Rhoads, M. Gaglani, A. A. Ginde, D. J Douin, et al., Comparative effectiveness of moderna, pfizer-biontech, and janssen (johnson & johnson) vaccines in preventing covid-19 hospitalizations among adults without immunocompromising conditions—united states, march–august 2021, Morb. Mortal. Wkly. Rep., 70 (2021), 1337. https://doi.org/10.15585/mmwr.mm7038e1 doi: 10.15585/mmwr.mm7038e1
[21]	L. Wang, G. Cheng, Sequence analysis of the emerging sars-cov-2 variant omicron in south africa, J. Med. Virol., 94 (2022), 1728–1733. https://doi.org/10.1002/jmv.27516 doi: 10.1002/jmv.27516
[22]	L. T. Brandal, E. MacDonald, L. Veneti, T. Ravlo, H. Lange, U. Naseer, et al., Outbreak caused by the sars-cov-2 omicron variant in norway, november to december 2021, Eurosurveillance, 26 (2021), 2101147. https://doi.org/10.2807/1560-7917.ES.2021.26.50.2101147 doi: 10.2807/1560-7917.ES.2021.26.50.2101147
[23]	R. Antia, M. E. Halloran, Transition to endemicity: Understanding covid-19, Immunity, 54 (2021), 2172–2176. https://doi.org/10.1016/j.immuni.2021.09.019 doi: 10.1016/j.immuni.2021.09.019
[24]	J. S. Lavine, O. N. Bjornstad, R. Antia, Immunological characteristics govern the transition of covid-19 to endemicity, Science, 371 (2021), 741–745. https://doi.org/10.1126/science.abe6522 doi: 10.1126/science.abe6522
[25]	Y. Tang, S. Wang, Mathematic modeling of covid-19 in the united states. Emerging Microbes Infect., 9 (2020), 827–829. https://doi.org/10.1080/22221751.2020.1760146
[26]	A. R. Tuite, D. N. Fisman, A. L. Greer, Mathematical modelling of covid-19 transmission and mitigation strategies in the population of ontario, canada, CMAJ, 192 (2020), E497–E505. https://doi.org/10.1503/cmaj.200476 doi: 10.1503/cmaj.200476
[27]	A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funk, et al., Early dynamics of transmission and control of covid-19: a mathematical modelling study, Lancet Infect. Dis., 20 (2020), 553–558. https://doi.org/10.1016/S1473-3099(20)30144-4 doi: 10.1016/S1473-3099(20)30144-4
[28]	S. A. Iyaniwura, R. C. Falcão, N. Ringa, P. A. Adu, M. Spencer, M. Taylor, et al., Mathematical modeling of covid-19 in british columbia: an age-structured model with time-dependent contact rates, Epidemics, (2022), 100559. https://doi.org/10.1016/j.epidem.2022.100559
[29]	F. Ndaïrou, I. Area, J. J. Nieto, D. F. M. Torres, Mathematical modeling of covid-19 transmission dynamics with a case study of wuhan, Chaos Solitons Fractals, 135 (2020), 109846. https://doi.org/10.1016/j.chaos.2020.109846 doi: 10.1016/j.chaos.2020.109846
[30]	D. K. Chu, E. A. Akl, S. Duda, K. Solo, S. Yaacoub, H. J. Schünemann, et al., Physical distancing, face masks, and eye protection to prevent person-to-person transmission of sars-cov-2 and covid-19: a systematic review and meta-analysis, Lancet, 395 (2020), 1973–1987. https://doi.org/10.1016/S0140-6736(20)31142-9 doi: 10.1016/S0140-6736(20)31142-9
[31]	M. A. Khan, Ab. Atangana, E. Alzahrani, Fatmawati, The dynamics of covid-19 with quarantined and isolation, Adv. Differ. Equations, 2020 (2020), 1–22. https://doi.org/10.1186/s13662-020-02882-9 doi: 10.1186/s13662-020-02882-9
[32]	S. C. Anderson, A. M. Edwards, M. Yerlanov, N. Mulberry, J. E. Stockdale, S. A. Iyaniwura, et al., Quantifying the impact of covid-19 control measures using a bayesian model of physical distancing, PLoS Comput. Biol., 16 (2020), e1008274. https://doi.org/10.1371/journal.pcbi.1008274 doi: 10.1371/journal.pcbi.1008274
[33]	S. A. Iyaniwura, M. A. Rabiu, J. David, J. D. Kong, Assessing the impact of adherence to non-pharmaceutical interventions and indirect transmission on the dynamics of covid-19: a mathematical modelling study, medRxiv, 2021. https://doi.org/10.1101/2021.10.23.21265421
[34]	M. Sadarangani, B. A. Raya, J. M. Conway, S. A. Iyaniwura, R. C. Falcao, C. Colijn, et al., Importance of covid-19 vaccine efficacy in older age groups, Vaccine, 39 (2021), 2020–2023. https://doi.org/10.1016/j.vaccine.2021.03.020 doi: 10.1016/j.vaccine.2021.03.020
[35]	N. Mulberry, P. Tupper, E. Kirwin, C. McCabe, C. Colijn, Vaccine rollout strategies: The case for vaccinating essential workers early, PLOS Global Public Health, 1 (2021), e0000020. https://doi.org/10.1371/journal.pgph.0000020 doi: 10.1371/journal.pgph.0000020
[36]	J. Demongeot, Q. Griette, P. Magal, G. Webb, Modeling vaccine efficacy for covid-19 outbreak in new york city, Biology, 11 (2022), 345. https://doi.org/10.3390/biology11030345 doi: 10.3390/biology11030345
[37]	L. M. Pinto, V. Nanda, A. Sunavala, C. Rodriques, Reinfection in covid-19: A scoping review, Med. J. Armed Forces India, 77, S257–S263. https://doi.org/10.1016/j.mjafi.2021.02.010
