Mathematical modeling of postcoinfection with influenza A virus and Streptococcus pneumoniae, with implications for pneumonia and COPD-risk assessment

Figures & data

Figure 1 Study framework and computational algorithms used in this study.

Notes: (A) Applied published study data. (B, C) Implemented mathematical within-host coinfection dynamic model and associated parameterization, as well as uncertainty and sensitivity analyses (refer to Table 1 for specific symbol meanings). (D) Methodology and algorithm in transferring and normalizing bacterial loads, cytokine levels, and inflammatory responses. (E) Probabilistic risk assessment by jointing. (F) Implications of control-measure implements.
Abbreviations: D_M, bacterial load in mice; D_H, bacterial load in humans; CL_M, level of inflammatory cytokines in mice; CL_H, level of inflammatory cytokines in humans; K_M, body-surface area-related transfer factors for mice; K_H, body-surface area-related transfer factors for humans; IE, inflammatory effect; CL_Pna,2.5, the 2.5% tile of inflammatory cytokine levels estimated from patients with influenza-associated pneumonia; P(D_H), prior probability; P(IE|D_H), conditional probability; P(R_CoI(IE)), posterior probability.

Table 1 Descriptions of governing equations used for influenza A virus and Streptococcus pneumoniae coinfection dynamic model¹⁵

Description	Equation
Influenza A virus
Uninfected epithelial cells (T)	$\frac{dT}{dt}=-\beta TV$	(T1)
Unproductive infected cells (I)	$\frac{dI}{dt}=-\beta TV-kI$	(T2)
Productive infected cells (J)	$\frac{dJ}{dt}=kI-\delta J$	(T3)
Influenza (V)	$\frac{dV}{dt}=pJ-cV$	(T4)
Influenza A virus postcoinfection
Unproductive infected cells (I)	$\frac{dI}{dt}=\beta TV-kI-\mu IB$	(T5)
Productive infected cells (J)	$\frac{dJ}{dt}=kI-\delta J-\mu JB$	(T6)
Influenza (V)	$\frac{dV}{dt}=pJ\left(I+\alpha B^{\mathrm{Z}}\right)-cV$	(T7)
Streptococcus pneumoniae postcoinfection
S. pneumoniae (B)	$\begin{array}{c}\frac{dB}{dt}=rB\left(I-\frac{B}{K_{\mathrm{B}}\left(I+\psi V_{\mathrm{C}}\right)}\right)\\ -\gamma M_{\mathrm{A}}F{M}_{\mathrm{A}}^{*}B\left(I-\frac{\varphi V_{\mathrm{C}}}{K_{\mathrm{B}\mathrm{V}}+V_{\mathrm{C}}}\right)\end{array}$	(T8)
Phagocytosis-decreasing function (F)	$F=f\left(B,M_{\mathrm{A}}\right)=\frac{n^2{M}_{\mathrm{A}}}{B^2+n^2{M}_{\mathrm{A}}}$	(T9)

Abbreviations: β, virus infectivity; k, transition rate of infected cells in eclipse phase; δ, infected cell-death rate; p, virus-production rate; c, virus-clearance rate; r, bacteria-growth rate; K_B, carrying capacity; M_A*, steady-state macrophages; γM_A, phagocytosis rate; n, maximum bacteria (phagocytosis) per M_A; coinfection-associated param eters; μ, toxic death of infected cells; a, virion production/release-increase rate; z, nonlinearity of virion production/release; ψ, increase in carrying capacity; ϕ, decrease in phagocytosis rate; K_BV, half-saturation constant.

