Table 1 The biological significance of the parameters of system \((2)\).

\({g_{1}}\)

Intrinsic growth rate of tumor cells

[0.1, 0.9]

²⁸

\({s_{1}}\)

Carrying capacity of the environment for tumor cells

1000

²⁸

\({s_{2}}\)

Half-saturation constant for macrophages

50

²⁸

\({s_{3}}\)

Half-saturation constant for dendritic cells

50

²⁸

\({s_{4}}\)

Half-saturation constant for \(\hbox {CD4}^{+}\)T cells

50

²⁸

\({r_{1}}\)

The production rate of Dendritic cells

0.3

²⁸

\({r_{2}}\)

The production rate of \(\hbox {CD4}^{+}\)T cells

0.7

²⁸

\({\beta _{1}}\)

Rate at which tumor cells are killed by macrophages

0.1

²⁸

\({\omega }\)

Rate at which tumor cells are killed by \(\hbox {CD4}^{+}\)T cells

0.3

²²

\({g_{2}}\)

Recruitment rate of macrophages by tumor cells

[0.2, 0.9]

²⁸

\({\beta _{2}}\)

Natural death rate of macrophages

0.5

²⁸

\({g_{3}}\)

Recruitment rate of dendritic cells by tumor cells

[0.2, 0.9]

²⁸

\({\beta _{3}}\)

Natural death rate of dendritic cells

0.4

²⁸

\({g_{4}}\)

Activation rate of \(\hbox {CD4}^{+}\)T cells by dendritic cells

[0.1, 0.9]

²⁸

\({\beta _{4}}\)

Natural death rate of \(\hbox {CD4}^{+}\)T cells

0.7

²⁸

\({d_{1}}\)

Represents the death rate of macrophages that

lose their activity under the effect of chemotherapy

0.008

²²

\({d_{2}}\)

Represents the death rate of dendritic cells that

lose their activity under the effect of chemotherapy

0.0008

²²

\({d_{3}}\)

Represents the death rate of \(\hbox {CD4}^{+}\)T helper cells that

lose their activity under the effect of chemotherapy

0.0001

²²

\({\vartheta }\)

Decay rate of P

1

²²

\({\delta }\)

Recruitment rate of \(\hbox {CD4}^{+}\)T helper cells by P

[0.15, 0.60]

²²

b

Half-saturation constant for a probiotic drug

0.25

²²

\({u_{1}}\)

Represents the dosage of the probiotic drug

–

–

\({u_{2}}\)

Represents the dosage of the immunotherapeutic drug

–

–

\({u_{3}}\)

Represents the dosage of the chemotherapy drug

–

–