[38]	S, Roy, Covid-19 reinfection: myth or truth?, SN Compr. Clin. Med., 2 (2020), 710–713. https://doi.org/10.1007/s42399-020-00335-8 doi: 10.1007/s42399-020-00335-8
[39]	D. de A. Torres, L. Ribeiro, A. de F. Riello, D. D. G. Horovitz, L. F. R. Pinto, J. Croda, Reinfection of covid-19 after 3 months with a distinct and more aggressive clinical presentation: Case report, J. Med. Virol., 2020. https://doi.org/10.1002/jmv.26637
[40]	CDC Covid, Vaccine Breakthrough Case Investigations Team, CDC COVID, Vaccine Breakthrough Case Investigations Team, CDC COVID, Vaccine Breakthrough Case Investigations Team, et al., Covid-19 vaccine breakthrough infections reported to cdc—united states, january 1–april 30, 2021, Morb. Mortal. Wkly. Rep., 70 (2021), 792. http://dx.doi.org/10.15585/mmwr.mm7021e3.
[41]	M. Bergwerk, T. Gonen, Y. Lustig, S. Amit, M. Lipsitch, C. Cohen, et al., Covid-19 breakthrough infections in vaccinated health care workers, N. Engl. J. Med., 385 (2021), 1474–1484. http://doi.org/10.1056/NEJMoa2109072 doi: 10.1056/NEJMoa2109072
[42]	R. K. Gupta, E. J. Topol, Covid-19 vaccine breakthrough infections, Science, 374 (2021), 1561–1562. http://doi.org/10.1126/science.abl8487 doi: 10.1126/science.abl8487
[43]	Y. Goldberg, M. Mandel, Y. M. Bar-On, O. Bodenheimer, L. Freedman, E. J. Haas, et al., Waning immunity after the bnt162b2 vaccine in israel, N. Engl. J. Med., 385 (2021), e85. http://doi.org/10.1056/NEJMoa2114228 doi: 10.1056/NEJMoa2114228
[44]	Elie Dolgin, Covid vaccine immunity is waning-how much does that matter, Nature, 597 (2021), 606–607. https://doi.org/10.1038/d41586-021-02532-4 doi: 10.1038/d41586-021-02532-4
[45]	E. G. Levin, Y. Lustig, C. Cohen, R. Fluss, V. Indenbaum, S. Amit, et al., Waning immune humoral response to bnt162b2 covid-19 vaccine over 6 months, N. Engl. J. Med., 385 (2021), e84. https://doi.org/10.1056/NEJMoa2114583 doi: 10.1056/NEJMoa2114583
[46]	E. B. Are, Y. Song, J. E. Stockdale, P. Tupper, C. Colijn, Covid-19 endgame: from pandemic to endemic? vaccination, reopening and evolution in a well-vaccinated population, medRxiv, 2021. https://doi.org/10.1101/2021.12.18.21268002
[47]	A. V. Tkachenko, S. Maslov, T. Wang, A. Elbana, G. N. Wong, N. Goldenfeld, Stochastic social behavior coupled to covid-19 dynamics leads to waves, plateaus, and an endemic state, Elife, 10 (2021), e68341. https://doi.org/10.7554/eLife.68341 doi: 10.7554/eLife.68341
[48]	M. Rabiu, S. A Iyaniwura, Assessing the potential impact of immunity waning on the dynamics of covid-19 in south africa: an endemic model of covid-19, Nonlinear Dyn., (2022), 1–21. https://doi.org/10.1007/s11071-022-07225-9
[49]	N. Nuraini, K. Khairudin, M. Apri, Modeling simulation of covid-19 in indonesia based on early endemic data, Commun. Biomath. Sci., 3 (2020), 1–8.
[50]	W. Zhou, B. Tang, Y. Bai, Y. Shao, Y. Xiao, S. Tang, The resurgence risk of covid-19 in the presence of immunity waning and ade effect: A mathematical modelling study, medRxiv, 2021. https://doi.org/10.1101/2021.08.25.21262601
[51]	F. Inayaturohmat, R. N. Zikkah, A. K. Supriatna, N. Anggriani, Mathematical model of covid-19 transmission in the presence of waning immunity, J. Phys. Conf. Ser., 1722 (2021), 012038. https://doi.org/10.1088/1742-6596/1722/1/012038 doi: 10.1088/1742-6596/1722/1/012038
[52]	T. Xiang, B. Liang, Y. Fang, S. Lu, S. Li, H. Wang, et al., Declining levels of neutralizing antibodies against sars-cov-2 in convalescent covid-19 patients one year post symptom onset, Front. Immunol., 12 (2021), 2327. https://doi.org/10.3389/fimmu.2021.708523 doi: 10.3389/fimmu.2021.708523
[53]	B. Pell, M. D. Johnston, P. Nelson, A data-validated temporary immunity model of covid-19 spread in michigan, Math. Biosci. Eng., 19 (2022), 10122–10142. https://doi.org/10.3934/mbe.2022474 doi: 10.3934/mbe.2022474
[54]	S. Ghosh, M. Banerjee, V. Volpert, Immuno-epidemiological model-based prediction of further covid-19 epidemic outbreaks due to immunity waning, Math. Modell. Nat. Phenom., 17 (2022), 9. https://doi.org/10.1051/mmnp/2022017 doi: 10.1051/mmnp/2022017
[55]	C. P. Tadiri, J. D. Kong, G. F. Fussmann, M. E. Scott, H. Wang, A data-validated host-parasite model for infectious disease outbreaks, Front. Ecol. Evol., (2019), 307. https://doi.org/10.3389/fevo.2019.00307
[56]	Y. Kuang, Delay differential equations: with applications in population dynamics, Academic press, 1993.
[57]	T. Cassidy, Distributed delay differential equation representations of cyclic differential equations, SIAM J. Appl. Math., 81 (2021), 1742–1766. https://doi.org/10.1137/20M1351606 doi: 10.1137/20M1351606
