Modelagem matemática da interação entre o vírus da dengue e os anticorpos
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Data
2023-02-16
Autores
Camargo, Felipe de Almeida
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Universidade Estadual Paulista (Unesp)
Resumo
É fato que a circulação simultânea de diferentes sorotipos do vírus da dengue (DENV) em uma região aumenta o número de casos graves da doença (dengue hemorrágica, DH). Embora, na maioria das vezes, a DH ocorra em crianças e adultos infectados por um segundo sorotipo (heterólogo), ela também pode ocorrer em lactentes durante a infecção primária devido à presença de anticorpos maternos. Duas hipóteses são propostas para explicar tal aumento, ambas associadas a existência de memória imunológica: (i) Anticorpos não neutralizantes se ligam ao DENV formando complexos imunes que facilitam a entrada do vírus nas células-alvo, (ii) Células T com baixa afinidade para o sorotipo predominante levam a um controle sub-ótimo da infecção. Para explicar a exacerbação da infecção durante a infecção secundária com vírus heterólogo (crianças e adultos) ou primária (lactentes) consideramos a hipótese de aprimoramento dependente de anticorpos (caso (i)). Dois modelos de equações diferenciais ordinárias são propostos. No primeiro modelo, consideramos a interação entre as populações de células-alvo, do vírus e dos anticorpos. O estudo analítico mostrou a existência de dois pontos de equilíbrios, o equilíbrio livre da doença e o de persistência da doença. A análise de estabilidade assintótica dos pontos de equilíbrio permitiu a obtenção do número reprodutivo básico R0 e mostrou que se R0 < 1 o equilíbrio livre da doença é estável, e se R0 > 1 o equilíbrio endêmico é estável. Para discutir a gravidade da doença, consideramos o valor máximo obtido para a população de células infectadas durante a evolução temporal destas, e o instante de tempo em que esse valor máximo é atingido. Observamos que esse tempo corresponde ao momento em que a função de aprimoramento da infecção atinge seu máximo. Para o segundo modelo, propusemos uma reformulação das funções de neutralização e aprimoramento descritas em Adimy et al. (2020). Reproduzimos todos os resultados analíticos e numéricos de Adimy et al. (2020) e verificamos que as novas funções propostas se adéquam melhor ao problema. Para análise de sensibilidade, utilizamos uma técnica mais adequada, visto que a função resposta R0 é não monotônica com relação aos parâmetros do modelo. Acrescentamos um parâmetro a mais na estimação de parâmetros, a quantidade inicial de vírus, e os novos resultados foram semelhantes. As novas funções propostas são de fácil interpretação visto que são baseadas em conceitos de reações químicas.
It is a fact that the simultaneous circulation of different serotypes of the dengue virus (DENV) in a region increases the number of severe cases of the disease (dengue hemorrhagic fever, DHF). Although, DHF most often occurs in children and adults infected with a second (heterologous) serotype, it can also occur in infants during primary infection due to the presence of maternal antibodies. Two hypotheses are proposed to explain this increase, both associated with the existence of immunological memory: (i) Non-neutralizing antibodies bind to DENV forming immune complexes that facilitate the entry of the virus into target cells, and (ii) T cells with low affinity for current serotype lead to suboptimal infection control. To explain the exacerbation of infection during secondary infection with heterologous (children and adults) or primary (infants) virus, we considered the hypothesis of antibody-dependent enhancement (case (i)). Two ordinary differential equation models were proposed. In the first model, we considered the interaction between the target cell, virus, and antibody populations. The analytical study showed the existence of two equilibrium points, the disease-free equilibrium and the disease persistence equilibrium (endemic equilibrium). The asymptotic stability analysis of the equilibrium points allowed obtaining the basic reproductive number R0 and showed that if R0 < 1 the disease-free equilibrium is stable, and if R0 > 1 the endemic equilibrium is stable. To discuss the severity of the disease, we considered the maximum value obtained for the population of infected cells during their temporal evolution and the time instant in which this maximum value is reached. Wenoticed that this time corresponds to the moment when the infection enhancement function reaches its maximum. For the second model, we proposed a reformulation of the neutralization and enhancement functions described in Adimy et al. (2020). We reproduced all the analytical and numerical results and verified that the new proposed functions were better suited to the problem. For sensitivity analysis, we used a more adequate technique since the response function R0 is not monotonic with respect to the model parameters. We added one more parameter in the estimation of parameters, an initial amount of virus, and the new results were similar. The new proposed functions are easily interpreted as they are based on chemical reaction concepts.
It is a fact that the simultaneous circulation of different serotypes of the dengue virus (DENV) in a region increases the number of severe cases of the disease (dengue hemorrhagic fever, DHF). Although, DHF most often occurs in children and adults infected with a second (heterologous) serotype, it can also occur in infants during primary infection due to the presence of maternal antibodies. Two hypotheses are proposed to explain this increase, both associated with the existence of immunological memory: (i) Non-neutralizing antibodies bind to DENV forming immune complexes that facilitate the entry of the virus into target cells, and (ii) T cells with low affinity for current serotype lead to suboptimal infection control. To explain the exacerbation of infection during secondary infection with heterologous (children and adults) or primary (infants) virus, we considered the hypothesis of antibody-dependent enhancement (case (i)). Two ordinary differential equation models were proposed. In the first model, we considered the interaction between the target cell, virus, and antibody populations. The analytical study showed the existence of two equilibrium points, the disease-free equilibrium and the disease persistence equilibrium (endemic equilibrium). The asymptotic stability analysis of the equilibrium points allowed obtaining the basic reproductive number R0 and showed that if R0 < 1 the disease-free equilibrium is stable, and if R0 > 1 the endemic equilibrium is stable. To discuss the severity of the disease, we considered the maximum value obtained for the population of infected cells during their temporal evolution and the time instant in which this maximum value is reached. Wenoticed that this time corresponds to the moment when the infection enhancement function reaches its maximum. For the second model, we proposed a reformulation of the neutralization and enhancement functions described in Adimy et al. (2020). We reproduced all the analytical and numerical results and verified that the new proposed functions were better suited to the problem. For sensitivity analysis, we used a more adequate technique since the response function R0 is not monotonic with respect to the model parameters. We added one more parameter in the estimation of parameters, an initial amount of virus, and the new results were similar. The new proposed functions are easily interpreted as they are based on chemical reaction concepts.
Descrição
Palavras-chave
Análise de estabilidade local, DENV 1-4, Anticorpos neutralizantes e de aprimoramento, Casos graves em lactentes, Local stability analysis, Neutralizing and enhancing antibodies, Severe cases in infants