# A Within-Host Stochastic Model for Nematode Infection

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stochastic Within-Host Model and Control Criteria

#### 2.1. Grazing Management Strategies: A Stochastic Within-Host Model

**Strategy UM:**- The host is left untreated, but moved to a paddock with safe pasture at age ${t}_{0}$. The resulting process $\mathcal{Z}$ can be thought of as an age-dependent pure birth process with killing, the birth rates of which are given by ${\lambda}_{m}(t)=\lambda (t)$ if $t\in [0,{t}_{0})$, and ${\lambda}^{\prime}(t)$ if $t\in [{t}_{0},\tau ]$, and killing rates are defined by ${\delta}_{m}(t)=\delta (t)+{\gamma}_{m}(t)$ if $t\in [0,{t}_{0})$ and ${\delta}^{\prime}(t)+{\gamma}_{m}^{\prime}(t)$ if $t\in [{t}_{0},\tau ]$, for $m\in \mathcal{S}$.
**Strategy TS:**- The host is treated with anthelmintics and set-stocked at age ${t}_{0}$. Let ${\eta}_{m}^{\prime}(t)$ be the death rate of parasites when the infection level of the host is $m\in \mathcal{S}$ at time t with $t>{t}_{0}$. In this case, $\mathcal{Z}$ can be seen as an age-dependent birth and death process with killing. The birth and death rates are defined by ${\lambda}_{m}(t)=\lambda (t)$ if $t\in [0,\tau ]$, ${\eta}_{m}(t)=0$ if $t\in [0,{t}_{0})$ and ${\eta}_{m}^{\prime}(t)$ if $t\in [{t}_{0},\tau ]$, for $m\in \mathcal{S}$, respectively. Killing rates are defined identically to the rates ${\delta}_{m}(t)$ in strategy UM.
**Strategy TM:**- The host is treated with anthelmintics and moved to safe pasture at age ${t}_{0}$. In a similar manner to strategy TS, the process $\mathcal{Z}$ may be formulated as an age-dependent birth and death process with killing. Birth, death and killing rates are identical to those in strategy TS with the exception of ${\lambda}_{m}(t)$ for time instants $t\in [{t}_{0},\tau ]$, which has the form ${\lambda}_{m}(t)={\lambda}^{\prime}(t)$.

**scenario US**to reflect no intervention, that is the host is left untreated and set-stocked. Note that, in such a case, the process $\mathcal{Z}$ is an age-dependent pure birth process with killing, and its birth and killing rates are specified by ${\lambda}_{m}(t)=\lambda (t)$ and ${\delta}_{m}(t)=\delta (t)+{\gamma}_{m}(t)$ if $t\in [0,\tau ]$, for $m\in \mathcal{S}$. It follows then that the transient distribution of $\mathcal{Z}$ is readily derived from [20] for time instants $t\in (0,\tau ]$.

#### 2.2. Splitting Techniques

- (i)
- $m\to m+1$ at rate ${\lambda}_{m}(t)$, for levels $m\in \{0,1,\dots ,{M}_{0}-1\}$;
- (ii)
- $m\to m-1$ at rate ${\eta}_{m}(t)$, for levels $m\in \{1,2,\dots ,{M}_{0}\}$;
- (iii)
- $m\to -1$ at rate ${\delta}_{m}(t)$, for levels $m\in \{0,1,\dots ,{M}_{0}-1\}$;
- (iv)
- ${M}_{0}\to -1$ at rate ${\delta}_{{M}_{0}}(t)+{\lambda}_{{M}_{0}}(t)$.

#### 2.3. Control Criteria Based on Stochastic Principles

**Criterion 1:**- We select the intervention instant ${t}_{0}$ verifying $cos{t}^{s}({t}_{0};\tau )=\mathrm{inf}\left\{cos{t}^{s}(t;\tau ):t\in {J}_{\ge {m}^{\prime}}^{1}\right\}$, where the subset ${J}_{\ge {m}^{\prime}}^{1}$ consists of those potential intervention instants $t\in {I}_{\ge {m}^{\prime}}$ satisfying the inequality $ef{f}^{s}(t;\tau )\ge {p}_{1}$, for a certain probability ${p}_{1}\in (0,1)$.
**Criterion 2:**- We select the intervention instant ${t}_{0}$ such that $ef{f}^{s}({t}_{0};\tau )=\mathrm{sup}\left\{ef{f}^{s}(t;\tau ):t\in {J}_{\ge {m}^{\prime}}^{2}\right\}$, where the subset ${J}_{\ge {m}^{\prime}}^{2}$ is defined by those time instants $t\in {I}_{\ge {m}^{\prime}}$ verifying $cos{t}^{s}(t;\tau )\le {p}_{2}$, for a certain probability ${p}_{2}\in (0,1)$.