[58]	P. J. Hurtado, A. S. Kirosingh, Generalizations of the 'linear chain trick': incorporating more flexible dwell time distributions into mean field ode models, J. Math. Biol., 79, 1831–1883. https://doi.org/10.1007/s00285-019-01412-w
[59]	H. L. Smith, An introduction to delay differential equations with applications to the life sciences, Springer New York, 2011.
[60]	O. Diekmann, J. A. P. Heesterbeek, J. A. Metz, On the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[61]	P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[62]	J. H. Jones, Notes on r0, Calif. Dep. Anthropol. Sci., 323 (2007), 1–19.
[63]	C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361
[64]	M. Rabiu, R. Willie, N. Parumasur, Mathematical analysis of a disease-resistant model with imperfect vaccine, quarantine and treatment, Ric. Mat., 69 (2020), 603–627. https://doi.org/10.1007/s11587-020-00496-7 doi: 10.1007/s11587-020-00496-7
[65]	O. Dyer, Covid-19: Unvaccinated face 11 times risk of death from delta variant, cdc data show, BMJ, 374 (2021). http://dx.doi.org/10.1136/bmj.n2282
[66]	A. Khan, R. Zarin, G. Hussain, N. A. Ahmad, M. H. Mohd, A. Yusuf, Stability analysis and optimal control of covid-19 with convex incidence rate in khyber pakhtunkhawa (pakistan), Results Phys., 20 (2021), 103703. https://doi.org/10.1016/j.rinp.2020.103703 doi: 10.1016/j.rinp.2020.103703
[67]	J. P. La Salle, The stability of dynamical systems, SIAM, 1976.
[68]	U. C. Bureau, Census Bureau Reveals Fastest-Growing Large Cities, (accessed May 27, 2022). Available from: https://www.census.gov/newsroom/press-releases/2018/estimates-cities.html.
[69]	S. Mushayabasa, E. T. Ngarakana-Gwasira, J. Mushanyu, On the role of governmental action and individual reaction on covid-19 dynamics in south africa: A mathematical modelling study, Inf. Med. Unlocked, 20 (2020), 100387. https://doi.org/10.1016/j.imu.2020.100387 doi: 10.1016/j.imu.2020.100387
[70]	Z. Yang, Z. Zeng, K. Wang, S. Wong, W. Liang, M. Zanin, et al., Modified seir and ai prediction of the epidemics trend of covid-19 in china under public health interventions, J. Thorac. Dis., 12 (2020), 165. https://doi.org/10.21037/jtd.2020.02.64 doi: 10.21037/jtd.2020.02.64
[71]	A. Mahajan, R. Solanki, N. Sivadas, Estimation of undetected symptomatic and asymptomatic cases of covid-19 infection and prediction of its spread in the usa, J. Med. Virol., 93 (2021), 3202–3210. https://doi.org/10.1002/jmv.26897 doi: 10.1002/jmv.26897
[72]	S. Choi, M. Ki, Estimating the reproductive number and the outbreak size of covid-19 in korea, Epidemiol. Health, 42, (2020). https://doi.org/10.4178/epih.e2020011
[73]	S. Khajanchi, K. Sarkar, J. Mondal, Dynamics of the covid-19 pandemic in india, preprint, arXiv: 2005.06286.
[74]	L. Zou, F. Ruan, M. Huang, L. Liang, H. Huang, Z. Hong, et al., Sars-cov-2 viral load in upper respiratory specimens of infected patients, N. Engl. J. Med., 382 (2020), 1177–1179. https://doi.org/10.1056/NEJMc2001737 doi: 10.1056/NEJMc2001737
[75]	L. Tindale, M. Coombe, J. E. Stockdale, E. Garlock, W. Y. V. Lau, M. Saraswat, et al., Transmission interval estimates suggest pre-symptomatic spread of covid-19, MedRxiv, 2020. https://doi.org/10.1101/2020.03.03.20029983
[76]	T. Ganyani, C. Kremer, D. Chen, A. Torneri, C. Faes, J. Wallinga, et al., Estimating the generation interval for coronavirus disease (covid-19) based on symptom onset data, march 2020, Eurosurveillance, 25 (2020), 2000257. https://doi.org/10.2807/1560-7917.ES.2020.25.17.2000257 doi: 10.2807/1560-7917.ES.2020.25.17.2000257
[77]	Z. Zhao, X. Li, F. Liu, G. Zhu, C. Ma, L. Wang, Prediction of the covid-19 spread in african countries and implications for prevention and control: A case study in south africa, egypt, algeria, nigeria, senegal and kenya, Sci. Total Environ., 729 (2020), 138959. https://doi.org/10.1016/j.scitotenv.2020.138959 doi: 10.1016/j.scitotenv.2020.138959
[78]	L. Peng, W. Yang, D. Zhang, C. Zhuge, L. Hong, Epidemic analysis of covid-19 in china by dynamical modeling, preprint, arXiv: 2002.06563.
[79]	W. Jassat, C. Mudara, L. Ozougwu, S. Tempia, L. Blumberg, M. Davies, et al., Difference in mortality among individuals admitted to hospital with covid-19 during the first and second waves in south africa: a cohort study, Lancet Global Health, 2021. https://doi.org/10.1016/S2214-109X(21)00289-8
[80]	Webometer, Total Coronavirus Deaths in South Africa, (accessed August 28, 2021). Available from: https://www.worldometers.info/coronavirus/country/south-africa.
[81]	Organisation for Economic Co-operation and Development (OECD) Data, Urban population by city size, (accessed May 27, 2022). Available from: https://data.oecd.org/popregion/urban-population-by-city-size.htm.