Table 2 Parameter values used in influenza A virus and Streptococcus pneumoniae coinfection model

Parameter	Description	Unit	Value
Influenza A virus
β	Virus infectivity	1/(TCID50⋅mL^−1 × day)	2.8×10^−6
k	Transition rate of infected cells in eclipse phase	Day^−1	4
δ	Infected cell-death rate	Day^−1	0.89
p	Virus-production rate	1/(TCID50⋅mL^−1 × day)	25.1
c	Virus-clearance rate	Day^−1	28.4
T(0)	Initial uninfected cells	Cells	10^7
I(0)	Initial unproductive infected cells	Cells	0
J(0)	Initial productive infected cells	Cells	0
V(0)	Initial virus titer	TCID50⋅mL^−1	1 (0.35–2.96)a,b
Streptococcus pneumoniae
r	Bacteria-growth rate	Day^−1	27
KB	Carrying capacity	CFU⋅m L^−1	2.3×10^8
MA*	Steady-state macrophages	Cells	10^6
γMA	Phagocytosis rate	Cells^−1 day^−1	1.35×10^−4
n	Maximum bacteria per MA	CFU⋅m L^−1 cell^−1	5
B(0)	Initial bacteria load	CFU⋅m L^−1	10^2–10^5,c
Coinfection
ψ	Increase in carrying capacity	1/TCID50⋅mL^−1	1.2×10^−8
ϕ	Decrease in phagocytosis rate	–	0.87 (0.86–0.91)
KBV	Half-saturation constant	TCID50⋅mL^−1	1.8×10^3 (5.7×10^2–9.4×10^3)
μ	Toxic death of infected cells	1/CFU⋅ mL^−1	5.2×10^−10 (0–0.43)
a	Increase in virion production/release	(CFU⋅m L^−1)^−^z	1.2×10^−3 (1.4×10^−4–4.3×10^−1)
z	Nonlinearity of virion production/release	–	0.5 (0.14–0.61)

Notes: Values derived from Smith et al¹⁵ unless otherwise noted.

a Estimated from Chen et al;⁴¹

b median (95% CI);

c estimated from Smith et al¹⁵ and Shrestha et al.¹⁷

Abbreviations: β, virus infectivity; k, transition rate of infected cells in eclipse phase; δ, infected cell-death rate; p, virus-production rate; c, virus-clearance rate; r, bacteria-growth rate; K_B, carrying capacity; M_A*, steady-state macrophages; γM_A, phagocytosis rate; n, maximum bacteria (phagocytosis) per M_A; coinfection-associated parameters; μ, toxic death of infected cells; a, virion production/release-increase rate; z, nonlinearity of virion production/release; ψ, increase in carrying capacity;ϕ, decrease in phagocytosis rate; K_BV, half-saturation constant.

Figure 2 IAV and SP coinfection dynamics within hosts.

Notes: (A) Epithelial cells, including uninfected (T, brown line), unproductive infected (I, dashed line), and virus-productive infected (J, green line) cells; (B) pathogens consisting of IAV (V, blue line) and SP (B, pink line).
Abbreviations: IAV, influenza A virus; SP, Streptococcus pneumoniae.

Table 3 Conversion of bacteria load and cytokine level between mice and humans and normalized inflammatory effect

Bacteria load		Cytokine level		Normalized inflammatory effect for humans (fold)b
Mice (log CFU⋅mL^−1)	Humansa (log CFU⋅mL^−1)	Mice (log pg⋅mL^−1)	Humansa (log pg⋅mL^−1)
IL6
7.89	6.84	4.21	3.16	256
8.5	7.45	4.27	3.22	294
5.46	4.41	3.41	2.36	39
7.2	6.15	3.9	2.85	124
6.75	5.7	3.91	2.86	128
6.74	5.69	3.9	2.85	123
7.67	6.62	3.92	2.87	129
TNFα
7.89	6.84	4.05	3	2,419
7.94	6.89	4.06	3.01	2,506
5.46	4.41	3.68	2.63	1,035
7.2	6.15	3.92	2.87	1,790
6.75	5.7	3.93	2.88	1,832
6.74	5.69	3.87	2.82	1,605
7.87	6.82	3.93	2.88	1,850
7.67	6.62	3.91	2.86	1,766
3.61	2.56	2.75	1.7	122
IFNγ
8.5	7.45	2.93	1.88	329
3.61	2.56	2.03	0.98	41
8.16	7.11	2.71	1.66	196

Notes:

a Adjusted by using human equivalent dose – (D_H, CL_H) = (D_M, CL_M)× K_M K_H^20, where D_H, D_M, CL_H, and CL_M represent bacteria load (D) or cytokine level (CL) for humans and mice (Table S1), respectively, and K_M and K_H are the body surface area-related transfer factors for mice and human, respectively, of 3 and 37 kg m⁻²;

b normalized coinfection-induced inflammatory effects (IE) – IE = (CL_H − CL_Pna,2.5) CL_Pna,2.5, where CL_H is the level of inflammatory cytokines in humans and CL_Pna,2.5 represents the 2.5% tile of inflammatory cytokine levels estimated from patients with influenza-associated pneumonia (Table S2).