## 3. Empirical Data, Age-Dependent Rates and Results

#### 3.1. Preliminary Analysis

#### 3.2. Intervention Instants ${t}_{0}$

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. GI Nematode Infection in Growing Lambs

#### Appendix A.1. Empirical Data and Age-Dependent Rates

#### Appendix A.2. Intervention Instants t_{0}

**Table A1.**Intervention instants ${t}_{0}$ versus the index p and the lower bound ${p}_{1}$ for effectiveness (Criterion 1) and the upper bound ${p}_{2}$ for the cost of intervention (Criterion 2) for ${m}^{\prime}=4$. Grazing strategy UM.

p | ${\mathit{I}}_{\ge 4}$ | ${\mathit{p}}_{1}$ | ${\mathit{J}}_{\ge 4}^{1}$ | ${\mathit{t}}_{0}^{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{J}}_{\ge 4}^{2}$ | ${\mathit{t}}_{0}^{2}$ |
---|---|---|---|---|---|---|---|

0.1 | [170, 365) | 0.70 | —– | —– | 0.25 | [170, 299] | 170 |

0.60 | —– | —– | 0.20 | [170, 290] | 170 | ||

0.50 | [170, 194] | 170 | 0.15 | [170, 282] | 170 | ||

0.2 | [274, 365) | 0.70 | —– | —– | 0.25 | [274, 299] | 274 |

0.60 | —– | —– | 0.20 | [274, 290] | 274 | ||

0.50 | —– | —– | 0.15 | [274, 282] | 274 | ||

0.3 | [281, 365) | 0.70 | —– | —– | 0.25 | [281, 299] | 281 |

0.60 | —– | —– | 0.20 | [281, 290] | 281 | ||

0.50 | —– | —– | 0.15 | [281, 282] | 281 | ||

0.4 | [286, 365) | 0.70 | —– | —– | 0.25 | [286, 299] | 286 |

0.60 | —– | —– | 0.20 | [286, 290] | 286 | ||

0.50 | —– | —– | 0.15 | —– | —– | ||

0.5 | [290, 365) | 0.70 | —– | —– | 0.25 | [290, 299] | 290 |

0.60 | —– | —– | 0.20 | [290, 290] | 290 | ||

0.50 | —– | —– | 0.15 | —– | —– | ||

0.6 | [298, 365) | 0.70 | —– | —– | 0.25 | [298, 299] | 298 |

0.60 | —– | —– | 0.20 | —– | —– | ||

0.50 | —– | —– | 0.15 | —– | —– | ||

0.7 | [308, 365) | 0.70 | —– | —– | 0.25 | —– | —– |

0.60 | —– | —– | 0.20 | —– | —– | ||

0.50 | —– | —– | 0.15 | —– | —– |

**Figure A1.**The age-dependent probability ${P}_{\ge {m}^{\prime}}(t)$ as a function of the age $t\in (0,\tau )$ with $\tau =1$ year, for ${m}^{\prime}=1$ (

**broken**line), 4 (

**dotted**line) and 8 (

**solid**line), and increments in the number of ${L}_{3}$ infective larvae on the small intestine (

**shaded**area,

**right**vertical axis).

**Table A2.**Intervention instants ${t}_{0}$ versus the index p and the lower bound ${p}_{1}$ for effectiveness (Criterion 1) for ${m}^{\prime}=4$. Grazing strategies TS and TM with the anthelmintics ivermectin (B), fenbendazole (C) and albendazole (D).

p | ${\mathit{I}}_{\ge 4}$ | ${\mathit{p}}_{1}$ | ${\mathit{J}}_{\ge 4}^{1,\mathit{B}}$ | ${\mathit{t}}_{0}^{\mathit{B}}$ | ${\mathit{J}}_{\ge 4}^{1,\mathit{C}}$ | ${\mathit{t}}_{0}^{\mathit{C}}$ | ${\mathit{J}}_{\ge 4}^{1,\mathit{D}}$ | ${\mathit{t}}_{0}^{\mathit{D}}$ | |
---|---|---|---|---|---|---|---|---|---|

0.1 | [170, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | [170, 278] | 170 | [170, 319] | 273 | [170, 308] | 272 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | [170 ,301] | 170 | [170, 344] | 273 | [170, 342] | 272 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [170, 343] | 170 | [170, 348] | 273 | [170, 346] | 272 | |||

