Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays
In this paper, we investigate two within-host mathematical models that are based on differential equations. These mathematical models include healthy cells, tachyzoites, and bradyzoites. The first model is based on ordinary differential equations and the second one includes a discrete time delay. We...
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| Format: | Article |
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MDPI AG
2023-10-01
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| author | Sharmin Sultana Gilberto González-Parra Abraham J. Arenas |
| author_facet | Sharmin Sultana Gilberto González-Parra Abraham J. Arenas |
| author_sort | Sharmin Sultana |
| collection | DOAJ |
| description | In this paper, we investigate two within-host mathematical models that are based on differential equations. These mathematical models include healthy cells, tachyzoites, and bradyzoites. The first model is based on ordinary differential equations and the second one includes a discrete time delay. We found the models’ steady states and computed the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mn>0</mn></msub></semantics></math></inline-formula>. Two equilibrium points exist in both models: the first is the disease-free equilibrium point and the second one is the endemic equilibrium point. We found that the initial quantity of uninfected cells has an impact on the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mn>0</mn></msub></semantics></math></inline-formula>. This threshold parameter also depends on the contact rate between tachyzoites and uninfected cells, the contact rate between encysted bradyzoite and the uninfected cells, the conversion rate from tachyzoites to bradyzoites, and the death rate of the bradyzoites- and tachyzoites-infected cells. We investigated the local and global stability of the two equilibrium points for the within-host models that are based on differential equations. We perform numerical simulations to validate our analytical findings. We also demonstrated that the disease-free equilibrium point cannot lose stability regardless of the value of the time delay. The numerical simulations corroborated our analytical results. |
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| id | doaj.art-1096b91885f149a380471c6fc0e3f2d1 |
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| language | English |
| last_indexed | 2024-03-11T11:25:00Z |
| publishDate | 2023-10-01 |
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| series | Mathematics |
| spelling | doaj.art-1096b91885f149a380471c6fc0e3f2d12023-11-10T15:07:58ZengMDPI AGMathematics2227-73902023-10-011121446910.3390/math11214469Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time DelaysSharmin Sultana0Gilberto González-Parra1Abraham J. Arenas2Department of Mathematics, New Mexico Tech, Socorro, NM 87801, USADepartment of Mathematics, New Mexico Tech, Socorro, NM 87801, USADepartamento de Matemáticas y Estadística, Universidad de Córdoba, Monteria 230002, ColombiaIn this paper, we investigate two within-host mathematical models that are based on differential equations. These mathematical models include healthy cells, tachyzoites, and bradyzoites. The first model is based on ordinary differential equations and the second one includes a discrete time delay. We found the models’ steady states and computed the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mn>0</mn></msub></semantics></math></inline-formula>. Two equilibrium points exist in both models: the first is the disease-free equilibrium point and the second one is the endemic equilibrium point. We found that the initial quantity of uninfected cells has an impact on the basic reproduction number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">R</mi><mn>0</mn></msub></semantics></math></inline-formula>. This threshold parameter also depends on the contact rate between tachyzoites and uninfected cells, the contact rate between encysted bradyzoite and the uninfected cells, the conversion rate from tachyzoites to bradyzoites, and the death rate of the bradyzoites- and tachyzoites-infected cells. We investigated the local and global stability of the two equilibrium points for the within-host models that are based on differential equations. We perform numerical simulations to validate our analytical findings. We also demonstrated that the disease-free equilibrium point cannot lose stability regardless of the value of the time delay. The numerical simulations corroborated our analytical results.https://www.mdpi.com/2227-7390/11/21/4469within-hosttoxoplasmosisdiscrete delaystability analysis |
| spellingShingle | Sharmin Sultana Gilberto González-Parra Abraham J. Arenas Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays Mathematics within-host toxoplasmosis discrete delay stability analysis |
| title | Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays |
| title_full | Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays |
| title_fullStr | Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays |
| title_full_unstemmed | Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays |
| title_short | Analysis of Within-Host Mathematical Models of Toxoplasmosis That Consider Time Delays |
| title_sort | analysis of within host mathematical models of toxoplasmosis that consider time delays |
| topic | within-host toxoplasmosis discrete delay stability analysis |
| url | https://www.mdpi.com/2227-7390/11/21/4469 |
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