Figure 3 Cytokine-specific inflammatory effects postcoinfection in response to various human bacteria loads.

Notes: (A) IL6; (B) TNFα; (C) IFNγ.

Figure 4 Sensitivity analysis represented with bacteria-load distributions corresponding to changes in IAV and SP coinfection-related parameters.

Abbreviations: IAV, influenza A virus; SP, Streptococcus pneumoniae; μ, toxic death of infected cells; a, virion production/release-increase rate; z, nonlinearity of virion production/release; φ, decrease in phagocytosis rate; K_BV, half-saturation constant.

Figure 5 Bacteria-load estimates.

Notes: (A) Various initial virus titers; (B) various initial bacteria loads; (C) various initial virus titers and bacteria loads with SP inoculating at days 1, 7, and 9 post-IAV infection.
Abbreviations: IAV, influenza A virus; SP, Streptococcus pneumoniae.

Figure 6 Human bacterial load and corresponded exceedance risk estimates due to coinfection.

Notes: (A–C) Human bacteria-load distributions given coinfection beginning at days 1, 7, and 9 post-IAV infection, respectively. (D–F) Cytokine-specific exceedance risk profiles of inflammation based on coinfection. (G–I) Severity levels in coinfection-associated inflammation for IL6, TNFα, and IFNγ, respectively. (D–I) Days 1 (square), 7 (triangle), and 9 (circle) of coinfection taking place. (G–I) I, II, and III represent mild, moderate, and severe inflammatory responses, respectively, and ER 0.8, ER 0.5, and ER 0.2 indicate whether it is, respectively, more likely, likely, or less likely for inflammatory response to exceed a certain severity level.
Abbreviation: ER, exceedance risk.

Figure S1 Relationships between bacterial loads and different cytokine levels.

Note: Linear regression analysis for various inflammatory cytokines (A) IL6, (B) IL10, (C) KC, (D) TNFα, and (E) IFNγ.

Table S1 Summary of extracted data for mice coinfected with IAV and SP from previously published studies

Age (weeks)	Species	Virus		Bacteria		Cytokines	Reference
		Strain	Dose (TCID50 mL^−1)	Strain	Dose (CFU⋅mL^−1)
8	C57BL/6	PR8	10	Type 3 (ATCC 6303)	10^4	IL6, IL10, KC, TNFα, IFNγ	^Citation1
8	C57BL/6	PR8	10	Type 3 (ATCC 6303)	2×10^5	IL6, IL10, KC, TNFα	^Citation2
6–12	C57BL/6	PR8	8.6×10^2	Type 4 (ATCC 6304)	10^7	IL6, IL10, TNFα, IFNγ	^Citation3
6	BALB/c	PR8	1.8	Type 2 (D39)	10^2–10^5	IL6, IL10, KC, TNFα	^Citation4
6–8	C57BL/6 without PAFR	PR8	10^2	Type 2 (D39), type 3 (A66.1), type 4 (T4)	10^2–10^4	IL6, IL10, KC, TNFα	^Citation5
8–10	C57BL/6	PR8	10	Type 3 (ATCC 6303)	2×10^4	IL10, KC, TNFα, IFNγ	^Citation6
7–10	C57BL/6	A/FM/1/47 (H1N1)	1.4×10^5	Type 3 (ATCC 6303)	10^3	KC, TNFα, IFNγ	^Citation7

Abbreviations: IAV, influenza A virus; SP, Streptococcus pneumoniae; PAFR, platelet-activating factor receptor; KC, keratinocyte chemoattractant.