0.2 | [274, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | [274, 278] | 274 | [274, 319] | 274 | [274, 308] | 274 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | [274, 301] | 274 | [274, 344] | 274 | [274, 342] | 274 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [274, 343] | 274 | [274, 348] | 274 | [274, 346] | 274 | |||

0.3 | [281, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | —– | —– | [281, 319] | 281 | [281, 308] | 281 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | [281, 301] | 281 | [281, 344] | 281 | [281, 342] | 281 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [281, 343] | 281 | [281, 348] | 281 | [281, 346] | 281 | |||

0.4 | [286, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | —– | —– | [286, 319] | 286 | [286, 308] | 286 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | [286, 301] | 286 | [286, 344] | 286 | [286, 342] | 286 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [286, 343] | 286 | [286, 348] | 286 | [286, 346] | 286 | |||

0.5 | [290, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | —– | —– | [290, 319] | 290 | [290, 308] | 290 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | [290, 301] | 290 | [290, 344] | 290 | [290, 342] | 290 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [290, 343] | 290 | [290, 348] | 290 | [290, 346] | 290 | |||

0.6 | [298, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | —– | —– | [298, 319] | 298 | [298, 308] | 298 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | [298, 301] | 298 | [298, 344] | 298 | [298, 342] | 298 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [298, 343] | 298 | [298, 348] | 298 | [298, 346] | 298 | |||

0.7 | [308, 365) | 0.70 | TS | —– | —– | —– | —– | —– | —– |

TM | —– | —– | [308, 319] | 308 | [308, 308] | 308 | |||

0.60 | TS | —– | —– | [336, 339] | 336 | —– | —– | ||

TM | —– | —– | [308, 344] | 308 | [308, 342] | 308 | |||

0.50 | TS | —– | —– | [308, 344] | 308 | [313, 343] | 313 | ||

TM | [308, 343] | 308 | [308, 348] | 308 | [308, 346] | 308 |

**Table A3.**Intervention instants ${t}_{0}$ versus the index p and the upper bound ${p}_{2}$ for the cost of intervention (Criterion 2) for ${m}^{\prime}=4$. Grazing strategies TS and TM with the anthelmintics ivermectin (B), fenbendazole (C) and albendazole (D).

p | ${\mathit{I}}_{\ge 4}$ | ${\mathit{p}}_{2}$ | ${\mathit{J}}_{\ge 4}^{2,\mathit{B}}$ | ${\mathit{t}}_{0}^{\mathit{B}}$ | ${\mathit{J}}_{\ge 4}^{2,\mathit{C}}$ | ${\mathit{t}}_{0}^{\mathit{C}}$ | ${\mathit{J}}_{\ge 4}^{2,\mathit{D}}$ | ${\mathit{t}}_{0}^{\mathit{D}}$ | |
---|---|---|---|---|---|---|---|---|---|

