**5. Conclusions**

Overall, data analysis without adjustment resulted in substantial agreement between the self-collected and clinic-collected samples, with PPA values of 83.7% and 90.7% for vaginal and urine samples, respectively. With correction for excess dilution in the sample preparation, there was almost perfect agreement, with PPA values of 100% and 97.7% for vaginal and urine samples, respectively. In conclusion, data from this pilot study are promising for the employment of the BD Onclarity™ HPV assay on vaginal samples and first-void urine samples, with an accuracy almost equivalent to clinician-collected cervical samples.

**Author Contributions:** Conceptualization, C.E.C.; methodology, M.M., I.S., F.P., N.M. and F.B.; formal analysis, M.M., F.B. and C.G.; data curation, M.M., C.G., R.M., N.M. and C.E.C.; writing—original draft preparation, M.M., C.G., A.D.I., F.B. and C.E.C.; writing—review and editing, M.M., C.G., A.D.I., F.B., R.M., R.F., F.L., N.M., A.P., I.S. and C.E.C.; supervision, C.E.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. BD Onclarity™ HPV assays were donated by BD; FLOQSwab® L-shape and vaginal self-samples were donated by Copan; Colli-pee® devices were donated by Novosanis.

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki and approved by the Ethics Committee of the University of Milano-Bicocca (Protocol n. 0037320/2017 and 0086409/2018).

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Conflicts of Interest:** The authors declare no conflict of interest.


*2.3. Gene Structure and Conserved Motifs of NnDofs*

We investigated the conserved motifs of the genes using MEME to evaluate the diversity and conservation among all 29 *NnDofs* genes. Ten different motifs were predicted and they ranged in size from 16 to 134 amino acids (Figure 3B). The conserved structural domain of Dof consists of Motif 1, and Motif 1 corresponds to the CX2CX21CX2C single zinc finger structure in the Dof structural domain, which is a highly homologous core region of the *Dof* family. NnDofs proteins associated on evolutionary branches of the phylogenetic tree have identical or comparable motif structures. Some motifs, such as Motif 2, Motif 3, Motif 5, and Motif 8, are only found in certain subgroups and may be connected to different functions.

Introns and exons constitute genes, and their numbers and distribution patterns serve as an evolutionary indicator for a gene family. Thus, we completed a comparison of the intron-exon structures of each *NnDofs* (Figure 3C). In general, the number of introns per *NnDof* gene varied very little, ranging from 0 to 1. A total of 11 *NnDof* gene members had no introns (37.9%) and 18 members had only one intron (62.1%). The exon-intron structure pattern of the *NnDofs* gene is similar to that of Arabidopsis, rice, and tomato. Most introns are usually located upstream of the *Dof* structural domain, and only seven genes (*NnDof2*, *NnDof3*, *NnDof8*, *NnDof16*, *NnDof20*, *NnDof26*, and *NnDof28*) have introns located downstream of the *Dof* structural domain. Most of the genes in the same groups showed similar exon and intron patterns. For example, most *Dof* genes in group III do not contain introns. In contrast, groups II, V, VI, and VII almost all contain one intron.

**Figure 1.** Phylogenetic tree of Dof transcription factors in *N. nucifera* and *A. thaliana*. The phylogenetic tree was created using MEGA 6.0, with the neighbor-joining method and bootstrap value set at 1000. Black circles represent the 36 AtDof proteins, and the 29 NnDof proteins are marked with black squares. The resulting phylogenetic tree was clustered into seven main groups, labeled I, II, III, IV, V, VI, and VII.


4.2 Extraction procedure

Like all the information hiding papers, only the receiver can extract the message. But it is worth pointing out the receiver only uses the stego image to extract the secret text in our scheme that means it is a blind scheme. The extracting procedure can be described as follows.

#### Advances in Quantum Communication and Information

Figure 10. Embedding circuit: (a) specific embedding block circuit and (b) the whole circuit.

#### 4.2.1 Extraction

As can be seen from the above, the eight bit-planes of the secret text can be extracted from the eight blocks of the stego image. Accordingly, we can consider one block as an example. If the position of the stego image and the auxiliary blank quantum text are equal, and GRAY S7 YXS6 YXS5 YX � � is even and C0 YX ¼ 1 or GRAY S7 YXS6 YXS5 YX � � is odd and C0 YX ¼ 0, then T^i � � � E ¼ j i1 . The corresponding circuit is shown in Figure 11a, and a whole extracting circuit that combines the extraction of all eight blocks together is exposed in Figure 11b.