Table S2 Inflammatory cytokines in IAV-associated pneumonia patients

Age (years)	Size	Virus	IL6 (pg/mL)	TNFα (pg/mL)	IFNγ (pg/mL)	Reference
≥17	93	IAV	14.6 (6.2–40.2)a	1.7 (1.2–2.3)	0.5 (0.5–10.8)	^Citation8
			14.3 (6.3–38.9)	1.5 (1.1–2.6)	0.5 (0.5–10.9)
			20.4 (4.7–93.3)	1.8 (1.7–2.2)	0.5 (0.5–12.1)
			12.5 (6.6–19.6)	1.7 (1.4–2.7)	18.9 (6.1–30.5)
≥17	63	H1N1pdm	45.8 (7.8–167.6)	0.5 (0.5–1.4)	0.5 (0.5–0.5)
			2.7 (1.4–4.7)	0.5 (0.5–1.3)	0.5 (0.5–0.5)
47 (19–66)	16	H1N1pdm	66.1 (102.9)b	36.3 (56.6)b	77.3 (153.4)b	^Citation9
35 (13)b	42	H1N1pdm	16.4		105.1	^Citation10
32 (3)b	45	H1N1pdm	210.8 (270.4)b	128.6 (157.5)b	32.1 (34.9)b	^Citation11
40 (12)b	15	H1N1pdm	32.3 (37.9)c	49.1 (55)c	126 (138.6)c	^Citation12
45 (17)b	57	H1N1pdm	191 (507.9)b	20.9 (47.8)b		^Citation13
			1,074.1 (918.8)b	43 (43.2)b
31 (25–36)	23	IAV	64.9 (87.8)b		927.7 (1,153.2)b	^Citation14
31 (22–39)	17	H1N1pdm	47.1 (75.1)b		754.8 (1,055.5)b
29	–	H1N1pdm	245.5 (276.6)b		33.2 (34.9)b	^Citation15
			101.4 (132.5)b		34.2 (40.6)b
23 (11–40)	63	H1N1pdm	200.5 (209.8)b	160.8 (182.8)b	160 (182.8)b	^Citation16
			215.4 (241.5)b	185 (227.6)b	185.6 (227.6)b
			226.6 (250.8)b	165.8 (193.8)b	165.5 (193.8)b
			201.2 (218.6)b	219.1 (264.1)b	218.6 (264.1)b
			187.5 (208.7)b	204.9 (238.6)b	205 (238.6)b
			182.5 (212.4)b	231.5 (273.2)b	231.4 (273.2)b
35 (11–40)	47	IAV	456.5 (490)b	182.7 (195.6)b	182.8 (195.6)b
			436 (513.3)b	211.9 (249.5)b	213 (249.5)b
			478.8 (517)b	203.7 (240.3)b	203.8 (240.3)b
			475.3 (576.3)b	190.3 (202.3)b	190.4 (202.3)b
			455.4 (516.5)b	176.6 (183.1)b	175.9 (183.1)b
			462.9 (522.7)b	193.5 (203.2)b	195.9 (203.2)b

Notes:

a Interquartile range;

b standard deviation;

c standard error.

Abbreviation: IAV, influenza A virus.

Table S3 IAV and SP coinfection-associated inflammatory effect (fold) varied with cytokines, day of introducing SP post-IAV infection, and exceedance risks at 0.8, 0.5, and 0.2

Cytokine	Exceedance risk
	0.8 (more likely)	0.5 (likely)	0.2 (less likely)
	Day 1
IL6	0 (0–127)a	0 (0–147)	165 (101–230)
TNFα	48 (0–790)	760 (286–1,233)	1,895 (1,601–2,189)
IFNγ	27 (0–575)	99 (0–494)	214 (0–526)
	Day 7
IL6	0 (0–151)	166 (101–231)	408 (177–639)
TNFα	907 (483–1,331)	1,899 (1,604–2,194)	3,339 (2,568–4,109)
IFNγ	114 (0–483)	215 (0–527)	360 (0–933)
	Day 9
IL6	0 (0–133)	16 (0–155)	213 (131–294)
TNFα	261 (0–920)	1,006 (613–1,399)	2,176 (1,822–2,531)
IFNγ	49 (0–547)	124 (0–478)	243 (0–583)

Note:

a Median (95% CI).

Abbreviations: IAV, influenza A virus; SP, Streptococcus pneumoniae.

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