0.1 | [170, 365) | 0.25 | TS | [286, 363] | 358 | [268, 363] | 338 | [270, 363] | 338 |

TM | [170, 363] | 170 | [170, 363] | 273 | [170, 363] | 272 | |||

0.20 | TS | [299, 362] | 358 | [279, 362] | 338 | [281, 361] | 338 | ||

TM | [170, 362] | 170 | [170, 362] | 273 | [170, 362] | 272 | |||

0.15 | TS | —– | —– | [290, 346] | 338 | [292, 344] | 338 | ||

TM | [170, 350] | 170 | [170, 350] | 273 | [170, 348] | 272 | |||

0.2 | [274, 365) | 0.25 | TS | [286, 363] | 358 | [274, 363] | 338 | [274, 363] | 338 |

TM | [274, 363] | 274 | [274, 363] | 274 | [274, 363] | 274 | |||

0.20 | TS | [299, 362] | 358 | [279, 362] | 338 | [281, 361] | 338 | ||

TM | [274, 362] | 274 | [274, 362] | 274 | [274, 362] | 274 | |||

0.15 | TS | —– | —– | [290, 346] | 338 | [292, 344] | 338 | ||

TM | [274, 350] | 274 | [274, 350] | 274 | [274, 348] | 274 | |||

0.3 | [281, 365) | 0.25 | TS | [286, 363] | 358 | [281, 363] | 338 | [281, 363] | 338 |

TM | [281, 363] | 281 | [281, 363] | 281 | [281, 363] | 281 | |||

0.20 | TS | [299, 362] | 358 | [281, 362] | 338 | [281, 361] | 338 | ||

TM | [281, 362] | 281 | [281, 362] | 281 | [281, 362] | 281 | |||

0.15 | TS | —– | —– | [290, 346] | 338 | [292, 344] | 338 | ||

TM | [281, 350] | 281 | [281, 350] | 281 | [281, 348] | 281 | |||

0.4 | [286, 365) | 0.25 | TS | [286, 363] | 358 | [286, 363] | 338 | [286, 363] | 338 |

TM | [286, 363] | 286 | [286, 363] | 286 | [286, 363] | 286 | |||

0.20 | TS | [299, 362] | 358 | [286, 362] | 338 | [286, 361] | 338 | ||

TM | [286, 362] | 286 | [286, 362] | 286 | [286, 362] | 286 | |||

0.15 | TS | —– | —– | [290, 346] | 338 | [292, 344] | 338 | ||

TM | [286, 350] | 286 | [286, 350] | 286 | [286, 348] | 286 | |||

0.5 | [290, 365) | 0.25 | TS | [290, 363] | 358 | [290, 363] | 338 | [290, 363] | 338 |

TM | [290, 363] | 290 | [290, 363] | 290 | [290, 363] | 290 | |||

0.20 | TS | [299, 362] | 358 | [290, 362] | 338 | [290, 361] | 338 | ||

TM | [290, 362] | 290 | [290, 362] | 290 | [290, 362] | 290 | |||

0.15 | TS | —– | —– | [290, 346] | 338 | [292, 344] | 338 | ||

TM | [290, 350] | 290 | [290, 350] | 290 | [290, 348] | 290 | |||

0.6 | [298, 365) | 0.25 | TS | [298, 363] | 358 | [298, 363] | 338 | [298, 363] | 338 |

TM | [298, 363] | 298 | [298, 363] | 298 | [298, 363] | 298 | |||

0.20 | TS | [299, 362] | 358 | [298, 362] | 338 | [298, 361] | 338 | ||

TM | [298, 362] | 298 | [298, 362] | 298 | [298, 362] | 298 | |||

0.15 | TS | —– | —– | [298, 346] | 338 | [298, 344] | 338 | ||

TM | [298, 350] | 298 | [298, 350] | 298 | [298, 348] | 298 | |||

0.7 | [308, 365) | 0.25 | TS | [308, 363] | 358 | [308, 363] | 338 | [308, 363] | 338 |

TM | [308, 363] | 308 | [308, 363] | 308 | [308, 363] | 308 | |||

0.20 | TS | [308, 362] | 358 | [308, 362] | 338 | [308, 361] | 338 | ||

TM | [308, 362] | 308 | [308, 362] | 308 | [308, 362] | 308 | |||

0.15 | TS | —– | —– | [308, 346] | 338 | [308, 344] | 338 | ||

TM | [308, 350] | 308 | [308, 350] | 308 | [308, 348] | 308 |

- (i)
- In applying Criterion 1 (respectively, Criterion 2) to grazing strategy TM, values of the lower bound ${p}_{1}\in \{0.5,0.6,0.7\}$ for effectiveness (respectively, the upper bound ${p}_{2}\in \{0.15,0.2,0.25\}$ for the cost of intervention) result in identical intervention instants ${t}_{0}$, irrespective of the anthelmintic drug, with the exception of the case $p=0.1$. More concretely, we observe that, in the case $p=0.1$, identical intervention instants ${t}_{0}$ are derived for each fixed anthelmintic drug, but a replacement of the predetermined drug by another anthelmintic results in different intervention instants.
- (ii)
- For every anthelmintic drug and fixed index p, Criteria 1 and 2 applied to grazing strategy TM yield identical intervention instants ${t}_{0}$, with the exception of those pairs $(p,{p}_{1})$ for the anthelmintic ivermectin leading us to empty subsets ${J}_{\ge 4}^{1,B}$. In order to maintain high values of the minimum level of effectiveness (Criterion 1), we have therefore to handle smaller values of the index p ($0.1$ and $0.2$ in Table A2) for grazing strategy TM, which means that low-risk intervention instants should become potential intervention instants.
- (iii)
- For every anthelmintic, the intervention instant ${t}_{0}$ derived in grazing strategy TM behaves as an increasing function of the index p, regardless of the control criterion.
- (iv)
- For every anthelmintic and fixed value ${p}_{1}$, the intervention instant ${t}_{0}$ in grazing strategy TS appears to be constant as a function of the index p. This is in agreement with the fact that the maximum levels of effectiveness (Figure 2) and the minimum costs of intervention (Figure 3) are observed at the end of the year (November–December), in such a way that this period of time always consists of potential intervention instants (Figure A1) for the index p ranging between $0.1$ and $0.7$.
- (v)
- In contrast to grazing strategies TS and TM, the values ${p}_{1}\in \{0.5,0.6,0.7\}$ for grazing strategy UM lead us to empty subsets ${J}_{\ge 4}^{1}$ of potential intervention instants, with the exception of the pair $(p,{p}_{1})=(0.1,0.5)$. This observation is closely related to the monotonic behaviour of the effectiveness (Figure 2) and cost (Figure 3) functions, which links the first months of the year to the highest effectiveness and the minimum cost of intervention.
- (vi)
- The upper limit of the set ${I}_{\ge 4}$ in Table A1, Table A2 and Table A3 is always at Day 365, which can be readily explained from the monotone behaviour (Figure A1) of the age-dependent probability ${P}_{\ge {m}^{\prime}}(t)$ in the case ${m}^{\prime}=4$. It is clear that other thresholds ${m}^{\prime}$ will not necessarily yield Day 365; for example, ${I}_{\ge {m}^{\prime}}=(280,360)$ in the case ${m}^{\prime}=1$ with $p=0.85$.
- (vii)
- For strategies UM (Table A1) and TM (Table A2 and Table A3), the lower limits of the resulting sets ${J}_{\ge 4}^{1}$ and ${J}_{\ge 4}^{2}$ always coincide with the lower limit of the set ${I}_{\ge 4}$ of potential intervention instants ${t}_{0}$, but this is not the case for strategy TS. This means that an early movement of the host to safe pasture should lead to feasible intervention instants.

