# A Note on an Epidemic Model with Cautionary Response in the Presence of Asymptomatic Individuals

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The SAI Model without Vertical Transmission

#### Model Equations

#### 2.2. Boundedness

#### 2.3. Model Equilibria

#### 2.3.1. Endemic Coexistence

**The first surface ${\Sigma}^{\left(1\right)}$**

- If $h\ge {I}_{0}$, it follows that ${S}_{0}^{h}\le 0$; then, ${A}_{h}\left(S\right)>0$ if and only if $0<S<{S}_{\infty}^{h}$. The conic is positive and increasing only from the origin to the vertical asymptote.
- If $h<{I}_{0}$, the conic crosses the point $\left({S}_{0}^{h},0\right)$. In such case, we have the following three possibilities.
- -
- If ${S}_{0}^{h}<{S}_{\infty}^{h}$, then ${A}_{h}\left(S\right)>0$ if and only if ${S}_{0}^{h}<S<{S}_{\infty}^{h}$. The conic is a hyperbola which is positive and increasing in the ($I=h$) S-A plane only from the zero to the asymptote.
- -
- If ${S}_{\infty}^{h}<{S}_{0}^{h}$, then ${A}_{h}\left(S\right)>0$ if and only if ${S}_{\infty}^{h}<S<{S}_{0}^{h}$. The conic is a hyperbola which is positive and decreasing only from the asymptote to $\left({S}_{0}^{h},0\right)$.
- -
- Finally, if ${S}_{0}^{h}={S}_{\infty}^{h}$, then the conic is degenerate because (6) fails to hold, having, in this case,$$r=\frac{(m+{c}_{SI}h)({c}_{SA}+\tilde{\alpha})}{{c}_{SA}+\tilde{\alpha}-{c}_{SS}}\phantom{\rule{0.166667em}{0ex}}.$$

**The second surface ${\Theta}^{\left(1\right)}$**

**The third surface $\mathsf{\Gamma}$**

**The possible intersections of ${\Sigma}^{\left(1\right)}$, ${\Theta}^{\left(1\right)}$ and $\mathsf{\Gamma}$**

#### 2.4. Equilibria Stability

#### Local Stability

#### 2.5. SAI Model with Vertical Transmission

#### 2.6. Preliminary Analysis

#### 2.6.1. Equilibria

#### 2.6.2. The Endemic Coexistence Equilibrium

**The first surface ${\Sigma}^{\left(2\right)}$**

**The second surface ${\Theta}^{\left(2\right)}$**

**The possible intersections of ${\Sigma}^{\left(2\right)}$, ${\Theta}^{\left(2\right)}$ and $\mathsf{\Gamma}$**

#### 2.6.3. Local Stability

## 3. Results

## 4. Global Behavior of the Systems

#### 4.1. Application of Sotomayor’s Theorem for Model (1)

#### 4.1.1. Transcritical Bifurcation ${E}_{0}\to {E}_{S}$

#### 4.1.2. Transcritical Bifurcation ${E}_{S}\to {E}_{SAI}$

**Remark 1.**

#### 4.2. Application of Sotomayor’s Theorem for Model (23)

#### 4.2.1. Transcritical Bifurcation ${E}_{0}\to {E}_{S}$

#### 4.2.2. Transcritical Bifurcation ${E}_{S}\to {E}_{SAI}$

**Remark 2.**

#### 4.3. Numerical Simulations for the Bifurcations

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**(

**Left**): transcritical bifurcations for model (1) obtained with parameter values (72), (55) and initial conditions (50). (

**Right**): zoom of the left image showing two bifurcations as r changes; the first from ${E}_{0}$ to ${E}_{S}$ when $r={r}_{0}=2$ and the second from ${E}_{S}$ to ${E}_{SAI}$ when $r={r}^{\u2020}=2.4898$.

**Figure 9.**(

**Left**): transcritical bifurcations for model (23) obtained with parameter values (72), (55) and initial conditions (50). (

**Right**): zoom of the left image showing two bifurcations as r changes; the first one from ${E}_{0}$ to ${E}_{S}$ when $r={r}_{0}=2$ and the second from ${E}_{S}$ to ${E}_{SAI}$ when $r={r}^{\u2021}=2.3019$.

**Figure 10.**Here, the disease transmission rate is fixed and lower than the progression to the symptomatic/removed classes, with $\alpha =0.5<\pi =\gamma \in [1,\dots ,10]$. Comparison between (1) (or equivalently (23)) on the left column, and the Capasso–Serio model (73) on the right column, in terms of the progression from asymptomatic to symptomatic $\pi $ for (1) (as well as (23)) and of the removal rate $\gamma $ for (73), with parameter values (74) and initial conditions (50). Left frame: the simulations to show the settling of the systems. Right frame: the blow up of the initial instants to better show the transients.