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Mathematical Biosciences and Engineering

A generalized distributed delay model of COVID-19: An endemic model with immunity waning

Related Papers:

Abstract

1. Introduction

2. Single dispersion Cole bioimpedance model

3. Specially designed electrode-pair

4. Experiments

5. Results and discussions

5.1. ANOVA on apple and potato samples

5.2. Physiological difference between apple and potato

6. Conclusions

References

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Abstract

1. Introduction

2. Single dispersion Cole bioimpedance model

3. Specially designed electrode-pair

4. Experiments

5. Results and discussions

5.1. ANOVA on apple and potato samples

5.2. Physiological difference between apple and potato

6. Conclusions

References

Mathematical Biosciences and Engineering

A generalized distributed delay model of COVID-19: An endemic model with immunity waning

Related Papers:

Abstract

1. Introduction

2. Single dispersion Cole bioimpedance model

3. Specially designed electrode-pair

4. Experiments

5. Results and discussions

5.1. ANOVA on apple and potato samples

5.2. Physiological difference between apple and potato

6. Conclusions

References

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Abstract

1. Introduction

2. Single dispersion Cole bioimpedance model

3. Specially designed electrode-pair

4. Experiments

5. Results and discussions

5.1. ANOVA on apple and potato samples

5.2. Physiological difference between apple and potato

6. Conclusions

References