## References

- Sutherland, I.; Scott, I. Gastrointestinal Nematodes of Sheep and Cattle. Biology and Control; Wiley-Blackwell: Chichester, UK, 2010. [Google Scholar]
- Taylor, M.A.; Coop, R.L.; Wall, R.L. Veterinary Parasitology, 3rd ed.; Blackwell: Oxford, UK, 2007. [Google Scholar]
- Bjørn, H.; Monrad, J.; Nansen, P. Anthelmintic resistance in nematode parasites of sheep in Denmark with special emphasis on levamisole resistance in Ostertagia circumcincta. Acta Vet. Scand.
**1991**, 32, 145–154. [Google Scholar] [PubMed] - Entrocasso, C.; Alvarez, L.; Manazza, J.; Lifschitz, A.; Borda, B.; Virkel, G.; Mottier, L.; Lanusse, C. Clinical efficacy assessment of the albendazole-ivermectin combination in lambs parasited with resistant nematodes. Vet. Parasitol.
**2008**, 155, 249–256. [Google Scholar] [CrossRef] [PubMed] - Stear, M.J.; Doligalska, M.; Donskow-Schmelter, K. Alternatives to anthelmintics for the control of nematodes in livestock. Parasitology
**2007**, 134, 139–151. [Google Scholar] [CrossRef] [PubMed] - Hein, W.R.; Shoemaker, C.B.; Heath, A.C.G. Future technologies for control of nematodes of sheep. N. Z. Vet. J.
**2001**, 49, 247–251. [Google Scholar] [CrossRef] [PubMed] - Knox, D.P. Technological advances and genomics in metazoan parasites. Int. J. Parasitol.
**2004**, 34, 139–152. [Google Scholar] [CrossRef] [PubMed] - Sayers, G.; Sweeney, T. Gastrointestinal nematode infection in sheep—A review of the alternatives to anthelmintics in parasite control. Anim. Health Res. Rev.
**2005**, 6, 159–171. [Google Scholar] [CrossRef] [PubMed] - Waller, P.J.; Thamsborg, S.M. Nematode control in ‘green’ ruminant production systems. Trends Parasitol.
**2004**, 20, 493–497. [Google Scholar] [CrossRef] [PubMed][Green Version] - Smith, G.; Grenfell, B.T.; Isham, V.; Cornell, S. Anthelmintic resistance revisited: Under-dosing, chemoprophylactic strategies, and mating probabilities. Int. J. Parasitol.
**1999**, 29, 77–91. [Google Scholar] [CrossRef] - Praslička, J.; Bjørn, H.; Várady, M.; Nansen, P.; Hennessy, D.R.; Talvik, H. An in vivo dose-response study of fenbendazole against Oesophagostomum dentatum and Oesophagostomum quadrispinulatum in pigs. Int. J. Parasitol.
**1997**, 27, 403–409. [Google Scholar] [CrossRef] - Coles, G.C.; Roush, R.T. Slowing the spread of anthelmintic resistant nematodes of sheep and goats in the United Kingdom. Vet. Res.
**1992**, 130, 505–510. [Google Scholar] [CrossRef] - Prichard, R.K.; Hall, C.A.; Kelly, J.D.; Martin, I.C.A.; Donald, A.D. The problem of anthelmintic resistance in nematodes. Aust. Vet. J.
**1980**, 56, 239–250. [Google Scholar] [CrossRef] [PubMed] - Anderson, R.M.; May, R.M. Infectious Diseases of Humans: Dynamics and Control; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Marion, G.; Renshaw, E.; Gibson, G. Stochastic effects in a model of nematode infection in ruminants. IMA J. Math. Appl. Med. Biol.
**1998**, 15, 97–116. [Google Scholar] [CrossRef] [PubMed] - Cornell, S.J.; Isham, V.S.; Grenfell, B.T. Stochastic and spatial dynamics of nematode parasites in farmed ruminants. Proc. R. Soc. B Biol. Sci.
**2004**, 271, 1243–1250. [Google Scholar] [CrossRef] [PubMed][Green Version] - Roberts, M.G.; Grenfell, B.T. The population dynamics of nematode infections of ruminants: Periodic perturbations as a model for management. IMA J. Math. Appl. Med. Biol.