**Figure 11.**Here, $\pi =\gamma =5$. Comparison between (1) (or, equivalently, (23)) on the left column, and the Capasso–Serio model (73) on the right column, in terms of the disease transmission rate $\alpha \in [0.1,\cdots ,1.0]$ so that, again, it is below the progression to symptomatic or removal rates, $\alpha <\pi =\gamma $ with parameter values (74) and initial conditions (50). Left frame: the simulations to show the settling of the systems. Right frame: the blow up of the initial instants to better show the transients.

**Figure 12.**Here, we illustrate the case $0=\pi <\alpha \in [0.1\dots ,1.0]$, $\gamma =0.05$. In this case, in the SAI model, the whole population is quickly affected, while for (73), the disease propagates at a lower speed with a lower transmission rate. In the long run, here as well, the susceptible class is depleted. Left frame: the simulations to show the settling of the systems. Right frame: the time interval is shorter to better show the transients in the SAI model. Other parameter values are (74) and initial conditions (50).

**Figure 13.**Here, $\pi =\gamma =0$, so that both systems become SI models. Comparison between (1) (or, equivalently, (23)) on the left column, and the Capasso–Serio model (73), on the right column. Left frame: $\alpha \in [10,\dots ,100]$. Right frame: $\alpha \in [0.1,\dots ,1.0]$. The other parameter values are given in (74) and initial conditions (50).

**Figure 14.**Here, $\pi =5$, $\gamma =0$ and $\alpha \in [1.0,10]$. It is clearly seen that for $\alpha <5$ in the SAI model the susceptibles are preserved, while for transmission rates higher than this threshold, $\alpha >\pi =5$, all individuals eventually become symptomatic. The same occurs in the (73) model independently of $\alpha $, at a lower pace. Other parameter values are (74) and initial conditions (50).

Parameters | Interpretation | Dimensions |
---|---|---|

${c}_{XY}$, $X,Y\in \{S,A,I\}$ | competition pressure of Y on X | $\frac{1}{\left[t\right]}$ |

m | natural mortality rate | $\frac{1}{\left[t\right]}$ |

r | natural birth rate | $\frac{1}{\left[t\right]}$ |

Parameters | Interpretation | Dimensions |
---|---|---|

$\alpha $ | transmission rate | $\frac{1}{\left[t\right]}$ |

$\beta $ | inhibitory effect coefficient | - |

$\mu $ | disease-related mortality rate | $\frac{1}{\left[t\right]}$ |

$\pi $ | progression rate from asymptomatics to symptomatics | $\frac{1}{\left[t\right]}$ |

**Table 3.**Summary of equilibria and local stability for model (1).

Equilibria | Existence Conditions | Stability |
---|---|---|

${E}_{0}=(0,0,0)$ | - | stable if $r<m$ |

${E}_{S}=\left({\displaystyle \frac{r-m}{{c}_{SS}}},0,0\right)$ | $r>m$ | stable if (22) |

${E}_{SAI}=({S}^{*},{A}^{*},{I}^{*})$ | sufficient: (20) | numerical simulations |

**Table 4.**Summary of equilibria and their feasibility for model (23).

Equilibria | Existence Conditions |
---|---|

${E}_{0}=(0,0,0)$ | - |

${E}_{S}=\left({\displaystyle \frac{r-m}{{c}_{SS}}},0,0\right)$ | $r>m$ |

${E}_{AI}^{\pm}=\left(0,{\displaystyle \frac{r-m-{c}_{AI}{I}_{\pm}-\pi}{{c}_{AA}}},{I}_{\pm}\right)$ | (27), (28) |

${E}_{SAI}=({S}_{*},{A}_{*},{I}_{*})$ | Sufficient: (38), (40) and either one of (44) or (45) |

**Table 5.**Summary of equilibria and their local stability for model (23).

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**MDPI and ACS Style**

Acotto, F.; Venturino, E.
A Note on an Epidemic Model with Cautionary Response in the Presence of Asymptomatic Individuals. *Axioms* **2023**, *12*, 62.
https://doi.org/10.3390/axioms12010062

**AMA Style**

Acotto F, Venturino E.
A Note on an Epidemic Model with Cautionary Response in the Presence of Asymptomatic Individuals. *Axioms*. 2023; 12(1):62.
https://doi.org/10.3390/axioms12010062

**Chicago/Turabian Style**

Acotto, Francesca, and Ezio Venturino.
2023. "A Note on an Epidemic Model with Cautionary Response in the Presence of Asymptomatic Individuals" *Axioms* 12, no. 1: 62.
https://doi.org/10.3390/axioms12010062