# On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. Models and Theoretical Results

#### 2.1. Warm-Up: One-Stage Modeling

**Remark 1.**

**Remark 2.**

**Theorem 1.**

**Theorem 2.**

**Theorem 3.**

**Theorem 4.**

**Remark 3.**

#### 2.2. Several-Stage Modeling: Trivial Case

**Theorem 5.**

**Theorem 6.**

#### 2.3. Several-Stage Modeling: Advanced Case

**Theorem 7.**

**Theorem 8.**

**Theorem 9.**

**Remark 4.**

**Theorem 10.**

## 3. Numerical Results with Real Life Data

#### 3.1. One-Stage Modeling with the North American Ticks

^{−1}).

^{2}per month. Thus, for the critical patch size we obtain

#### 3.2. One-Stage Modeling with the Taiga Ticks

^{2}per year.

^{−1}, which is the approximate value used in [11], where it was shown to produce a reasonable fit to the experimental data available. Thus, for the critical patch size we obtain

#### 3.3. Two-Stage Modeling with the Taiga Ticks

**Remark 5.**

## 4. Simulations

## 5. Proofs

**Proof Theorem 1.**

**Proof of Theorem 2.**

**Proof of Theorem 3.**

**Proof of Theorem 4.**

**Proof of Theorems 5 and 6.**

**Proof of Theorem 7.**

**Proof of Theorem 8.**

**Proof of Theorem 9.**

**Proof of Theorem 10.**

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Kolokoltsov, V.N.
On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model. *Mathematics* **2023**, *11*, 478.
https://doi.org/10.3390/math11020478

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Kolokoltsov VN.
On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model. *Mathematics*. 2023; 11(2):478.
https://doi.org/10.3390/math11020478

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Kolokoltsov, Vassili N.
2023. "On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model" *Mathematics* 11, no. 2: 478.
https://doi.org/10.3390/math11020478