**1991**, 8, 83–93. [Google Scholar] [CrossRef] [PubMed] - Roberts, M.G.; Grenfell, B.T. The population dynamics of nematode infections of ruminants: The effect of seasonality in the free-living stages. IMA J. Math. Appl. Med. Biol.
**1992**, 9, 29–41. [Google Scholar] [CrossRef] [PubMed] - Allen, L.J.S. An Introduction to Stochastic Processes with Applications to Biology; Pearson Education: Hoboken, NJ, USA, 2003. [Google Scholar]
- Gómez-Corral, A.; López García, M. Control strategies for a stochastic model of host-parasite interaction in a seasonal environment. J. Theor. Biol.
**2014**, 354, 1–11. [Google Scholar] [CrossRef] [PubMed] - Abbott, K.A.; Taylor, M.; Stubbings, L.A. Sustainable Worm Control Strategies for Sheep, 4th ed.; A Technical Manual for Veterinary Surgeons and Advisers; SCOPS: Worcestershire, UK, 2012; Available online: http://www.scops.org.uk/workspace/pdfs/scops-technical-manual-4th-edition-updated-september-2013.pdf (accessed on 1 June 2018).
- Uriarte, J.; Llorente, M.M.; Valderrábano, J. Seasonal changes of gastrointestinal nematode burden in sheep under an intensive grazing system. Vet. Parasitol.
**2003**, 118, 79–92. [Google Scholar] [CrossRef] [PubMed] - Faragó, I.; Havasi, A.; Horváth, R. On the order of operator splitting methods for time-dependent linear systems of differential equations. Int. J. Numer. Anal. Model. Ser. B
**2011**, 2, 142–154. [Google Scholar] - Nasreen, S.; Jeelani, G.; Sheikh, F.D. Efficacy of different anthelmintics against gastro-intestinal nematodes of sheep in Kashmir Valley. VetScan
**2007**, 2, 1. [Google Scholar] - Kassai, T. Veterinary Helminthology; Butterworth-Heinemann: Oxford, UK, 1999. [Google Scholar]
- Barger, I.A. Genetic resistance of hosts and its influence on epidemiology. Vet. Parasitol.
**1989**, 32, 21–35. [Google Scholar] [CrossRef] - Barger, I.A.; Le Jambre, L.F.; Georgi, J.R.; Davies, H.I. Regulation of Haemonchus contortus populations in sheep exposed to continuous infection. Int. J. Parasitol.
**1985**, 15, 529–533. [Google Scholar] [CrossRef] - Dobson, R.J.; Waller, P.J.; Donald, A.D. Population dynamics of Trichostrongylus colubriformis in sheep: The effect of infection rate on the establishment of infective larvae and parasite fecundity. Int. J. Parasitol.
**1990**, 20, 347–352. [Google Scholar] [CrossRef] - Bailey, J.N.; Kahn, L.P.; Walkden-Brown, S.W. Availability of gastro-intestinal nematode larvae to sheep following winter contamination of pasture with six nematode species on the Northern Tablelands of New South Wales. Vet. Parasitol.
**2009**, 160, 89–99. [Google Scholar] [CrossRef] [PubMed] - Valderrábano, J.; Delfa, R.; Uriarte, J. Effect of level of feed intake on the development of gastrointestinal parasitism in growing lambs. Vet. Parasitol.
**2002**, 104, 327–338. [Google Scholar] [CrossRef] - Grennan, E.J. Lamb Growth Rate on Pasture: Effect of Grazing Management, Sward Type and Supplementation; Teagasc Research Centre: Athenry, Ireland, 1999. [Google Scholar]

**Figure 1.**State space and transitions at post-intervention instants $t\in [{t}_{0},\tau ]$. Grazing strategies TS and TM.