Figure 11.

Figure 12. Inverse transform.

23

Extracting circuit: (a) specific extracting block circuit and (b) the whole circuit.

A Novel Quantum Steganography Scheme Based on ASCII

DOI: http://dx.doi.org/10.5772/intechopen.86413

#### 4.2.2 Descrambling

Due to the operators used in embedding process are unitary, for this step of extracting process, we can use the inverse transpose of operators used in the embedding process. Figure 12 affords the quantum circuit to extract the secret image j i T from T^ � � � .

A Novel Quantum Steganography Scheme Based on ASCII DOI: http://dx.doi.org/10.5772/intechopen.86413

Figure 12. Inverse transform.

4.2.1 Extraction

Figure 10.

GRAY S7

YXS6 YXS5 YX � � is odd and C0

4.2.2 Descrambling

image j i T from T^

22

� � � .

quantum text are equal, and GRAY S7

of all eight blocks together is exposed in Figure 11b.

Embedding circuit: (a) specific embedding block circuit and (b) the whole circuit.

Advances in Quantum Communication and Information

As can be seen from the above, the eight bit-planes of the secret text can be extracted from the eight blocks of the stego image. Accordingly, we can consider one block as an example. If the position of the stego image and the auxiliary blank

> YXS6 YXS5 YX � � is even and C0

YX ¼ 0, then T^i �

is shown in Figure 11a, and a whole extracting circuit that combines the extraction

Due to the operators used in embedding process are unitary, for this step of extracting process, we can use the inverse transpose of operators used in the embedding process. Figure 12 affords the quantum circuit to extract the secret

� � E YX ¼ 1 or

¼ j i1 . The corresponding circuit


**3. Results and Discussion**

The orientation control system was implemented in the heliostat shown in Figure 18, whereas the printed circuit board (PCB) of the embedded system and the motor driver are shown in Figure 19. It is an azimuth–elevation mechanism heliostat, with a worm drive mechanism driven by a DC gear motor model ZYT6590-01 at each axis. The heliostat has a gap which allows directing the facets to the ground. The parameters of the heliostat and the DC motors are presented in Table 3.

**Figure 18.** Heliostat.

**Figure 19.** Printed circuit board of the embedded system (**a**) and the motor driver (**b**).

**Table 3.** Parameters of the heliostat and DC motors.

As mentioned already, the control algorithms were designed for the position control of a DC motor at no load. Afterwards, the control algorithms were implemented in the orientation control of the heliostat using the same controller parameters of the position control of the DC motor at no load.

Figure 20 shows the comparison of the consumption time of the PID controller (Figure 20a) and the FLC using the CoS (Figure 20b) and the CoG (Figure 20c) defuzzification methods, where the period of the signals represents the sampling time of the control algorithms. The results show that the FLC with the CoG defuzzification method does not accomplish with the design parameters because of its computational complexity.

The output response of the control algorithms for the position control of a DC motor at no load is shown in Figure 21 for the minimum change in the reference value of 0.087 degrees (1.533 mrad) and a reference value of 180 degrees (π rad). Both control algorithms accomplish with the design parameters for the position control of the DC motor at no load. However, Figure 21d shows that the FLC control signal decreases when the position of the DC motor is reaching the reference value.

**Figure 20.** Consumption time of the PID controller (**a**), and the fuzzy logic controller with the CoS (**b**) and the CoG (**c**) defuzzification methods.

**Figure 21.** Output response (**a**) and control signal (**c**) of the DC motor at no load at a minimum reference value. Output response (**b**) and control signal (**d**) of the DC motor at no load at a reference value of 180 degrees (π rad).

Finally, Figures 22 and 23 show the output response of the control algorithms for the orientation control for the DC motors at no load and the axes of the heliostat, respectively. The desired angles of the heliostat were calculated using the parameters of Table 4, whereas the error values are shown in Table 5.

**Figure 22.** Output response of the orientation control of the DC motors at no load for the azimuth axis (**a**) and the elevation axis (**b**).

**Figure 23.** Output response of the orientation control for the azimuth axis (**a**) and the elevation axis (**b**) of the heliostat. Desired reference error values of the orientation control of the heliostat for the fuzzy logic controller (**c**) and the PID controller (**d**). Final reference error values of the orientation control of the heliostat for the fuzzy logic controller (**e**) and the PID controller (**f**).