**Figure 2.**Effectiveness $ef{f}^{s}({t}_{0};\tau )$ as a function of the intervention age ${t}_{0}$ for $\tau =1$ year and increments in the number of ${L}_{3}$ infective larvae in the small intestine (shaded area, right vertical axis). Scenario US, and grazing strategies UM, TS and TM with the anthelmintics ivermectin, fenbendazole and albendazole (from top to bottom).

**Figure 3.**Cost $cos{t}^{s}({t}_{0};\tau )$ of intervention as a function of the intervention age ${t}_{0}$ for $\tau =1$ year and increments in the number of ${L}_{3}$ infective larvae in the small intestine (shaded area, right vertical axis). Scenario US, and grazing strategies UM, TS and TM with the anthelmintics ivermectin, fenbendazole and albendazole (from top to bottom).

**Figure 4.**Expected proportions ${\tau}^{-1}{E}^{s}({t}_{0};\tau )$ (

**top**) and ${\tau}^{-1}{C}^{s}({t}_{0};\tau )$ (

**bottom**) versus the intervention age ${t}_{0}$ for $\tau =1$ year and increments in the number of ${L}_{3}$ infective larvae in the small intestine (shaded area, right vertical axis). Scenario US, and grazing strategies UM, TS and TM with the anthelmintic fenbendazole.

**Figure 5.**The mass function of the parasite burden $M(\tau )$ at age $\tau =1$ year. Scenario US and grazing strategies UM, TS and TM (from left to right) with the anthelmintic ivermectin as the intervention prescribed at age ${t}_{0}=170$.

**Figure 6.**The mass function of the parasite burden $M(\tau )$ at age $\tau =1$ year. Scenario US and grazing strategy TM with the anthelmintics ivermectin, fenbendazole and albendazole (from left to right) as the intervention prescribed at ages ${t}_{0}=170$, 273 and 272, respectively.

**Table 1.**Effectiveness and cost of intervention. Scenario US and grazing strategies UM, TS and TM with the anthelmintics ivermectin, fenbendazole and albendazole.

Strategy (s) | Anthelmintic | ${\mathit{t}}_{0}$ | Criteria | ${\mathit{eff}}^{\mathit{s}}({\mathit{t}}_{0};\mathit{\tau})$ | ${\mathit{cost}}^{\mathit{s}}({\mathit{t}}_{0};\mathit{\tau})$ | ${\mathit{\tau}}^{-1}{\mathit{E}}^{\mathit{s}}({\mathit{t}}_{0};\mathit{\tau})$ | ${\mathit{\tau}}^{-1}{\mathit{C}}^{\mathit{s}}({\mathit{t}}_{0};\mathit{\tau})$ |
---|---|---|---|---|---|---|---|

US | — | — | — | 0.06072 | 0.49951 | 0.68645 | 0.14746 |

UM | 170 | 1 & 2 | 0.54431 | 0.11049 | 0.79996 | 0.09726 | |

274 | 2 | 0.45540 | 0.12524 | 0.76629 | 0.09983 | ||

281 | 2 | 0.38981 | 0.14216 | 0.74973 | 0.10267 | ||

286 | 2 | 0.32115 | 0.16811 | 0.73306 | 0.10715 | ||

290 | 2 | 0.26634 | 0.19763 | 0.72023 | 0.11233 | ||

298 | 2 | 0.20886 | 0.24130 | 0.70769 | 0.11984 | ||

TS | ivermectin | 358 | 2 | 0.41766 | 0.16608 | 0.69160 | 0.14217 |

fenbendazole | 308 | 1 | 0.50340 | 0.12350 | 0.75871 | 0.10433 | |

336 | 1 | 0.60161 | 0.13144 | 0.71941 | 0.12421 | ||

338 | 2 | 0.60604 | 0.13209 | 0.71613 | 0.12578 | ||

albendazole | 313 | 1 | 0.50240 | 0.12842 | 0.74908 | 0.10793 | |

338 | 2 | 0.57385 | 0.13407 | 0.71312 | 0.12626 | ||

TM | ivermectin | 170 | 1 & 2 | 0.73224 | 0.09721 | 0.86987 | 0.09525 |

274 | 1 & 2 | 0.71025 | 0.09797 | 0.82480 | 0.09580 | ||

281 | 1 & 2 | 0.69119 | 0.09877 | 0.81634 | 0.09602 | ||

286 | 1 & 2 | 0.66653 | 0.10011 | 0.80686 | 0.09644 | ||

290 | 1 & 2 | 0.64110 | 0.10197 | 0.79743 | 0.09713 | ||

298 | 1 & 2 | 0.61142 | 0.10528 | 0.78209 | 0.09911 | ||

308 | 1 & 2 | 0.56977 | 0.11374 | 0.76202 | 0.10372 | ||

fenbendazole | 273 | 1 & 2 | 0.79086 | 0.09589 | 0.83891 | 0.09557 | |

274 | 1 & 2 | 0.79080 | 0.09589 | 0.83820 | 0.09558 | ||

281 | 1 & 2 | 0.78559 | 0.09601 | 0.83107 | 0.09573 | ||

286 | 1 & 2 | 0.77604 | 0.09636 | 0.82304 | 0.09605 | ||

290 | 1 & 2 | 0.76467 | 0.09707 | 0.81476 | 0.09662 | ||

298 | 1 & 2 | 0.75182 | 0.09895 | 0.79922 | 0.09852 | ||

308 | 1 & 2 | 0.72721 | 0.10573 | 0.77734 | 0.10310 | ||

albendazole | 272 | 1 & 2 | 0.78128 | 0.09605 | 0.83749 | 0.09558 | |

274 | 1 & 2 | 0.78102 | 0.09606 | 0.83605 | 0.09560 | ||

281 | 1 & 2 | 0.77361 | 0.09623 | 0.82838 | 0.09576 | ||

286 | 1 & 2 | 0.76132 | 0.09666 | 0.81971 | 0.09610 | ||

290 | 1 & 2 | 0.74737 | 0.09747 | 0.81089 | 0.09671 | ||

298 | 1 & 2 | 0.73134 | 0.09945 | 0.79492 | 0.09867 | ||

308 | 1 & 2 | 0.70211 | 0.10641 | 0.77271 | 0.10336 |

**Table 2.**Indexes reduction in the mean infection level (RMIL) and reduction in the total lost probability (RTLP) for strategies UM, TS and TM with the anthelmintic fenbendazole.

Strategy (s) | Criteria | ${\mathit{t}}_{0}$ | ${\mathit{RMIL}}^{\mathit{s},{\mathit{t}}_{0}}$ | ${\mathit{RTLP}}^{\mathit{s},{\mathit{t}}_{0}}$ |
---|---|---|---|---|

UM | $1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 170 | $53.50\%$ | $39.34\%$ |

2 | 274 | $46.98\%$ | $39.17\%$ | |

2 | 281 | $41.85\%$ | $38.88\%$ | |

2 | 286 | $36.00\%$ | $38.21\%$ | |

2 | 290 | $30.78\%$ | $37.12\%$ | |

2 | 298 | $24.54\%$ | $34.86\%$ | |

Midsummer | 195 | $50.28\%$ | $39.28\%$ | |

TS | 1 | 308 | $51.01\%$ | $33.27\%$ |

1 | 336 | $59.19\%$ | $19.77\%$ | |

2 | 338 | $59.56\%$ | $19.10\%$ | |

Maximum pasture contamination | 287 | $37.78\%$ | $38.17\%$ | |

TM | $1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 273 | $72.36\%$ | $39.41\%$ |

$1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 274 | $72.36\%$ | $39.41\%$ | |

$1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 281 | $71.88\%$ | $39.39\%$ | |

$1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 286 | $71.03\%$ | $39.26\%$ | |

$1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 290 | $70.07\%$ | $38.92\%$ | |

$1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 298 | $69.09\%$ | $37.82\%$ | |

$1\phantom{\rule{3.33333pt}{0ex}}\&\phantom{\rule{3.33333pt}{0ex}}2$ | 308 | $67.47\%$ | $33.63\%$ | |

Midsummer | 195 | $70.47\%$ | $39.42\%$ |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gómez-Corral, A.; López-García, M. A Within-Host Stochastic Model for Nematode Infection. *Mathematics* **2018**, *6*, 143.
https://doi.org/10.3390/math6090143

**AMA Style**

Gómez-Corral A, López-García M. A Within-Host Stochastic Model for Nematode Infection. *Mathematics*. 2018; 6(9):143.
https://doi.org/10.3390/math6090143

**Chicago/Turabian Style**

Gómez-Corral, Antonio, and Martín López-García. 2018. "A Within-Host Stochastic Model for Nematode Infection" *Mathematics* 6, no. 9: 143.
https://doi.org/10.3390/math6090143