**Table 4.** Parameters of the orientation control test.

**Table 5.** Reference error values of the orientation control.

The experimental results show a similar Mean Squared Error (MSE) for the orientation control of the DC motors at no load and a similar output response between the orientation control of the heliostat and the final reference value for the FLC (Figure 23a) and the PID controller (Figure 23b), despite the load of the wind over the mechanical structure and the backlash in the axis mechanisms. However, for the orientation control of the heliostat, the FLC shows less dispersed error values (Figure 23c) and smaller final reference error values (Figure 23e) than the PID controller (Figure 23d,f).


**1. Introduction**

Mathematical modeling of infectious diseases attracts much attention due to successive epidemics, such as HIV, emerging in the 1980s and still continuing [1,2], SARS epidemic in 2002–2003 [3,4], H5N1 influenza in 2005 [5,6] and H1N1 in 2009 [7,8], Ebola in 2014 [9,10]. The ongoing COVID-19 pandemic has stimulated unprecedented efforts of mathematical modeling in epidemiology. A wide variety of mathematical approaches are developed to study epidemiological problems. However, sufficiently simple and validated models still remain in the focus of mathematical modeling in epidemiology.

Modern studies in mathematical epidemiology begin with the SIR model developed in the works by W. O. Kermack and A. G. McKendrick [11–13], stimulated by the Spanish flu epidemic in 1918–1919. Among many developments of such models, we can cite multi-compartment models [14–16], models with a time-varying or nonlinear disease transmission rate [17,18], multi-patch models [19–21], multi-group models incorporating the effect of the heterogeneity of the population [22], and epidemic models with vaccination and other control measures [23,24]. Random movement of individuals in the population is considered in spatiotemporal models in order to describe spatial distributions of susceptible and infected individuals [25,26]. A more detailed literature review can be found in the monographs [27,28] and review articles [29,30].

The conventional SIR model, which includes susceptible (*S*), infected (*I*), and recovered (*R*) compartments, and similar models assume that recovery and death rates at time *t* are proportional to the number of actively infected individuals *I*(*t*) at the same moment of

**Citation:** Ghosh, S.; Volpert, V.; Banerjee, M. An Epidemic Model with Time Delay Determined by the Disease Duration. *Mathematics* **2022**, *10*, 2561. https://doi.org/10.3390/ math10152561

Academic Editor: Alexandra Kashchenko

Received: 9 June 2022 Accepted: 15 July 2022 Published: 22 July 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

time. This assumption does not take into account disease duration, and it can lead to a large error. In our previous work [31], we showed that this assumption leads to an overestimation of actual recoveries and deaths. Instead, if we use distributed recovery and death rates, properly chosen from real data, the description of the epidemic progression becomes more precise. However, since distributed recovery and death rates are not easily available, we develop a simpler delay model in this work. It gives close results, but it does not require precise immunological data. The model considered in [31] involves distributed recovery and death rates. The model considered in [32] is an extension of the model by incorporating the vaccinated compartment, and the resulting model is an immuno-epidemic model. The delay model, with disease duration delay, considered here is derived from the model proposed in [31] with an appropriate assumption on the recovery and death rate functions. The present model is quite different from the delay model considered in [33] as the delay parameter involved with the earlier model was the measure of the incubation period and the departure of infected individuals from the infected compartment was due to the imposition of quarantine measure. The model proposed and analyzed here is solely dependent on the disease duration period and without imposed quarantine.

Most of the existing delay epidemic models consider time delay either in the disease incidence function or in the susceptible recruitment function (Appendix A). The delay in the recovery and death rates has not been studied yet thoroughly. In this work, we introduce time delay in recovery and death rates with the average disease duration considered as the delay parameter.

In Section 2, we discuss the distributed model and derive the delay model where the discrete time delay estimates average disease duration. We obtain epidemic characterization of the delay model in Section 3. Then, in Section 4, we perform a numerical comparison among the distributed model, delay model, and conventional SIR model with the equivalent parameter values. Next, we discuss a method to estimate the value of disease duration using the real data of disease incidence in Section 5. In Section 6, we validate our delay model with epidemiological data collected during the COVID-19 epidemic. The main outcomes of the proposed model and its epidemiological implications are discussed in the concluding section.